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�� MN ⊥   PQ for M (2, 1), N (-3, 0), P (x, 4), and Q (3, y). Find a set of possible values for x and y. SPIRAL REVIEW Find the x- and y-intercepts of the line that contains each pair of points. (Previous course) 34. (-5, 0) and (0, -5) 35. (0, 1) and (2, -7) 36. (1, -3) and (3, 3) Use the given paragraph proof to write a two-column proof. (Lesson 2-7) 37. Given: ∠1 is supplementary to ∠3. Prove: ∠2 ≅ ∠3 Proof: It is given that ∠1 is supplementary to ∠3. ∠1 and ∠2 are a linear pair by the definition of a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 are supplementary. Thus ∠2 ≅ ∠3 by the Congruent Supplements Theorem. Given that m∠2 = 75°, tell whether each statement is true or false. Justify your answer with a postulate or theorem. (Lesson 3-2) 38. ∠1 ≅ ∠8 39. ∠2 ≅ ∠6 40. ∠3 ≅ ∠5 3- 5 Slopes of Lines 187 187 �� 3-6 Use with Lesson 3-6 Explore Parallel and Perpendicular Lines A graphing calculator can help you explore graphs of parallel and perpendicular lines. To graph a line on a calculator, you can enter the equation of the line in slope-intercept form. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. For example, the line y = 2x + 3 has a slope of 2 and crosses the y-axis at (0, 3). TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines,.... KEYWORD: MG7 Lab3 Activity 1 1 On a graphing calculator, graph the lines y = 3
x – 4, y = –3x – 4, and y = 3x + 1. Which lines appear to be parallel? What do you notice about the slopes of the parallel lines? 2 Graph y = 2x. Experiment with other equations to find a line that appears parallel to y = 2x. If necessary, graph y = 2x on graph paper and construct a parallel line. What is the slope of this new line? 3 Graph y = - 1__ 2x + 3. Try to graph a line that appears parallel to y = - 1 __ 2 x + 3. What is the slope of this new line? Try This 1. Create two new equations of lines that you think will be parallel. Graph these to confirm your conjecture. 2. Graph two lines that you think are parallel. Change the window settings on the calculator. Do the lines still appear parallel? Describe your results. 3. Try changing the y-intercepts of one of the parallel lines. Does this change whether the lines appear to be parallel? 188 188 Chapter 3 Parallel and Perpendicular Lines On a graphing calculator, perpendicular lines may not appear to be perpendicular on the screen. This is because the unit distances on the x-axis and y-axis can have different lengths. To make sure that the lines appear perpendicular on the screen, use a square window, which shows the x-axis and y-axis as having equal unit distances. One way to get a square window is to use the Zoom feature. On the Zoom menu, the ZDecimal and ZSquare commands change the window to a square window. The ZStandard command does not produce a square window. Activity 2 1 Graph the lines y = x and y = -x in a square window. Do the lines appear to be perpendicular? 2 Graph y = 3x - 2 in a square window. Experiment with other equations to find a line that appears perpendicular to y = 3x - 2. If necessary, graph y = 3x - 2 on graph paper and construct a perpendicular line. What is the slope of this new line? 3 Graph y = 2 __ 3 x in a square window. Try to graph a line that appears perpendicular to y = 2 __ 3 x. What is the slope of this new line? Try This 4. Create two new equations of lines that you think will be perpendicular. Graph these in a square window to confirm your conjecture. 5. Graph two lines that you think are perpendicular. Change the window settings on the calculator.
Do the lines still appear perpendicular? Describe your results. 6. Try changing the y-intercepts of one of the perpendicular lines. Does this change whether the lines appear to be perpendicular? 3- 6 Technology Lab 189 189 3-6 Lines in the Coordinate Plane TEKS G.7.B Dimensionality and the geometry of location: use... equations of lines to investigate... parallel lines, perpendicular lines,.... Also G.3.C, G.3.E, G.7.A, G.7.C Objectives Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding. Vocabulary point-slope form slope-intercept form Why learn this? The cost of some health club plans includes a one-time enrollment fee and a monthly fee. You can use the equations of lines to determine which plan is best for you. (See Example 4.) The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line © Forms of the Equation of a Line FORM EXAMPLE The point-slope form of a line is y - y 1 = m (x - x 1 ), where m is the slope and ( x 1, y 1 ) is a given point on the line. y - 3 = 2 (x - 4 ) m = 23, 4 ) The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. y = 3x + 6 m = 3, b = 6 The equation of a vertical line is x = a, where a is the x-intercept. The equation of a horizontal line is y = b, where b is the y-intercept. x = 5 y = 2 You will prove the slope-intercept form of a line in Exercise 48. PROOF PROOF Point-Slope Form of a Line Given: The slope of a line through points ( x 1, y 1 ) and ( x 2, y 2 ) is m = Prove: The equation of the line through ( x 1, y 1 ) with slope m is x - x 1 ). Proof: Let (x, y) be any point on
the linex - x 1 ) m = (x - x 1 ) m (x - x 1 ) = ( Slope formula Substitute (x, y) for ( x 2, y 2 ). Multiply both sides by (x - x 1 ). Simplify. y - y 1 = m (x - x 1 ) Sym. Prop. of = 190 190 Chapter 3 Parallel and Perpendicular Lines E X A M P L E 1 Writing Equations of Lines Write the equation of each line in the given form. A the line with slope 3 through (2, 1) in point-slope form y - y 1 = m (x - 2) Point-slope form Substitute 3 for m, 2 for x 1, and 1 for y 1. B the line through (0, 4) and (-1, 2) in slope-intercept form = -2 _ -1 - 0 y = mx + b 4 = 2 (0) + b 4 = b y = 2x + 4 Find the slope. Slope-intercept form Substitute 2 for m, 0 for x, and 4 for y to find b. Simplify. Write in slope-intercept form using m = 2 and b = 4. A line with y-intercept b contains the point (0, b). A line with x-intercept a contains the point (a, 0). C the line with x-intercept 2 and y-intercept 3 in point-slope form x - 2) 2 y = - 3 _ (x - 2) 2 Use the points (2, 0) and (0, 3) to find the slope. Point-slope form 3 _ for m, 2 for x 1, and 0 for y 1. Substitute - 2 Simplify. Write the equation of each line in the given form. 1a. the line with slope 0 through (4, 6) in slope-intercept form 1b. the line through (-3, 2) and (1, 2) in point-slope form E X A M P L E 2 Graphing Lines Graph each line The equation is given in slope-intercept form, with a slope of 3 __ 2 and a y-intercept of 3. Plot the point (0, 3) and then rise 3 and run 2 to find another point. Draw the line containing the two points. B y + 3 = -2 (
x - 1) The equation is given in point-slope form, with a slope of -2 = -2 ___ 1 through the point (1, -3). Plot the point (1, -3) and then rise -2 and run 1 to find another point. Draw the line containing the two points. 3- 6 Lines in the Coordinate Plane 191 191 �� Graph the line. C x = 3 The equation is given in the form for a vertical line with an x-intercept of 3. The equation tells you that the x-coordinate of every point on the line is 3. Draw the vertical line through (3, 0). Graph each line. 2a. y = 2x - 3 2b. y - 1 = - 2 _ (x + 2) 3 2c. y = -4 A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. Pairs of Lines Parallel Lines Intersecting Lines Coinciding Lines y = 5x + 8 y = 5x - 4 Same slope different y-intercept y = 2x - 5 y = 4x + 3 Different slopes y = 2x - 4 y = 2x - 4 Same slope same y-intercept E X A M P L E 3 Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. A y = 2x + 3, y = 2x - 1 Both lines have a slope of 2, and the y-intercepts are different. So the lines are parallel. B y = 3x - 5, 6x - 2y = 10 Solve the second equation for y to find the slope-intercept form. 6x - 2y = 10 -2y = -6x + 10 y = 3x - 5 Both lines have a slope of 3 and a y-intercept of -5, so they coincide. C 3x + 2y = 7, 3y = 4x + 7 Solve both equations for y to find the slope-intercept form. 3x + 2y = 7 3y = 4x + The slope is 4 _. 3 2y = -3x +. The slope is - 2 The lines have different slopes, so they intersect. 3. Deter
mine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide. 192 192 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Problem-Solving Application Audrey is trying to decide between two health club plans. After how many months would both plans’ total costs be the same? Plan A Plan B Enrollment Fee $140 $60 Monthly Fee $35 $55 Understand the Problem The answer is the number of months after which the costs of the two plans would be the same. Plan A costs $140 for enrollment and $35 per month. Plan B costs $60 for enrollment and $55 per month. Make a Plan Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations. Solve Plan A: y = 35x + 140 Plan B: y = 55x + 60 0 = -20x + 80 Subtract the second x = 4 y = 35 (4) + 140 = 280 equation from the first. Solve for x. Substitute 4 for x in the first equation. The lines cross at (4, 280). Both plans cost $280 after 4 months. Look Back Check your answer for each plan in the original problem. For 4 months, plan A costs $140 plus $35 (4) = $140 + $140 = $280. Plan B costs $60 + $55 (4) = $60 + $220 = $280, so the plans cost the same. Use the information above to answer the following. 4. What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan? THINK AND DISCUSS 1. Explain how to use the slopes and y-intercepts to determine if two lines are parallel. 2. Describe the relationship between the slopes of perpendicular lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3- 6 Lines in the Coordinate Plane 193 193 1234�� 3-6 Exercises Exercises KEYWORD: MG7 3-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1
. Vocabulary How can you recognize the slope-intercept form of an equation Write the equation of each line in the given form. p. 191 2. the line through (4, 7) and (-2, 1) in slope-intercept form 3. the line through (-4, 2) with slope 3 _ 4 4. the line with x-intercept 4 and y-intercept -2 in slope-intercept form in point-slope form Graph each line. p. 191. 192 5. y = -3x + 4 6. y + 4 = 2 _ (x - 6) 3 Determine whether the lines are parallel, intersect, or coincide. 7. x = 5 8. y = -3x + 4, y = -3x + 1 x + 2 _ 10. y = 1 _ 3 3, 3y = x + 2 9. 6x - 12y = -24, 3y = 2x + 18 11. 4x + 2y = 10, y = -2x + 15. 193 12. Transportation A speeding ticket in Conroe costs $115 for the first 10 mi/h over the speed limit and $1 for each additional mi/h. In Lakeville, a ticket costs $50 for the first 10 mi/h over the speed limit and $10 for each additional mi/h. If the speed limit is 55 mi/h, at what speed will the tickets cost approximately the same? Homework Help See For Exercises Example 13–15 16–18 19–22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Write the equation of each line in the given form. 13. the line through (0, -2) and (4, 6) in point-slope form 14. the line through (5, 2) and (-2, 2) in slope-intercept form 15. the line through (6, -4) with slope 2 _ 3 in point-slope form Graph each line. 16. y - 7 = x + 4 17. y = 1 _ 2 x - 2 18. y = 2 Determine whether the lines are parallel, intersect, or coincide. 19. y = x - 7, y = -x + 3 21. x + 2y = 6 20. y = 5 _ 2 x + 4,
2y = 5x - 4 22. 7x + 2y = 10, 3y = 4x - 5 23. Business Chris is comparing two sales positions that he has been offered. The first pays a weekly salary of $375 plus a 20% commission. The second pays a weekly salary of $325 plus a 25% commission. How much must he make in sales per week for the two jobs to pay the same? Write the equation of each line in slope-intercept form. Then graph the line. 24. through (-6, 2) and (3, 6) 26. through (5, -2) with slope 2 _ 3 27. x-intercept 4, y-intercept -3 25. horizontal line through (2, 3) Write the equation of each line in point-slope form. Then graph the line. 28. slope - 1 _ 2 30. through (5, -1) with slope -1 29. slope 3 _ 4 31. through (4, 6) and (-2, -5), x-intercept -2, y-intercept 2 194 194 Chapter 3 Parallel and Perpendicular Lines 32. ////ERROR ANALYSIS///// Write the equation of the line with slope -2 through the point (-4, 3) in slope-intercept form. Which equation is incorrect? Explain. Determine whether the lines are perpendicular. 33. y = 3x - 5, y = -3x + 1 x + 5, y = 3 _ 35, 34. y = -x + 1, y = x + 2 36. y = -2x + 4 Multi-Step Given the equation of the line and point P not on the line, find the equation of a line parallel to the given line and a line perpendicular to the given line through the given point. 37. y = 3x + 7, P (2, 3) 38. y = -2x - 5, P (-1, 4) 39. 4x + 3y = 8, P (4, -2) 40. 2x - 5y = 7, P (-2, 4) Food Multi-Step Use slope to determine if each triangle is a right triangle. If so, which angle is the right angle? 41. A (-5, 3), B (0, -2), C (5, 3) 42. D (1, 0), E (2, 7), F (
5, 1) 43. G (3, 4), H (-3, 4), J (1, -2) 44. K (-2, 4), L (2, 1), M (1, 8) In 2004, the world’s largest pizza was baked in Italy. The diameter of the pizza was 5.19 m (about 17 ft) and it weighed 124 kg (about 273 lb). 45. Food A restaurant charges $8 for a large cheese pizza plus $1.50 per topping. Another restaurant charges $11 for a large cheese pizza plus $0.75 per topping. How many toppings does a pizza have that costs the same at both restaurants? 46. Estimation Estimate the solution of the system of equations represented by the lines in the graph. Write the equation of the perpendicular bisector of the segment with the given endpoints. 47. (2, 5) and (4, 9) 48. (1, 1) and (3, 1) 49. (1, 3) and (-1, 4) 50. (-3, 2) and (-3, -10) 51. Line ℓ has equation y = - 1 __ 2 x + 4, and point P has coordinates (3, 5). a. Find the equation of line m that passes through P and is perpendicular to ℓ. b. Find the coordinates of the intersection of ℓ and m. c. What is the distance from P to ℓ? 52. Line p has equation y = x + 3, and line q has equation y = x - 1. a. Find the equation of a line r that is perpendicular to p and q. b. Find the coordinates of the intersection of p and r and the coordinates of the intersection of q and r. c. Find the distance between lines p and q. 3- 6 Lines in the Coordinate Plane 195 195 �� 53. This problem will prepare you for the Multi-Step TAKS Prep on page 200. For a car moving at 60 mi/h, the equation d = 88t gives the distance in feet d that the car travels in t seconds. a. Graph the line d = 88t. b
. On the same graph you made for part a, graph the line d = 300. What does the intersection of the two lines represent? c. Use the graph to estimate the number of seconds it takes the car to travel 300 ft. 54. Prove the slope-intercept form of a line, given the point-slope form. Given: The equation of the line through ( x 1, y 1 ) with slope m is y - y 1 = m (x - x 1 ). Prove: The equation of the line through (0, b) with slope m is y = mx + b. Plan: Substitute (0, b) for ( x 1, y 1 ) in the equation y - y 1 = m (x - x 1 ) and simplify. 55. Data Collection Use a graphing calculator and a motion detector to do the following: Walk in front of the motion detector at a constant speed, and write the equation of the resulting graph. 56. Critical Thinking A line contains the points (-4, 6) and (2, 2). Write a convincing argument that the line crosses the x-axis at (5, 0). Include a graph to verify your argument. 57. Write About It Determine whether the lines are parallel. Use slope to explain your answer. 58. Which graph best represents a solution to this system of equations? ⎧ -3x + y = 7 ⎨ 2x + y = -3 ⎩ 196 196 Chapter 3 Parallel and Perpendicular Lines �� 59. Which line is parallel to the line with the equation y = -2x + 5?   AB through A (2, 3) and B (1, 1 4x + 2y = 10 x + 1 _ 2 y = 1 60. Which equation best describes the graph shown = 3x - � � � � � � � � 61. Which line includes the points (-4, 2) and (6, -3)? y = 2x - 4 y = 2x CHALLENGE AND EXTEND 62. A right triangle is formed by the x-axis, the y-axis, and the line y = -2x + 5. Find the length of the hypotenuse. 63. If the length of the hypotenuse of a right triangle is 17 units and the legs lie along the x-axis and y-axis
, find a possible equation that describes the line that contains the hypotenuse. 64. Find the equations of three lines that form a triangle with a hypotenuse of 13 units. 65. Multi-Step Are the points (-2, -4), (5, -2) and (2, -3) collinear? Explain the method you used to determine your answer. 66. For the line y = x + 1 and the point P (3, 2), let d represent the distance from P to a point (x, y) on the line. a. Write an expression for d 2 in terms of x and y. Substitute the expression x + 1 for y and simplify. b. How could you use this expression to find the shortest distance from P to the line? Compare your result to the distance along a perpendicular line. SPIRAL REVIEW 67. The cost of renting DVDs from an online company is $5.00 per month plus $2.50 for each DVD rented. Write an equation for the total cost c of renting d DVDs from the company in one month. Graph the equation. How many DVDs did Sean rent from the company if his total bill for one month was $20.00? (Previous course) Use the coordinate plane for Exercises 68–70. Find the coordinates of the midpoint of each segment. (Lesson 1-6) ̶̶ AB ̶̶ BC ̶̶ AC 70. 68. 69. Use the slope formula to find the slope of each segment. (Lesson 3-5) ̶̶ AB ̶̶ BC ̶̶ AC 73. 71. 72. � � � � �� 3- 6 Lines in the Coordinate Plane 197 197 Scatter Plots and Lines of Best Fit Data Analysis Recall that a line has an infinite number of points on it. You can compute the slope of a line if you can identify two distinct points on the line. See Skills Bank page S79 Example 1 The table shows several possible measures of an angle and its supplement. Graph the points in the table. Then draw the line that best represents the data and write the equation of the line. Step 1 Use the table to write ordered pairs (x, 180 - x) and then plot the points. (30, 150), (60, 120), (90, 90), (120, 60), (150, 30) Step 2 Draw a line that
passes through all the points. x 30 60 90 120 150 y = 180 - x 150 120 90 60 30 Step 3 Choose two points from the line, such as (30, 150) and (120, 60). m = Use them to find the slope = 60 - 150 _ 120 - 30 = -90 _ 90 = -1 Slope formula Substitute (30, 150) for ( x 1, y 1 ) and (120, 60) for ( x 2, y 2 ). Simplify. Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) Point-slope form y - 150 = -1 (x - 30) Substitute (30, 150) for ( x 1, y 1 ) and -1 for m. y = -x + 180 Simplify. 198 198 Chapter 3 Parallel and Perpendicular Lines �� If you can draw a line through all the points in a set of data, the relationship is linear. If the points are close to a line, you can approximate the relationship with a line of best fit. Example 2 A physical therapist evaluates a client’s progress by measuring the angle of motion of an injured joint. The table shows the angle of motion of a client’s wrist over six weeks. Estimate the equation of the line of best fit. Step 1 Use the table to write ordered pairs and then plot the points. (1, 30), (2, 36), (3, 46), (4, 48), (5, 54), (6, 62) Step 2 Use a ruler to estimate a line of best fit. Try to get the edge of the ruler closest to all the points on the line. Week Angle Measure 1 2 3 4 5 6 30 36 46 48 54 62 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Step 3 A line passing through (2, 36) and (
6, 62) seems to be closest to all the points. Draw this line. Use the points (2, 36) and (6, 62) to find the slope of the line = 62 - 36 _ _ 6 - 2 = 6.5 Substitute (2, 36) for ( x 1, y 1 ) and (6, 62) for ( x 2, y 2 ). Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) y - 36 = 6.5 (x - 2) Point-slope form Substitute (2, 36) for ( x 1, y 1 ) and 6.5 for m. y = 6.5x + 23 Simplify. Try This TAKS Grades 9–11 Obj. 2, 3, 9 Estimate the equation of the line of best fit for each relationship. 1. 2. the relationship between an angle and its complement 3. Data Collection Use a graphing calculator and a motion detector to do the following: Set the equipment so that the graph shows distance on the y-axis and time on the x-axis. Walk in front of the motion dector while varying your speed slightly and use the resulting graph. On Track for TAKS 199 199 �� SECTION 3B Coordinate Geometry Red Light, Green Light When a driver approaches an intersection and sees a yellow traffic light, she must decide if she can make it through the intersection before the light turns red. Traffic engineers use graphs and equations to study this situation. 1. Traffic engineers can set the duration of the yellow lights on Lincoln Road for any length of time t up to 10 seconds. For each value of t, there is a critical distance d. If a car moving at the speed limit is more than d feet from the light when it turns yellow, the driver will have to stop. If the car is less than d feet from the light, the driver can continue through the intersection. The graph shows the relationship between t and d. Find the speed limit on Lincoln Road in miles per hour. (Hint : 22 ft/s = 15 mi/h) 2. Traffic engineers use the equation d = 22 __ 15 st to determine the critical distance for various durations of a yellow light. In the equation, s is the speed limit. The speed limit on Porter Street is 45 mi/h. Write the equation
of the critical distance for a yellow light on Porter Street and then graph the line. Does this line intersect the line for Lincoln Road? If so, where? Is the line for Porter Street steeper or flatter than the line for Lincoln Road? Explain how you know. 200 200 Chapter 3 Parallel and Perpendicular Lines �� SECTION 3B Quiz for Lesson 3-5 Through 3-6 3-5 Slopes of Lines Use the slope formula to determine the slope of each line. 1.  AC 2.   CD 3.  AB 4.   BD Find the slope of the line through the given points. 6. F (-1, 4) and G (5, -1) 8. K (4, 2) and L (-3, 2) 5. M (2, 3) and N (0, 7) 7. P (4, 0) and Q (1, -3) 9. Sonia is walking 4 miles home from school. She leaves at 4:00 P.M., and gets home at 4:45 P.M. Graph the line that represents Sonia’s distance from school at a given time. Find and interpret the slope of the line. Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither. 10.  EF and   GH for E (-2, 3), F (6, 1), G (6, 4), and H (2, 5) JK and   LM for J (4, 3), K (5, -1), L (-2, 4), and M (3, -5) 11.  12.  NP and   QR for N (5, -3), P (0, 4), Q (-3, -2), and R (4, 3) 13
.  ST and   VW for S (0, 3), T (0, 7), V (2, 3), and W (5, 3) 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 14. the line through (3, 8) and (-3, 4) in slope-intercept form 15. the line through (-5, 4) with slope 2 _ 3 16. the line with y-intercept 2 through the point (4, 1) in slope-intercept form in point-slope form Graph each line. 17. y = -2x + 5 18. y + 3 = 1 _ (x - 4) 4 19. x = 3 Write the equation of each line. 20. 21. 22. Determine whether the lines are parallel, intersect, or coincide. 23. y = -2x + 5 y = -2x - 5 24. 3x + 2y = 25. y = 4x -5 3x + 4y = 7 Ready to Go On? 201 201 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary alternate exterior angles..... 147 parallel planes.............. 146 same-side interior angles.... 147 alternate interior angles..... 147 perpendicular bisector...... 172 skew lines.................. 146 corresponding angles....... 147 perpendicular lines......... 146 slope.........................182 distance from a point-slope form........... 190 slope-intercept form........ 190 point to a line............ 172 parallel lines............... 146 rise........................ 182 transversal.....
............ 147 run........................ 182 Complete the sentences below with vocabulary words from the list above. 1. Angles on opposite sides of a transversal and between the lines it intersects are?. ̶̶̶̶ 2. Lines that are in different planes are?. ̶̶̶̶ 3. A(n) 4. The? is a line that intersects two coplanar lines at two points. ̶̶̶̶? is used to write the equation of a line with a given slope that passes ̶̶̶̶ through a given point. 5. The slope of a line is the ratio of the? to the ̶̶̶̶?. ̶̶̶̶ 3-1 Lines and Angles (pp. 146–151) E X A M P L E S EXERCISES Identify each of the following. Identify each of the following. ■ a pair of parallel segments ̶̶ AB ǁ ̶̶ CD ■ a pair of parallel planes plane ABC ǁ plane EFG ■ a pair of perpendicular segments ̶̶ AB ⊥ ̶̶ AE ■ a pair of skew segments ̶̶ AB and ̶̶ FG are skew. 6. a pair of skew segments 7. a pair of parallel segments 8. a pair of perpendicular segments 9. a pair of parallel planes 202 202 Chapter 3 Parallel and Perpendicular Lines �� Identify the transversal and classify each angle pair. ■ ∠4 and ∠6 p, corresponding angles ■ ∠1 and ∠2 q, alternate interior angles ■ ∠3 and ∠4 p, alternate exterior angles ■ ∠6 and ∠7 r, same-side interior angles Identify the transversal and classify each angle pair. 10. ∠5 and ∠2 11. ∠6 and ∠3 12. ∠2 and ∠4 13. ∠1 and ∠2 3-2 Angles Formed by Parallel Lines and Transversals (pp. 155–161) TEKS G.2.A, E X A M P L E S Find each angle measure. ■ m∠TUV EXERCISES Find each angle measure. 14. m∠WYZ G.3.C, G.3
.E, G.9.A By the Same-Side Interior Angles Theorem, (6x + 10) + (4x + 20) = 180. 15. m∠KLM x = 15 Solve for x. Substitute the value for x into the expression for m∠TUV. m∠TUV = 4 (15) + 20 = 80° ■ m∠ABC 16. m∠DEF 17. m∠QRS By the Corresponding Angles Postulate, 8x + 28 = 10x + 4. x = 12 Solve for x. Substitute the value for x into the expression for one of the obtuse angles. 10 (12) + 4 = 124° ∠ABC is supplementary to the 124° angle, so m∠ABC = 180 - 124 = 56°. Study Guide Review 203 203 �� 3-3 Proving Lines Parallel (pp. 162–169) TEKS G.1.A, G.3.C, G.3.E, G.9.A EXERCISES Use the given information and theorems and postulates you have learned to show that c ǁ d. 18. m∠4 = 58°, m∠6 = 58° 19. m∠1 = (23x + 38) °, m∠5 = (17x + 56) °, x = 3 20. m∠6 = (12x + 6) °, m∠3 = (21x + 9) °, x = 5 21. m∠1 = 99°, m∠7 = (13x + 8) °, Use the given information and theorems and postulates you have learned to show that p ǁ q. ■ m∠2 + m∠3 = 180° ∠2 and ∠3 are supplementary, so p ǁ q by the Converse of the Same-Side Interior Angles Theorem. ■ ∠8 ≅ ∠6 ∠8 ≅ ∠6, so p ǁ q by the Converse of the Corresponding Angles Postulate. ■ m∠1 = (7x - 3) °, m∠5 = 5x +
15, x = 9 m∠1 = 60°, and m∠5 = 60°. So ∠1 ≅ ∠5. p ǁ q by the Converse of the Alternate Exterior Angles Theorem. 3-4 Perpendicular Lines (pp. 172–178) TEKS G.1.A, G.2.A, G.3.C, G.3.E, G.9.A EXERCISES 22. Name the shortest segment from point K to ̶̶ LN. 23. Write and solve an inequality for x. 24. Given: Prove: ̶̶ AD ǁ ̶̶ AB ǁ ̶̶ BC, ̶̶ CD ̶̶ AD ⊥ ̶̶ AB, ̶̶ DC ⊥ ̶̶ BC E X A M P L E S ■ Name the shortest segment from ̶̶ WY. point X to ̶̶ XZ ■ Write and solve an inequality for x. x + 3 > 3 x > 0 Subtract 3 from both sides. ■ Given: m ⊥ p, ∠1 and ∠2 are complementary. Prove: p ǁ q Proof: It is given that m ⊥ p. ∠1 and ∠2 are complementary, so m∠1 + m∠2 = 90°. Thus m ⊥ q. Two lines perpendicular to the same line are parallel, so p ǁ q. 204 204 Chapter 3 Parallel and Perpendicular Lines �� 3-5 Slopes of Lines (pp. 182–187) TEKS G.7.A, G.7.B, G.7.C E X A M P L E S ■ Use the slope formula to determine the slope of the line. EXERCISES Use the slope formula to determine the slope of each line. 25. 26. slope of   WX = - (-3) _ 2 - (-4) = 6 _ 6 = 1 slope of   AB = ■ Use slopes to determine whether    AB and    CD are parallel, perpendicular, or neither for A (-1, 5), B (-3, 4), C
(3, -1), and D (4, -33 - (-1) -3 - (-1) = -2 _ _ 4 - 3 1 The slopes are opposite reciprocals, so the slope of   CD = = -2 Use slopes to determine if the lines are parallel, perpendicular, or neither. 27. EF and GH for E (8, 2), F (-3, 4), G (6, 1), and H (-4, 3) 28. JK and LM for J (4, 3), K (-4, -2), L (5, 6), and M (-3, 1) 29. ST and UV for S (-4, 5), T (2, 3), U (3, 1), and V (4, 4) lines are perpendicular. 3-6 Lines in the Coordinate Plane (pp. 190–197) TEKS G.3.C, G.3.E, G.7.A, G.7.B, G.7.C E X A M P L E S EXERCISES ■ Write the equation of the line through (5, -2) with slope 3 __ in slope-intercept form. 5 y - (-2) = 3 _ (x - 5 Solve for y. Simplify. x - 3 x - 5 Point-slope form ■ Determine whether the lines y = 4x + 6 and 8x - 2y = 4 are parallel, intersect, or coincide. Solve the second equation for y to find the slope-intercept form. 8x - 2y = 4 y = 4x - 2 Both the lines have a slope of 4 and have different y-intercepts, so they are parallel. Write the equation of each line in the given form. 30. the line through (6, 1) and (-3, 5) in slope-intercept form 31. the line through (-3, -4) with slope 2 _ 3 in slope-intercept form 32. the line with x-intercept 1 and y-intercept -2 in point-slope form Determine whether
the lines are parallel, intersect, or coincide. 33. -3x + 2y = 5, 6x - 4y = 8 34. y = 4x - 3, 5x + 2y = 1 35. y = 2x + 1, 2x - y = -1 Study Guide Review 205 205 �� Identify each of the following. 1. a pair of parallel planes 2. a pair of parallel segments 3. a pair of skew segments Find each angle measure. � � � � � � � � �� 4. 5. 6. Use the given information and the theorems and postulates you have learned to show f ǁ g. 7. m∠4 = (16x + 20) °, m∠5 = (12x + 32) °, x = 3 8. m∠3 = (18x + 6) °, m∠5 = (21x + 18) °, x = 4 Write a two-column proof. 9. Given: ∠1 ≅ ∠2, n ⊥ ℓ Prove: n ⊥ m Use the slope formula to determine the slope of each line. 10. 11. 12. 13. Greg is on a 32-mile bicycle trail from Elroy, Wisconsin, to Sparta, Wisconsin. He leaves Elroy at 9:30 A.M. and arrives in Sparta at 2:00 P.M. Graph the line that represents Greg’s distance from Elroy at a given time. Find and interpret the slope of the line. 14. Graph   QR and   ST for Q (3, 3), R (6, -5), S (-4, 6), and T (-1, -2). Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15. Write the equation of the line through (-2, -5) with slope - 3 _ 4 16. Determine whether the lines 6x + y = 3 and 2x + 3y = 1 are parallel, intersect, in point-slope form. or coincide. 206 206 Chapter 3 Parallel and Perpendicular Lines �� FOC
US ON ACT When you take the ACT Mathematics Test, you receive a separate subscore for each of the following areas: • Pre-Algebra/Elementary Algebra, • Intermediate Algebra/Coordinate Geometry, and • Plane Geometry/Trigonometry. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. 1. Which of the following is an equation of the line that passes through the point (2, -3) and is parallel to the line 4x - 5y = 1? (A) -4x + 5y = -23 (B) -5x - 4y = 2 (C) -2x - 5y = 11 (D) -4x - 5y = 7 (E) -5x + 4y = -22 2. In the figure below, line t crosses parallel lines ℓ and m. Which of the following statements are true? I. ∠1 and ∠6 are alternate interior angles. II. ∠2 ≅ ∠4 III. ∠2 ≅ ∠8 (F) I only (G) II only (H) III only (J) I and II only (K) II and III only Find out what percent of questions are from each area and concentrate on content that represents the greatest percent of questions. 3. In the standard (x, y) coordinate plane, the line that passes through (1, -7) and (-8, 5) is perpendicular to the line that passes through (3, 6) and (-1, b). What is the value of b? (A) 2 (B) 3 (C) 7 (D) 9 (E) 10 4. Lines m and n are cut by a transversal so that ∠2 and ∠5 are corresponding angles. If m∠2 = (x + 18) ° and m∠5 = (2x - 28) °, which value of x makes lines m and n parallel? (F) 3 1 _ 3 (G) 33 1 _ 3 (H) 46 (J) 63 1 _ 3 (K) 72 5. What is the distance between point G (4, 2) and the line through the points E (1, -2) and F (7, -2)? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 College Entrance
Exam Practice 207 207 �� Any Question Type: Interpret Coordinate Graphs When test items refer to a coordinate plane, it is important to interpret the coordinate graphs correctly. It is also important to understand the relationship between an equation and its graph. Multiple Choice Which statement best describes the graph of the following equations? y = 2x + 6 2y = 4x + 6 The lines are parallel. The lines are perpendicular. The lines coincide. The lines have the same y-intercept. It may help to graph the lines or visualize the graph in order to answer the question. The lines appear to be parallel. Write the equations of both lines in slope-intercept form. y = 2x + 6 y = 2x + 3 The slope is 2 and the y-intercept is 6. The slope is 2 and the y-intercept is 3. The lines have the same slope and different y-intercepts, so they are parallel. The answer is A. Gridded Response What is the rate of change of the following graph? Remember that the rate of change of the graph of a line is its slope. Choose two points on the line and use their coordinates to calculate the slope of the line. Use the points (5, 0) and (0, -2). m = - The slope is 2 _ 5 = 0.4. Enter 0.4 in the answer grid. 208 208 Chapter 3 Parallel and Perpendicular Lines �� A quick look at the graph of a line can tell you whether the slope is positive or negative. This may help you eliminate answer choices. Item C Multiple Choice Which equation describes the line through the point (4, 2) that is perpendicular to the line 3x - y = 7? Read each test item and answer the questions that follow. Item A Multiple Choice The line segment on the graph shows the altitude of a hot air balloon during a landing. Which statement best describes the slope of the line segment? The balloon descends about 5 feet per 8 seconds. The balloon descends about 8 feet per 5 seconds. The balloon descends about 1 foot per 2 seconds. The balloon descends about 2 feet per second. 1. What are the coordinates of two points on the graph? 2. How does the scale of the graph affect the appearance of the slope? 3. How is the slope of the line related to the rate of descent? Item B Gridded Response What is the
y-intercept of the line through (5, 2) that is parallel to the line x - 4y = 8? y = 3x - 10 y = -3x + 14. Graph the line represented by 3x - y = 7. What is its slope? 8. Is the slope of a line perpendicular to the line represented by 3x - y = 7 positive or negative? Based on your answer, can you eliminate any answer choices? 9. What is the slope of a line perpendicular to the line represented by 3x - y = 7? Item D Multiple Choice Which graph best represents a solution to the following system of equations? -2x + y = 6 3x + y = -2 10. What is the slope of the line represented by -2x + y = 6? Based on your answer, can you eliminate any answer choices? 4. Graph the line x - 4y = 8. What is its slope? 11. What is the slope of the line represented 5. Write an equation in point-slope form for the line through (5, 2) that is parallel to x - 4y = 8. 6. How would you use the equation in point- slope form to find the y-intercept of the line? by 3x + y = -2? Based on your answer, can you eliminate any answer choices? TAKS Tackler 209 209 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–3 Multiple Choice Use the diagram below for Items 1 and 2. 1. What type of angle pair are ∠JKM and ∠KMN? Corresponding angles Alternate exterior angles Same-side interior angles Alternate interior angles 2. What is m∠KML? 57° 80° 102° 125° 3. What is a possible value of x in the diagram? 2 3 4 5 4. A graphic artist used a computer illustration program to draw a line connecting points with coordinates (3, -1) and (4, 6). She needs to draw a second line parallel to the first line. What slope should the second line have. Which term describes a pair of vertical angles that are also supplementary? Acute Obtuse Right Straight 6. What is the equation of the line that passes through the points (-1, 8) and (4, -2)? y =
-2x + = 2x + 10 7. Given the points R (-5, 3), S (-5, 4), T (-3, 4), and U (-3, 1), which line is perpendicular to   TU?   RS   RT   ST   SU 8. Which of following is true if   XY and   UV are skew?   XY and   UV are coplanar. X, Y, and U are noncollinear.   XY ǁ   UV   XY ⊥   UV �� Make sure that you answer the question that is asked. Some problems require more than one step. You must perform all of the steps to get the correct answer. 9. Point C is the midpoint of B (7, 2). What is the length of nearest tenth. ̶̶ AB for A (1, -2) and ̶̶ AC? Round to the 3.0 3.6 5.0 7.2 Use the diagram below for Items 10 and 11. 10.  AD bisects ∠CAE, and  AE bisects ∠CAF. If m∠DAF = 120°, what is m∠DAE? 40° 60° 80° 100° 210 210 Chapter 3 Parallel and Perpendicular Lines A F 11. What is the intersection of  AD?  AF and ̶̶ FD ∠DAF �� 12. Which statement is true by the Transitive Property of Equality? If x + 3 = y, then y = x + 3. If k = 6, then 2k = 12. If a = b and b = 8, then a = 8. If m = n, then m + 7
= n + 7. 13. Which condition guarantees that r \|\| s? STANDARDIZED TEST PREP Short Response 21. Given ℓ ǁ m with transversal t, explain why ∠1 and ∠8 are supplementary. ∠1 ≅ ∠2 ∠2 ≅ ∠7 ∠2 ≅ ∠3 ∠1 ≅ ∠4 22. Read the following conditional statement. If two angles are vertical angles, then they are congruent. a. Write the converse of this conditional statement. b. Give a counterexample to show that the 14. What is the converse of the following statement? converse is false. If x = 2, then x + 3 = 5. If x ≠ 2, then x + 3 = 5. If x = 2, then x + 3 ≠ 5. If x + 3 ≠ 5, then x ≠ 2. If x + 3 = 5, then x = 2. Gridded Response 15. Two lines a and b are cut by a transversal so that ∠1 and ∠2 are same-side interior angles. If m∠1 = (2x + 30) ° and m∠2 = (4x - 75) °, what value of x proves that a ǁ b? 23. Assume that the following statements are true when the bases are loaded in a baseball game. If a batter hits the ball over the fence, then the batter hits a home run. A batter hits a home run if and only if the result is four runs scored. a. If a batter hits the ball over the fence when the bases are loaded, can you conclude that four runs were scored? Explain your answer. b. If a batter hits a home run when the bases are loaded, can you conclude that the batter hit the ball over the fence? Explain your answer. 16. What is the slope of the line that passes through (3, 7) and (-5, 1)? Extended Response 17. ∠1 and ∠2 form a linear pair. m∠1 = (4x + 18)° and m∠2 = (3x - 6)°. What is the value of x? 18. Points A, B, and C are collinear, and B is between A and C. AB = 16 and AC = 27. What is the distance BC? 19. Ms
. Nelson wants to put a chain-link fence around 3 sides of a square-shaped lawn. Chain-link fencing is sold in sections that are each 6 feet wide. If Ms. Nelson’s lawn has an area of 3600 square feet, how many sections of fencing will she need? 20. What is the next number in this pattern? 67, 76, 83, 88,… 24. A car passes through a tollbooth at 8:00 A.M. and begins traveling east at an average speed of 45 miles per hour. A second car passes through the same tollbooth an hour later and begins traveling east at an average speed of 60 miles per hour. a. Write an equation for each car that relates the number of hours x since 8:00 A.M. to the distance in miles y the car has traveled. Explain what the slope of each equation represents. b. Graph the system of equations on the coordinate plane. c. If neither car stops, at what time will the second car catch up to the first car? Explain how you determined your answer. Cumulative Assessment, Chapters 1–3 211 211 �� Triangle Congruence 4A Triangles and Congruence 4-1 Classifying Triangles Lab Develop the Triangle Sum Theorem 4-2 Angle Relationships in Triangles 4-3 Congruent Triangles 4B Proving Triangle Congruence Lab Explore SSS and SAS Triangle Congruence 4-4 Triangle Congruence: SSS and SAS Lab Predict Other Triangle Congruence Relationships 4-5 Triangle Congruence: ASA, AAS, and HL 4-6 Triangle Congruence: CPCTC 4-7 Introduction to Coordinate Proof 4-8 Isosceles and Equilateral Triangles Ext Proving Constructions Valid KEYWORD: MG7 ChProj The Bob Bullock Texas State History Museum opened in Austin in April 2001. 212 212 Chapter 4 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. a statement that is accepted as true without proof 2. congruent segments B. an angle that measures greater than 90° and less than 180° 3. obtuse angle C. a statement that you can prove 4. postulate 5. triangle D. segments that have the same length E. a three-sided polygon F. an angle that measures greater than 0° and less than 90° Measure Angles Use a protractor to
measure each angle. 6. 7. Use a protractor to draw an angle with each of the following measures. 8. 20° 10. 105° 9. 63° 11. 158° Solve Equations with Fractions x + 7 = 25 Solve. 12. 9_ 2 14. x - 1_ 5 = 12_ 5 13. 3x - 2_ 3 = 4_ 3 15. 2y = 5y - 21_ 2 Connect Words and Algebra Write an equation for each statement. 16. Tanya’s age t is three times Martin’s age m. 17. Twice the length of a segment x is 9 ft. 18. The sum of 53° and twice an angle measure y is 90°. 19. The price of a radio r is $25 less than the price of a CD player p. 20. Half the amount of liquid j in a jar is 5 oz more than the amount of liquid b in a bowl. Triangle Congruence 213 213 Key Vocabulary/Vocabulario acute triangle triángulo acutángulo congruent polygons polígonos congruentes corollary corolario equilateral triangle triángulo equilátero exterior angle ángulo externo interior angle ángulo interno isosceles triangle triángulo isósceles obtuse triangle triángulo obtusángulo right triangle triángulo rectángulo scalene triangle triángulo escaleno Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The Latin word acutus means “pointed” or “sharp.” Draw a triangle that looks pointed or sharp. Do you think this is an acute triangle? 2. Consider the everyday meaning of the word exterior. Where do you think an exterior angle of a triangle is located? 3. You already know the definition of an obtuse angle. Use this meaning to make a conjecture about an obtuse triangle. Geometry TEKS G.1.A Geometric structure* develop an awareness of the structure of a mathematical system … G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures … G.2.B Geometric structure* make
conjectures about angles, lines, polygons … and determine validity of the conjectures, choosing from a variety of approaches … 4-2 Geo. Lab Les. 4-1 ★ Les. 4-2 Les. 4-3 ★ 4-4 Geo. Lab 4-5 Tech. Lab Les. 4-4 Les. 4-5 Les. 4-6 Les. 4-7 Les. 4-8 Ext.5.A Geometric patterns* use … geometric patterns ★ to develop algebraic expressions representing geometric properties G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures ★ G.9.B Congruence and the geometry of ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties and attributes of polygons … based on explorations … G.10.B Congruence and the geometry of size* justify ★ ★ ★ ★ ★ ★ ★ ★ and apply triangle congruence relationships * Knowledge and skills are written out completely on pages TX28–TX35. 214 214 Chapter 4 Reading Strategy: Read Geometry Symbols In Geometry we often use symbols to communicate information. When studying each lesson, read both the symbols and the words slowly and carefully. Reading aloud can sometimes help you translate symbols into words. � �� Throughout this course, you will use these symbols and combinations of these symbols to represent various geometric statements. Symbol Combinations Translated into Words   ST ǁ   UV ̶̶̶ ̶̶ GH BC ⊥ p → q Line ST is parallel to line UV. Segment BC is perpendicular to segment GH. If p, then q. m∠QRS = 45° The measure of angle QRS is 45 degrees. ∠CDE ≅ ∠LMN Angle CDE is congruent to angle LMN. Try This Rewrite each statement using symbols. 1. the absolute value of 2 times pi 2. The measure of angle 2 is 125 degrees.
3. Segment XY is perpendicular to line BC. 4. If not p, then not q. Translate the symbols into words. 5. m∠FGH = m∠VWX 6. ZA ǁ TU 7. ∼p → q 8. ST bisects ∠TSU. Triangle Congruence 215 215 4-1 Classifying Triangles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning... Objectives Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Who uses this? Manufacturers use properties of triangles to calculate the amount of material needed to make triangular objects. (See Example 4.) A triangle is a steel percussion instrument in the shape of an equilateral triangle. Different-sized triangles produce different musical notes when struck with a metal rod. Recall that a triangle (△) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. ̶̶ BC, and ̶̶ AB, A, B, and C are the triangle’s vertices. ̶̶ AC are the sides of △ABC. Triangle Classification By Angle Measures Acute Triangle Equiangular Triangle Right Triangle Obtuse Triangle Three acute angles Three congruent acute angles One right angle One obtuse angle E X A M P L E 1 Classifying Triangles by Angle Measures Classify each triangle by its angle measures. A △EHG ∠EHG is a right angle. So △EHG is a right triangle. B △EFH ∠EFH and ∠HFG form a linear pair, so they are supplementary. Therefore m∠EFH + m∠HFG = 180°. By substitution, m∠EFH + 60° = 180°. So m∠EFH = 120°. △EFH is an obtuse triangle by definition. 1. Use the diagram to classify △FHG by its angle measures. 216 216 Chapter 4 Triangle Congruence �� Triangle Classification By Side Length
s Equilateral Triangle Isosceles Triangle Scalene Triangle Three congruent sides At least two congruent sides No congruent sides E X A M P L E 2 Classifying Triangles by Side Lengths Classify each triangle by its side lengths. When you look at a figure, you cannot assume segments are congruent based on their appearance. They must be marked as congruent. A △ABC From the figure, and △ABC is equilateral. ̶̶ AB ≅ ̶̶ AC. So AC = 15, B △ABD By the Segment Addition Postulate, BD = BC + CD = 15 + 5 = 20. Since no sides are congruent, △ABD is scalene. 2. Use the diagram to classify △ACD by its side lengths. E X A M P L E 3 Using Triangle Classification Find the side lengths of the triangle. Step 1 Find the value of x. ̶̶ ̶̶ KL JK ≅ JK = KL (4x - 1.3) = (x + 3.2) 3x = 4.5 x = 1.5 Given Def. of ≅ segs. Substitute (4x - 13) for JK and (x + 3.2) for KL. Add 1.3 and subtract x from both sides. Divide both sides by 3. Step 2 Substitute 1.5 into the expressions to find the side lengths. JK = 4x - 1.3 = 4 (1.5) - 1.3 = 4.7 KL = x + 3.2 = (1.5) + 3.2 = 4.7 JL = 5x - 0.2 = 5 (1.5) - 0.2 = 7.3 3. Find the side lengths of equilateral △FGH. 4-1 Classifying Triangles 217 217 �� E X A M P L E 4 Music Application �� A manufacturer produces musical triangles by bending pieces of steel into the shape of an equilateral triangle. The triangles are available in side lengths of 4 inches, 7 inches, and 10 inches. How many 4-inch triangles can the manufacturer produce from a 100 inch piece of steel? The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3 (4) =
12 in. To find the number of triangles that can be made from 100 inches. of steel, divide 100 by the amount of steel needed for one triangle. 100 ÷ 12 = 8 1 _ 3 triangles There is not enough steel to complete a ninth triangle. So the manufacturer can make 8 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many triangles can be formed from a 100 in. piece of steel. 4a. 7 in. 4b. 10 in. THINK AND DISCUSS 1. For △DEF, name the three pairs of consecutive sides and the vertex formed by each. 2. Sketch an example of an obtuse isosceles triangle, or explain why it is not possible to do so. 3. Is every acute triangle equiangular? Explain and support your answer with a sketch. 4. Use the Pythagorean Theorem to explain why you cannot draw an equilateral right triangle. 5. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe each type of triangle. 218 218 Chapter 4 Triangle Congruence �� 4-1 Exercises Exercises KEYWORD: MG7 4-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, JK, KL, and JL are equal. How does this help you classify △JKL by its side lengths? 2. △XYZ is an obtuse triangle. What can you say about the types of angles in △XYZ Classify each triangle by its angle measures. p. 216 3. △DBC 4. △ABD 5. △ADC Classify each triangle by its side lengths. p. 217 6. △EGH 7. △EFH 8. △HFG Multi-Step Find the side lengths of each triangle. p. 217 9. 10. 218 11. Crafts A jeweler creates triangular earrings by bending pieces of silver wire. Each earring is an isosceles triangle with the dimensions shown. How many earrings can be made from a piece of wire that is 50 cm long? Independent Practice For See Exercises Example 12–14 15–17 18–20 21–22 1 2 3 4 TEKS TEKS T
AKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Classify each triangle by its angle measures. 12. △BEA 13. △DBC 14. △ABC Classify each triangle by its side lengths. 15. △PST 16. △RSP 17. △RPT Multi-Step Find the side lengths of each triangle. 18. 19. �� 20. Draw a triangle large enough to measure. Label the vertices X, Y, and Z. a. Name the three sides and three angles of the triangle. b. Use a ruler and protractor to classify the triangle by its side lengths and angle measures. 4-1 Classifying Triangles 219 219 �� Carpentry Use the following information for Exercises 21 and 22. A manufacturer makes trusses, or triangular supports, for the roofs of houses. Each truss is the shape of an isosceles triangle in which base ̶̶ ̶̶ PQ ≅ PR. The length of the ̶̶ QR is 4 __ 3 the length of each of the congruent sides. 21. The perimeter of each truss is 60 ft. Find each side length. 22. How many trusses can the manufacturer make from 150 feet of lumber? Draw an example of each type of triangle or explain why it is not possible. 23. isosceles right 24. equiangular obtuse 25. scalene right 26. equilateral acute 27. scalene equiangular 28. isosceles acute 29. An equilateral triangle has a perimeter of 105 in. What is the length of each side of the triangle? Architecture Classify each triangle by its angles and sides. 30. △ABC 31. △ACD Daniel Burnham designed and built the 22-story Flatiron Building in New York City in 1902. Source: www.greatbuildings.com 32. An isosceles triangle has a perimeter of 34 cm. The congruent sides measure (4x - 1) cm. The length of the third side is x cm. What is the value of x? 33. Architecture The base of the Flatiron Building is a triangle bordered by three streets: Broadway, Fifth Avenue, and East
Twenty-second Street. The Fifth Avenue side is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft. a. Find the two unknown side lengths. b. Classify the triangle by its side lengths. 34. Critical Thinking Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 35. An acute triangle is a scalene triangle. 36. A scalene triangle is an obtuse triangle. 37. An equiangular triangle is an isosceles triangle. 38. Write About It Write a formula for the side length s of an equilateral triangle, given the perimeter P. Explain how you derived the formula. 39. Construction Use the method for constructing congruent segments to construct an equilateral triangle. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. Marc folded a rectangular sheet of paper, ABCD, in half ̶̶ EF. He folded the resulting square diagonally and along then unfolded the paper to create the creases shown. a. Use the Pythagorean Theorem to find DE and CE. b. What is the m∠DEC? c. Classify △DEC by its side lengths and by its angle measures. 220 220 Chapter 4 Triangle Congruence �� 41. What is the side length of an equilateral triangle with a perimeter of 36 2 __ inches? 3 36 2 _ 3 18 1 _ 3 inches inches 12 1 _ 3 12 2 _ 9 inches inches 42. The vertices of △RST are R (3, 2), S (-2, 3), and T (-2, 1). Which of these best describes △RST? Isosceles Scalene Equilateral Right 43. Which of the following is NOT a correct classification of △LMN? Acute Isosceles Equiangular Right ̶̶ 44. Gridded Response △ABC is isosceles, and AB ≅ - x). What is the perimeter of △ABC? BC = ( 5 __ 2 ̶̶ AC. AB = ( 1 __ x + 1 __ 4 2 ), and CHALLENGE AND EXTEND 45. A triangle has
vertices with coordinates (0, 0), (a, 0), and (0, a), where a ≠ 0. Classify the triangle in two different ways. Explain your answer. 46. Write a two-column proof. Given: △ABC is equiangular. EF ǁ AC Prove: △EFB is equiangular. 47. Two sides of an equilateral triangle measure (y + 10) units and ( y 2 - 2) units. If the perimeter of the triangle is 21 units, what is the value of y? 48. Multi-Step The average length of the sides of △PQR is 24. How much longer then the average is the longest side? SPIRAL REVIEW Name the parent function of each function. (Previous course) 49. y = 5x 2 + 4 50. 2y = 3x + 4 51. y = 2 (x - 8) 2 + 6 Determine if each biconditional is true. If false, give a counterexample. (Lesson 2-4) 52. Two lines are parallel if and only if they do not intersect. 53. A triangle is equiangular if and only if it has three congruent angles. 54. A number is a multiple of 20 if and only if the number ends in a 0. Determine whether each line is parallel to, is perpendicular to, or coincides with y = 4x. (Lesson 3-6) 55. y = 4x + 2 57. 1 _ 2 y = 2x 56. 4y = -x + 8 58. -2y = 1 _ x 2 4-1 Classifying Triangles 221 221 �� 4-2 Use with Lesson 4-2 Activity Develop the Triangle Sum Theorem In this lab, you will use patty paper to discover a relationship between the measures of the interior angles of a triangle. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.3.D G.4.A, G.5.A 1 Draw and label △ABC on a sheet of notebook paper. 2 On patty paper draw a line ℓ and label a point P on the line. ̶̶ AB is on line ℓ and P and B 3
Place the patty paper on top of the triangle you drew. Align the papers so that coincide. Trace ∠B. Rotate the triangle and trace ∠C adjacent to ∠B. Rotate the triangle again and trace ∠A adjacent to ∠C. The diagram shows your final step. Try This 1. What do you notice about the three angles of the triangle that you traced? 2. Repeat the activity two more times using two different triangles. Do you get the same results each time? 3. Write an equation describing the relationship among the measures of the angles of △ABC. 4. Use inductive reasoning to write a conjecture about the sum of the measures of the angles of a triangle. 222 222 Chapter 4 Triangle Congruence 4-2 Angle Relationships in Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons …. Also G.1.A Objectives Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Vocabulary auxiliary line corollary interior exterior interior angle exterior angle remote interior angle Who uses this? Surveyors use triangles to make measurements and create boundaries. (See Example 1.) Triangulation is a method used in surveying. Land is divided into adjacent triangles. By measuring the sides and angles of one triangle and applying properties of triangles, surveyors can gather information about adjacent triangles. This engraving shows the county surveyor and commissioners laying out the town of Baltimore in 1730. Theorem 4-2-1 Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. m∠A + m∠B + m∠C = 180° The proof of the Triangle Sum Theorem uses an auxiliary line. An auxiliary line is a line that is added to a figure to aid in a proof. PROOF PROOF Triangle Sum Theorem Given: △ABC Prove: m∠1 + m∠2 + m∠3 = 180° Proof: Whenever you draw an auxiliary line, you must be able to justify its existence. Give this as the reason: Through any two points there is exactly one line. 4-2 Angle Relationships in Triangles 223 223 ��
�� E X A M P L E 1 Surveying Application The map of France commonly used in the 1600s was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures. A m∠NKM m∠KMN + m∠MNK + m∠NKM = 180° △ Sum Thm. 88 + 48 + m∠NKM = 180 136 + m∠NKM = 180 m∠NKM = 44° Substitute 88 for m∠KMN and 48 for m∠MNK. Simplify. Subtract 136 from both sides. B m∠JLK Step 1 Find m∠JKL. m∠NKM + m∠MKJ + m∠JKL = 180° Lin. Pair Thm. & ∠ Add. Post. 44 + 104 + m∠JKL = 180 148 + m∠JKL = 180 m∠JKL = 32° Substitute 44 for m∠NKM and 104 for m∠MKJ. Simplify. Subtract 148 from both sides. Step 2 Use substitution and then solve for m∠JLK. m∠JLK + m∠JKL + m∠KJL = 180° △ Sum Thm. m∠JLK + 32 + 70 = 180 Substitute 32 for m∠JKL and m∠JLK + 102 = 180 m∠JLK = 78° 70 for m∠KJL. Simplify. Subtract 102 from both sides. 1. Use the diagram to find m∠MJK. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Corollaries COROLLARY HYPOTHESIS CONCLUSION 4-2-2 The acute angles of a right triangle are complementary. 4-2-3 The measure of each angle of an equiangular triangle is 60°. ∠D and ∠E are complementary. m∠D + m∠E = 90° m∠A = m∠B = m∠C = 60° You will prove Corollaries 4-
2-2 and 4-2-3 in Exercises 24 and 25. 224 224 Chapter 4 Triangle Congruence 70°104°88°48°�� E X A M P L E 2 Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 22.9°. What is the measure of the other acute angle? Let the acute angles be ∠M and ∠N, with m∠M = 22.9°. m∠M + m∠N = 90 22.9 + m∠N = 90 Acute  of rt. △ are comp. Substitute 22.9 for m∠M. m∠N = 67.1° Subtract 22.9 from both sides. The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 2a. 63.7° 2b. x ° ° 2c. 48 2 _ 5 The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. ∠4 is an exterior angle. Its remote interior angles are ∠1 and ∠2. Theorem 4-2-4 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m∠4 = m∠1 + m∠2 You will prove Theorem 4-2-4 in Exercise 28. E X A M P L E 3 Applying the Exterior Angle Theorem Find m∠J. m∠J + m∠H = m∠FGH 5x + 17 + 6x - 1 = 126 11x + 16 = 126 11x = 110 x = 10 Ext. ∠ Thm. Substitute 5x + 17 for m∠J, 6x - 1 for m∠H, and 126 for m∠FGH. Simplify. Subtract 16 from both sides. Divide both sides by 11. m∠J = 5x + 17 = 5 (10) + 17 = 67° 3.
Find m∠ACD. 4-2 Angle Relationships in Triangles 225 225 �� Theorem 4-2-5 Third Angles Theorem THEOREM HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. ∠N ≅ ∠T You will prove Theorem 4-2-5 in Exercise 27. E X A M P L E 4 Applying the Third Angles Theorem Find m∠C and m∠F. ∠C ≅ ∠F m∠C = m∠F y 2 = 3y 2 - 72 Third  Thm. Def. of ≅ . Substitute y 2 for m∠C and 3 y 2 - 72 for m∠F. Subtract 3 y 2 from both sides. Divide both sides by -2. -2y 2 = -72 y 2 = 36 So m∠C = 36°. Since m∠F = m∠C, m∠F = 36°. 4. Find m∠P and m∠T. THINK AND DISCUSS 1. Use the Triangle Sum Theorem to explain why the supplement of one of the angles of a triangle equals in measure the sum of the other two angles of the triangle. Support your answer with a sketch. 2. Sketch a triangle and draw all of its exterior angles. How many exterior angles are there at each vertex of the triangle? How many total exterior angles does the triangle have? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write each theorem in words and then draw a diagram to represent it. 226 226 Chapter 4 Triangle Congruence �� 4-2 Exercises Exercises KEYWORD: MG7 4-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. To remember the meaning of remote interior angle, think of a television remote control. What is another way to remember the term remote? 2. An exterior angle is drawn at vertex
E of △DEF. What are its remote interior angles? 3. What do you call segments, rays, or lines that are added to a given diagram. 224 Astronomy Use the following information for Exercises 4 and 5. An asterism is a group of stars that is easier to recognize than a constellation. One popular asterism is the Summer Triangle, which is composed of the stars Deneb, Altair, and Vega. 4. What is the value of y? 5. What is the measure of each angle in the Summer Triangle. 225 The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 6. 20.8° 7 Find each angle measure. p. 225 9. m∠M ° 8. 24 2 _ 3 10. m∠L 11. In △ABC, m∠A = 65°, and the measure of an exterior angle at C is 117°. Find m∠B and the m∠BCA 12. m∠C and m∠F 13. m∠S and m∠U p. 226 14. For △ABC and △XYZ, m∠A = m∠X and m∠B = m∠Y. Find the measures of ∠C and ∠Z if m∠C = 4x + 7 and m∠Z = 3 (x + 5). 4-2 Angle Relationships in Triangles 227 227 �� Independent Practice For See Exercises Example 15 16–18 19–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 15. Navigation A sailor on ship A measures the angle between ship B and the pier and finds that it is 39°. A sailor on ship B measures the angle between ship A and the pier and finds that it is 57°. What is the measure of the angle between ships A and B? The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? ° 16. 76 1 _ 4 17. 2x° 18. 56.8° Find each
angle measure. 19. m∠XYZ 20. m∠C 21. m∠N and m∠P 22. m∠Q and m∠S 23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures.) 24. Complete the proof of Corollary 4-2-2. Given: △DEF with right ∠F Prove: ∠D and ∠E are complementary. Proof: Statements Reasons 1. △DEF with rt. ∠F 2. b.? ̶̶̶̶ 3. m∠D + m∠E + m∠F = 180° 4. m∠D + m∠E + 90° = 180° 5. e.? ̶̶̶̶ 6. ∠D and ∠E are comp. 1. a.? ̶̶̶̶ 2. Def. of rt. ∠ 3. c.? ̶̶̶̶? ̶̶̶̶ 5. Subtr. Prop. 4. d. 6. f.? ̶̶̶̶ 25. Prove Corollary 4-2-3 using two different methods of proof. Given: △ABC is equiangular. Prove: m∠A = m∠B = m∠C = 60° 26. Multi-Step The measure of one acute angle in a right triangle is 1 1 __ 4 times the measure of the other acute angle. What is the measure of the larger acute angle? 27. Write a two-column proof of the Third Angles Theorem. 228 228 Chapter 4 Triangle Congruence Ship AShip BPier57º39º�� 28. Prove the Exterior Angle Theorem. Given: △ABC with exterior angle ∠ACD Prove: m∠ACD = m∠A + m∠B (Hint: ∠BCA and ∠DCA form a linear pair.) Find each angle measure. 29. ∠UXW 31. ∠WZX 30. ∠UWY 32. ∠XYZ 33. Critical Thinking
What is the measure of any exterior angle of an equiangular triangle? What is the sum of the exterior angle measures? 34. Find m∠SRQ, given that ∠P ≅ ∠U, ∠Q ≅ ∠T, and m∠RST = 37.5°. 35. Multi-Step In a right triangle, one acute angle measure is 4 times the other acute angle measure. What is the measure of the smaller angle? 36. Aviation To study the forces of lift and drag, the Wright brothers built a glider, attached two ropes to it, and flew it like a kite. They modeled the two wind forces as the legs of a right triangle. a. What part of a right triangle is formed by each rope? b. Use the Triangle Sum Theorem to write an equation relating the angle measures in the right triangle. c. Simplify the equation from part b. What is the relationship between x and y? d. Use the Exterior Angle Theorem to write an expression for z in terms of x. e. If x = 37°, use your results from parts c and d to find y and z. 37. Estimation Draw a triangle and two exterior angles at each vertex. Estimate the measure of each angle. How are the exterior angles at each vertex related? Explain. 38. Given: Prove: ̶̶ AB ⊥ ̶̶ AD ǁ ̶̶ BD, ̶̶ CB ̶̶ BD ⊥ ̶̶ DC, ∠A ≅ ∠C 39. Write About It A triangle has angle measures of 115°, 40°, and 25°. Explain how to find the measures of the triangle’s exterior angles. Support your answer with a sketch. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. One of the steps in making an origami crane involves folding a square sheet of paper into the shape shown. a. ∠DCE is a right angle. ̶̶ FC bisects ∠DCE, and ̶̶ BC bisects ∠FCE. Find m∠FCB. b. Use the Triangle Sum Theorem to find m∠CBE. 4-2 Angle Relationships in Triangles 229 229 ��DragLiftRopezºyºxºge07sec04ll02005aABoehm��
41. What is the value of x? 19 52 42. Find the value of s. 23 28 57 71 34 56 43. ∠A and ∠B are the remote interior angles of ∠BCD in △ABC. Which of these equations must be true? m∠A - 180° = m∠B m∠A = 90° - m∠B m∠BCD = m∠BCA - m∠A m∠B = m∠BCD - m∠A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117°. Its remote interior angles measure ( 2y 2 + 7) ° and (61 - y 2 ) °. Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two same-side interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two same-side interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30°. What is the probability that the triangle has at least two congruent angles? 49. In △ABC, m∠B is 5° less than 1 1 __ 2 times m∠A. m∠C is 5° less than 2 1 __ 2 times m∠A. What is m∠A in degrees? SPIRAL REVIEW Make a table to show the value of each function when x is -2, 0, 1, and 4. (Previous course) 52. f (x) = (x - 3) 2 + 5 50. f (x) = 3x - 4 51. f (x) = x 2 + 1 53. Find the length of ̶̶̶ NQ. Name the theorem or postulate that justifies your answer. (Lesson 2-7) Classify each triangle by its side lengths. (Lesson 4-1) 54. △ACD 55. △BCD
56. △ABD 57. What if…? If CA = 8, What is the effect on the classification of △ACD? 230 230 Chapter 4 Triangle Congruence �� 4-3 Congruent Triangles TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.B Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Who uses this? Machinists used triangles to construct a model of the International Space Station’s support structure. Vocabulary corresponding angles corresponding sides congruent polygons Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent. Thus triangles that are the same size and shape are congruent. Properties of Congruent Polygons DIAGRAM CORRESPONDING ANGLES CORRESPONDING SIDES Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. △ABC ≅ △DEF polygon PQRS ≅ polygon WXYZ ∠A ≅ ∠D ∠B ≅ ∠E ∠C ≅ ∠F ∠P ≅ ∠W ∠Q ≅ ∠X ∠R ≅ ∠Y ∠S ≅ ∠ Z ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ AC ≅ ̶̶ DE ̶̶ EF ̶̶ DF ̶̶ PQ ≅ ̶̶ QR ≅ ̶̶ RS ≅ ̶̶ PS ≅ ̶̶̶ WX ̶̶ XY ̶̶ YZ ̶̶ WZ To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts. E X A M P L E 1 Naming Congruent Corresponding Parts △RST and △
XYZ represent the triangles of the space station’s support structure. If △RST ≅ △XYZ, identify all pairs of congruent corresponding parts. Angles: ∠R ≅ ∠X, ∠S ≅ ∠Y, ∠T ≅ ∠Z ̶̶ ST ≅ Sides: ̶̶ RT ≅ ̶̶ RS ≅ ̶̶ XY, ̶̶ YZ, ̶̶ XZ 1. If polygon LMNP ≅ polygon EFGH, identify all pairs of corresponding congruent parts. 4-3 Congruent Triangles 231 231 �� E X A M P L E 2 Using Corresponding Parts of Congruent Triangles Given: △EFH ≅ △GFH When you write a statement such as △ABC ≅ △DEF, you are also stating which parts are congruent. A Find the value of x. ∠FHE and ∠FHG are rt. . Def. of ⊥ lines ∠FHE ≅ ∠FHG m∠FHE = m∠FHG (6x - 12) ° = 90° 6x = 102 x = 7 B Find m∠GFH. Rt. ∠ ≅ Thm. Def. of ≅  Substitute values for m∠FHE and m∠FHG. Add 12 to both sides. Divide both sides by 6. m∠EFH + m∠FHE + m∠E = 180° △ Sum Thm. m∠EFH + 90 + 21.6 = 180 Substitute values for m∠FHE m∠EFH + 111.6 = 180 m∠EFH = 68.4 and m∠E. Simplify. Subtract 111.6 from both sides. ∠GFH ≅ ∠EFH m∠GFH = m∠EFH m∠GFH = 68.4° Corr.  of ≅  are ≅. Def. of ≅  Trans. Prop. of = Given: △ABC ≅ △DEF 2a. Find the value of x. 2b. Find m∠F. E X A M P L E 3 Proving Triangles Congruent
Given: ∠P and ∠M are right angles. R is the midpoint of ̶̶ ̶̶ NR PQ ≅ ̶̶ PM. ̶̶̶ MN, ̶̶ QR ≅ Prove: △PQR ≅ △MNR Proof: Statements Reasons 1. ∠P and ∠M are rt.  1. Given 2. ∠P ≅ ∠M 3. ∠PRQ ≅ ∠MRN 4. ∠Q ≅ ∠N ̶̶̶ PM. 6. 5. R is the mdpt. of ̶̶̶ MR ̶̶̶ MN ; ̶̶ PR ≅ ̶̶ PQ ≅ ̶̶ QR ≅ 7. ̶̶ NR 2. Rt. ∠ ≅ Thm. 3. Vert.  Thm. 4. Third  Thm. 5. Given 6. Def. of mdpt. 7. Given 8. △PQR ≅ △MNR 8. Def. of ≅  3. Given: ̶̶ AD bisects ̶̶ BE bisects ̶̶ AB ≅ Prove: △ABC ≅ △DEC ̶̶ BE. ̶̶ AD. ̶̶ DE, ∠A ≅ ∠D 232 232 Chapter 4 Triangle Congruence �� Overlapping Triangles “With overlapping triangles, it helps me to redraw the triangles separately. That way I can mark what I know about one triangle without getting confused by the other one.” Cecelia Medina Lamar High School E X A M P L E 4 Engineering Application Engineering The Rattler is one of the tallest wood-tracked roller coasters in the world. The ride sits on the side of a cliff at Six Flags Fiesta Texas. It is 5080 ft long and 179 ft high. The coaster travels at a speed of 65 mi/h. The bars that give structural support to a roller coaster form triangles. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. ̶̶ ML ⊥ Given: ̶̶ JL ≅ Prove: △JKL ≅ △MLK Proof: ̶̶ KL, ∠KLJ ≅ �
�LKM, ̶̶ MK ̶̶ JK ⊥ ̶̶ JK ≅ ̶̶ KL, ̶̶ ML, ̶̶ JK ⊥ ̶̶ KL, 1. Statements ̶̶̶ ML ⊥ ̶̶ KL Reasons 1. Given 2. ∠JKL and ∠MLK are rt. . 2. Def. of ⊥ lines 3. ∠JKL ≅ ∠MLK 4. ∠KLJ ≅ ∠LKM 5. ∠KJL ≅ ∠LMK ̶̶ JK ≅ ̶̶ KL ≅ ̶̶̶ ML, ̶̶ LK 6. 7. ̶̶ JL ≅ ̶̶̶ MK 8. △JKL ≅ △MLK 3. Rt. ∠ ≅ Thm. 4. Given 5. Third  Thm. 6. Given 7. Reflex. Prop. of ≅ 8. Def. of ≅  4. Use the diagram to prove the following. ̶̶ JK ≅ ̶̶̶ MK bisects ̶̶ JL bisects ̶̶ JL. Given: Prove: △JKN ≅ △LMN ̶̶̶ MK. ̶̶̶ ML, ̶̶ JK ǁ ̶̶̶ ML THINK AND DISCUSS 1. A roof truss is a triangular structure that supports a roof. How can you be sure that two roof trusses are the same size and shape? 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, name the congruent corresponding parts. 4-3 Congruent Triangles 233 233 �� 4-3 Exercises Exercises KEYWORD: MG7 4-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An everyday meaning of corresponding is “matching.” How can this help you find the corresponding parts of two triangles? 2. If △ABC ≅ △RST, what angle corresponds to ∠S. 231 Given: △RST ≅ △LMN
. Identify the congruent corresponding parts. ̶̶ RS ≅ ̶̶ TS ≅ 3. 6.? ̶̶̶̶? ̶̶̶̶ ̶̶ LN ≅ 4. 7. ∠L ≅? ̶̶̶̶? ̶̶̶̶ 5. ∠S ≅ 8. ∠N ≅? ̶̶̶̶? ̶̶̶̶ Given: △FGH ≅ △JKL. Find each value. p. 232 9. KL 10. 232. 233 11. Given: E is the midpoint of ̶̶ AB ≅ ̶̶ CD, ̶̶ AB ǁ ̶̶ CD ̶̶ AC and ̶̶ BD. Prove: △ABE ≅ △CDE Proof: Statements Reasons 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 5. Def. of mdpt. 4. d. 3. c. 6. f. 7. g.? ̶̶̶̶? ̶̶̶̶ ̶̶ AB ǁ ̶̶ CD 1. ̶̶ AB ≅ ̶̶ CD 3. 2. ∠ABE ≅ ∠CDE, ∠BAE ≅ ∠DCE 4. E is the mdpt. of ̶̶ AC and ̶̶ BD. 5. e.? ̶̶̶̶ 6. ∠AEB ≅ ∠CED 7. △ABE ≅ △CDE 12. Engineering The McDonald Observatory has four research telescopes and is a leading center for astronomical study. Prove that the triangles that make up the observatory dome are congruent. Given: ̶̶ ̶̶ ̶̶ SU ≅ SR, ST ≅ ∠UST ≅ ∠RST, and ∠U ≅ ∠R ̶̶ TU ≅ ̶̶ TR, Prove: △RTS ≅ △UTS 234 234 Chapter 4 Triangle Congruence �� Independent Practice Given: Polygon CDEF ≅ polygon KLMN. Identify the congruent corresponding parts. PRACTICE AND PROBLEM SOLVING 13. ̶̶ DE ≅ 15
. ∠F ≅? ̶̶̶̶? ̶̶̶̶ 14. ̶̶ KN ≅ 16. ∠L ≅? ̶̶̶̶? ̶̶̶̶ For See Exercises Example 13–16 17–18 19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 Given: △ABD ≅ △CBD. Find each value. 17. m∠C 18. y 19. Given: ̶̶̶ MP bisects ∠NMR. P is the midpoint of ̶̶ NR. ̶̶̶ MR, ∠N ≅ ∠R ̶̶̶ MN ≅ Prove: △MNP ≅ △MRP Proof: Statements Reasons 1. ∠N ≅ ∠R ̶̶̶ MP bisects ∠NMR. 2. 3. c. 4. d.? ̶̶̶̶? ̶̶̶̶ 5. P is the mdpt. of ̶̶ NR. 6. f.? ̶̶̶̶ ̶̶̶ MN ≅ ̶̶̶ MP ≅ ̶̶̶ MR ̶̶̶ MP 7. 8. 9. △MNP ≅ △MRP 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Def. of ∠ bisector 4. Third  Thm. 5. e.? ̶̶̶̶ 6. Def. of mdpt. 7. g. 8. h.? ̶̶̶̶? ̶̶̶̶ 9. Def. of ≅  20. Hobbies In a garden, triangular flower beds are separated by straight rows of grass as shown. Given: ∠ADC and ∠BCD are right angles. ̶̶ AD ≅ ̶̶ BC ̶̶ ̶̶ AC ≅ BD, ∠DAC ≅ ∠CBD Prove: △ADC ≅ △BCD 21. For two triangles, the following corresponding parts are given: ̶̶ ̶̶ ̶̶ PH, KH, GS ≅ ∠S ≅ ∠P, ∠G ≅ ∠K, and ∠R ≅ ∠H. Write three
different congruence statements. ̶̶ GR ≅ ̶̶ SR ≅ ̶̶ KP, 22. The two polygons in the diagram are congruent. Complete the following congruence statement for the polygons. polygon R? ≅ polygon V ̶̶̶̶? ̶̶̶̶ Write and solve an equation for each of the following. 23. △ABC ≅ △DEF. AB = 2x - 10, and DE = x + 20. Find the value of x and AB. 24. △JKL ≅ △MNP. m∠L = ( x 2 + 10) °, and m∠P = ( 2x 2 + 1) °. What is m∠L? 25. Polygon ABCD ≅ polygon PQRS. BC = 6x + 5, and QR = 5x + 7. Find the value of x and BC. 4-3 Congruent Triangles 235 235 ��ge07se_c04l03005aAEBDC�� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 238. ̶̶ JL and Many origami models begin with a square piece of paper, JKLM, that is folded along both diagonals to make the ̶̶̶ MK are perpendicular bisectors creases shown. of each other, and ∠NML ≅ ∠NKL. ̶̶ a. Explain how you know that KL and b. Prove △NML ≅ △NKL. ̶̶̶ ML are congruent. 27. Draw a diagram and then write a proof. ̶̶ BD ⊥ ̶̶ AC. D is the midpoint of ̶̶ AC. ̶̶ AB ≅ ̶̶ CB, and ̶̶ BD bisects ∠ABC. Given: Prove: △ABD ≅ △CBD 28. Critical Thinking Draw two triangles that are not congruent but have an area of 4 cm 2 each. 29. /////ERROR ANALYSIS///// Given △MPQ ≅ △EDF. Two solutions for finding m∠E are shown. Which is incorrect? Explain the error. 30. Write About It Given the diagram of the triangles, is there enough information to prove that △
HKL is congruent to △YWX? Explain. 31. Which congruence statement correctly indicates that the two given triangles are congruent? △ABC ≅ △DEF △ABC ≅ △FED △ABC ≅ △EFD △ABC ≅ △FDE 32. △MNP ≅ △RST. What are the values of x and y? x = 26, y = 21 1 _ 3 x = 27, y = 20 x = 25, y = 20 2 _ 3, y = 16 2 _ x = 30 1 _ 3 3 33. △ABC ≅ △XYZ. m∠A = 47.1°, and m∠C = 13.8°. Find m∠Y. 13.8 42.9 76.2 119.1 34. △MNR ≅ △SPQ, NL = 18, SP = 33, SR = 10, RQ = 24, and QP = 30. What is the perimeter of △MNR? 79 85 87 97 236 236 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 35. Multi-Step Given that the perimeter of TUVW is 149 units, find the value of x. Is △TUV ≅ △TWV? Explain. 36. Multi-Step Polygon ABCD ≅ polygon EFGH. ∠A is a right angle. m∠E = ( y 2 - 10) °, and m∠H = ( 2y 2 - 132) °. Find m∠D. 37. Given: ̶̶ RS ≅ ̶̶ RT, ∠S ≅ ∠T Prove: △RST ≅ △RTS SPIRAL REVIEW Two number cubes are rolled. Find the probability of each outcome. (Previous course) 38. Both numbers rolled are even. 39. The sum of the numbers rolled is 5. Classify each angle by its measure. (Lesson 1-3) 40. m∠DOC = 40° 41. m∠BOA = 90° 42. m∠
COA = 140° Find each angle measure. (Lesson 4-2) 43. ∠Q 44. ∠P 45. ∠QRS KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra 1 and 2, Geometry, Precalculus Q: What kind of degree or certification will you receive? A: I will receive an associate’s degree in applied science. Then I will take an exam to be certified as an EMT or paramedic. Q: How do you use math in your hands-on training? A: I calculate dosages based on body weight and age. I also calculate drug doses in milligrams per kilogram per hour or set up an IV drip to deliver medications at the correct rate. Q: What are your future career plans? A: When I am certified, I can work for a private ambulance service or with a fire department. I could also work in a hospital, transporting critically ill patients by ambulance or helicopter. 4-3 Congruent Triangles 237 237 Jordan Carter Emergency Medical Services Program �� SECTION 4A Triangles and Congruence Origami Origami is the Japanese art of paper folding. The Japanese word origami literally means “fold paper.” This ancient art form relies on properties of geometry to produce fascinating and beautiful shapes. Each of the figures shows a step in making an origami swan from a square piece of paper. The final figure shows the creases of an origami swan that has been unfolded. Step 1 Step 2 � Step 3 � � � � � � � � � � � � Fold the paper in half diagonally and crease it. Turn it over. Fold corners A and C to the center line and crease. Turn it over. � Step 4 Step 5 � � �� Fold in half along the center crease so that and ̶̶ DF are together. ̶̶ DE Step 6 � �� Fold the narrow point upward at a 90° angle and crease. Push in the fold so that the neck is inside the body. Fold the tip downward and crease. Push in the fold so that the head is inside the neck. Fold up the flap to form the wing. � 1. Use the fact that ABCD is a square � 2. to classify △ABD by its side lengths
and by its angle measures. ̶̶ ̶̶ DB bisects ∠ABC and ∠ADC. DE bisects ∠ADB. Find the measures of the angles in △EDB. Explain how you found the measures. ̶̶ 3. Given that DB bisects ∠ABC and ̶̶ ∠EDF, DE ≅ prove that △EDB ≅ △ FDB. ̶̶ BF, and ̶̶ BE ≅ ̶̶ DF, � � � � � � 238 238 Chapter 4 Triangle Congruence SECTION 4A Quiz for Lessons 4-1 Through 4-3 4-1 Classifying Triangles Classify each triangle by its angle measures. 1. △ACD 2. △ABD 3. △ADE Classify each triangle by its side lengths. 4. △PQR 5. △PRS 6. △PQS 4-2 Angle Relationships in Triangles Find each angle measure. 7. m∠M 8. m∠ABC 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 37° and 55°. Find the measure of ∠RTP, the angle formed by the roof of the house and the roof of the patio. 4-3 Congruent Triangles Given: △JKL ≅ △DEF. Identify the congruent corresponding parts. 10. ̶̶ KL ≅? ̶̶̶̶ 11. ̶̶ DF ≅? ̶̶̶̶ 12. ∠K ≅? ̶̶̶̶ 13. ∠F ≅? ̶̶̶̶ Given: △PQR ≅ △STU. Find each value. 14. PQ 15. y 16. Given:   AB ǁ   CD, ̶̶ CD, ̶̶ AB ≅ ̶̶ ̶̶ DB ⊥ AC ⊥ Prove: △ACD ≅ △DBA ̶̶ CD, ̶̶ AB ̶̶ AC ≅ ̶̶ BD, Proof: Statements Reasons 1.   AB ǁ   CD 2. ∠BAD ≅ �
�CDA ̶̶ AC ⊥ ̶̶ CD, ̶̶ DB ⊥ ̶̶ AB 3. 4. ∠ACD and ∠DBA are rt.  5. e.? ̶̶̶̶̶? ̶̶̶̶̶ ̶̶ CD, 6. f. ̶̶ AB ≅ 7. ̶̶ AC ≅ ̶̶ BD 8. h.? ̶̶̶̶̶ 9. △ACD ≅ △DBA 1. a. 2. b. 3. c. 4. d.? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶ 5. Rt. ∠ ≅ Thm. 6. Third  Thm. 7. g.? ̶̶̶̶̶ 8. Reflex Prop. of ≅ 9. i.? ̶̶̶̶̶ Ready to Go On? 239 239 �� 4-4 Explore SSS and SAS Triangle Congruence Use with Lesson 4-4 In Lesson 4-3, you used the definition of congruent triangles to prove triangles congruent. To use the definition, you need to prove that all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. In this lab, you will discover some shortcuts for proving triangles congruent. Activity 1 TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.4.A G.10.B 1 Measure and cut six pieces from the straws: two that are 2 inches long, two that are 4 inches long, and two that are 5 inches long. 2 Cut two pieces of string that are each about 20 inches long. 3 Thread one piece of each size of straw onto a piece of string. Tie the ends of the string together so that the pieces of straw form a triangle. 4 Using the remaining pieces, try to make another triangle with the same side lengths that is not congruent to the first triangle. Try This 1. Repeat Activity 1 using side lengths of your choice. Are your results the same? 2. Do you think it is possible
to make two triangles that have the same side lengths but that are not congruent? Why or why not? 3. How does your answer to Problem 2 provide a shortcut for proving triangles congruent? 4. Complete the following conjecture based on your results. Two triangles are congruent if?. ̶̶̶̶̶̶̶̶̶̶̶̶̶ 240 240 Chapter 4 Triangle Congruence Activity 2 1 Measure and cut two pieces from the straws: one that is 4 inches long and one that is 5 inches long. 2 Use a protractor to help you bend a paper clip to form a 30° angle. 3 Place the pieces of straw on the sides of the 30° angle. The straws will form two sides of your triangle. 4 Without changing the angle formed by the paper clip, use a piece of straw to make a third side for your triangle, cutting it to fit as necessary. Use additional paper clips or string to hold the straws together in a triangle. Try This 5. Repeat Activity 2 using side lengths and an angle measure of your choice. Are your results the same? 6. Suppose you know two side lengths of a triangle and the measure of the angle between these sides. Can the length of the third side be any measure? Explain. 7. How does your answer to Problem 6 provide a shortcut for proving triangles congruent? 8. Use the two given sides and the given angle from Activity 2 to form a triangle that is not congruent to the triangle you formed. (Hint: One of the given sides does not have to be adjacent to the given angle.) 9. Complete the following conjecture based on your results. Two triangles are congruent if?. ̶̶̶̶̶̶̶̶̶̶̶̶̶ 4- 4 Geometry Lab 241 241 4-4 Triangle Congruence: SSS and SAS TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.A, G.3.B, G.3.E Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles
congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. △ABC ≅ △FDE E X A M P L E 1 Using SSS to Prove Triangle Congruence Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why △PQR ≅ △PSR. ̶̶ ̶̶ SR. By QR ≅ It is given that ̶̶ PR ≅ the Reflexive Property of Congruence, Therefore △PQR ≅ △PSR by SSS. ̶̶ PS and that ̶̶ PQ ≅ ̶̶ PR. 1. Use SSS to explain why △ABC ≅ △CDA. An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides ̶̶ AB and ̶̶ BC. 242 242 Chapter 4 Triangle Congruence �� It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. Postulate 4-4-2 Side-Angle-Side (SAS) Congruence POSTULATE HYPOTHESIS CONCLUSION If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. △ABC ≅ △EFD E X A M P L E 2 Engineering Application The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. The figure shows part of the support structure of the Statue of Liberty. Use SAS to explain
why △KPN ≅ △LPM. It is given that ̶̶ NP ≅ and that By the Vertical Angles Theorem, ∠KPN ≅ ∠LPM. Therefore △KPN ≅ △LPM by SAS. ̶̶ KP ≅ ̶̶̶ MP. ̶̶ LP 2. Use SAS to explain why △ABC ≅ △DBC. The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angle, you can construct one and only one triangle. Construction Congruent Triangles Using SAS Use a straightedge to draw two segments and one angle, or copy the given segments and angle.    ̶̶ AB congruent to one Construct of the segments. Construct ∠A congruent to the given angle. ̶̶ Construct AC congruent to the other segment. Draw to complete △ABC. ̶̶ CB 4-4 Triangle Congruence: SSS and SAS 243 243 KPMNLge07se_c04l04009a3rd pass4/25/5cmurphy�� E X A M P L E 3 Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. A △UVW ≅ △YXW, x = 3 ZY = x - 1 = 3 - 1 = 2 XZ = x = 3 XY = 3x - 5 = 3 (3) - 5 = 4 ̶̶ ̶̶̶ ̶̶ UV ≅ XZ, and VW ≅ So △UVW ≅ △YXZ by SSS. ̶̶ YX. ̶̶̶ UW ≅ ̶̶ YZ. B △DEF ≅ △JGH, y = 7 JG = 2y + 1 = 2 (7) + 1 = 15 GH = y 2 - 4y + 3 = (7) 2 - 4 (7) + 3 = 24 m∠G = 12y + 42 = 12 (7) + 42 = 126° ̶̶ ̶̶ ̶̶ DE ≅ EF ≅ JG. So △DEF ≅ △JGH by SAS. ̶̶̶ GH, and ∠E ≅ ∠G. 3. △ADB
≅ △CDB Proving Triangles Congruent ̶̶ EG ≅ ̶̶ Given: ℓ ǁ m, HF Prove: △EGF ≅ △HFG Proof: Statements Reasons ̶̶ EG ≅ ̶̶ HF 1. 2. ℓ ǁ m 3. ∠EGF ≅ ∠HFG ̶̶ FG ≅ ̶̶ FG 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex Prop. of ≅ 5. △EGF ≅ △HFG 5. SAS Steps 1, 3, 4 4. Given: Prove: △RQP ≅ △SQP  QP bisects ∠RQS. ̶̶ QR ≅ ̶̶ QS 244 244 Chapter 4 Triangle Congruence �� THINK AND DISCUSS 1. Describe three ways you could prove that △ABC ≅ △DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 Exercises Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In △RST which angle is the included angle of sides ̶̶ ST and ̶̶ TR? Use SSS to explain why the triangles in each pair are congruent. p. 242 2. △ABD ≅ △CDB 3. △MNP ≅ △MQP. Design This Texas flag consists of a blue, p. 243 perpendicular stripe with a white star in the center. The star consists of five triangles. GJ = LG = 20 in., and GK = GH = 13 in. Use SAS to explain why △JGK ≅ △LGH Show that the triangles are congruent for the given value of the variable. p. 244 5. △GHJ ≅ △IHJ, x = 4 6. △RST ≅ △TUR, x =
18 4-4 Triangle Congruence: SSS and SAS 245 245 ��. Given: ̶̶ JK ≅ ̶̶̶ ML, ∠JKL ≅ ∠MLK p. 244 Prove: △JKL ≅ △MLK Proof: Statements Reasons ̶̶ JK ≅ ̶̶̶ ML 1. 2. b. ̶̶ KL ≅? ̶̶̶̶ ̶̶ LK 3. 4. △JKL ≅ △MLK 1. a.? ̶̶̶̶ 2. Given 3. c. 4. d.? ̶̶̶̶? ̶̶̶̶ Independent Practice Use SSS to explain why the triangles in each pair are congruent. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 8. △BCD ≅ △EDC 9. △GJK ≅ △GJL 10. Theater The lights shining on a stage appear to form two congruent right triangles. Given △ECB ≅ △DBC. ̶̶ DB, use SAS to explain why ̶̶ EC ≅ Show that the triangles are congruent for the given value of the variable. 11. △MNP ≅ △QNP, y = 3 12. △XYZ ≅ △STU, t = 5 13. Given: B is the midpoint of ̶̶ DC. ̶̶ AB ⊥ ̶̶ DC Prove: △ABD ≅ △ABC Proof: Statements ̶̶ DC. 1. B is the mdpt. of 2. b. 3. c.? ̶̶̶̶? ̶̶̶̶ 4. ∠ABD and ∠ABC are rt. . 5. ∠ABD ≅ ∠ABC 6. f.? ̶̶̶̶ 7. △ABD ≅ △ABC 246 246 Chapter 4 Triangle Congruence Reasons 1. a.? ̶̶̶̶ 2. Def. of mdpt
. 3. Given 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶ 6. Reflex. Prop. of ≅ 7. g.? ̶̶̶̶ �� Which postulate, if any, can be used to prove the triangles congruent? 14. 16. 15. 17. 18. Explain what additional information, if any, you would need to prove △ABC ≅ △DEC by each postulate. b. SAS a. SSS Multi-Step Graph each triangle. Then use the Distance Formula and the SSS Postulate to determine whether the triangles are congruent. 19. △QRS and △TUV 20. △ABC and △DEF Q (-2, 0), R (1, -2), S (-3, -2) T (5, 1), U (3, -2), V (3, 2) A (2, 3), B (3, -1), C (7, 2) D (-3, 1), E (1, 2), F (-3, 5) 21. Given: ∠ZVY ≅ ∠WYV, ∠ZVW ≅ ∠WYZ, ̶̶ ̶̶̶ YZ VW ≅ Prove: △ZVY ≅ △WYV Proof: Statements Reasons 1. ∠ZVY ≅ ∠WYV, ∠ZVW ≅ WYZ 2. m∠ZVY = m∠WYV, m∠ZVW = m∠WYZ 3. m∠ZVY + m∠ZVW = m∠WYV + m∠WYZ 4. c.? ̶̶̶̶ 5. ∠WVY ≅ ∠ZYV ̶̶̶ VW ≅ ̶̶ YZ 6. 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Add. Prop. of = 4. ∠ Add. Post. 5. d. 6. e.? ̶̶̶̶? ̶̶̶̶ 7. f.? ̶̶̶̶ 8. �
�ZVY ≅ △WYV 7. Reflex. Prop. of ≅ 8. g.? ̶̶̶̶ 22. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that ∠A and ∠D are right angles. Explain how you could make just two additional measurements on each truss to ensure that the trusses are congruent. b. You verify that the trusses are congruent and find ̶̶ EF to the that AB = AC = 2.5 ft. Find the length of nearest tenth. Explain. 4-4 Triangle Congruence: SSS and SAS 247 247 �� 23. Critical Thinking Draw two isosceles triangles that are not congruent but that have a perimeter of 15 cm each. Ecology 24. △ABC ≅ △ADC for what value of x? Explain why the SSS Postulate can be used to prove the two triangles congruent. 25. Ecology A wing deflector is a triangular structure made of logs that is filled with large rocks and placed in a stream to guide the current or prevent erosion. Wing deflectors are often used in pairs. Suppose an engineer wants to build two wing deflectors. The logs that form the sides of each wing deflector are perpendicular. How can the engineer make sure that the two wing deflectors are congruent? Wing deflectors are designed to reduce the width-to-depth ratio of a stream. Reducing the width increases the velocity of the stream. 26. Write About It If you use the same two sides and included angle to repeat the construction of a triangle, are your two constructed triangles congruent? Explain. 27. Construction Use three segments (SSS) to construct a scalene triangle. Suppose you then use the same segments in a different order to construct a second triangle. Will the result be the same? Explain. 28. Which of the three triangles below can be proven congruent by SSS or SAS? I and II II and III I and III I, II, and III 29. What is the perimeter of polygon ABCD? 29.9 cm 39.8 cm 49.8 cm 59.8 cm 30. Jacob wants to prove that △FGH ≅ �
�JKL using SAS. ̶̶ ̶̶ JL. What additional JK and ̶̶ FH ≅ ̶̶ He knows that FG ≅ piece of information does he need? ∠H ≅ ∠L ∠F ≅ ∠G ∠F ≅ ∠J ∠G ≅ ∠K 31. What must the value of x be in order to prove that △EFG ≅ △EHG by SSS? 1.5 4.25 4.67 5.5 248 248 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 32. Given:. ∠ADC and ∠BCD are ̶̶ AD ≅ supplementary. ̶̶ CB Prove: △ADB ≅ △CBD (Hint: Draw an auxiliary line.) 33. Given: ∠QPS ≅ ∠TPR, ̶̶ PQ ≅ ̶̶ PT, ̶̶ PR ≅ ̶̶ PS Prove: △PQR ≅ △PTS Algebra Use the following information for Exercises 34 and 35. Find the value of x. Then use SSS or SAS to write a paragraph proof showing that two of the triangles are congruent. 34. m∠FKJ = 2x° m∠KFJ = (3x + 10) ° KJ = 4x + 8 HJ = 6 (x - 4) 35. ̶̶ FJ bisects ∠KFH. m∠KFJ = (2x + 6) ° m∠HFJ = (3x - 21) ° FK = 8x - 45 FH = 6x + 9 SPIRAL REVIEW Solve and graph each inequality. (Previous course) 36. x _ 2 37. 2a + 4 > 3a - 8 ≤ 5 38. -6m - 1 ≤ -13 Solve each equation. Write a justification for each step. (Lesson 2-5) 40. a _ 4 39. 4x - 7 = 21 + 5 = -8 41. 6r = 4r + 10 Given: △EFG ≅ △GHE. Find each value. (Lesson 4-3) 42. x 43. m∠F
EG 44. m∠FGH Using Technology Use geometry software to complete the following. 1. Draw a triangle and label the vertices A, B, and C. Draw a point and label it D. Mark a vector from A to B and translate D by the marked vector. Label the image E. Draw   DE. Mark ∠BAC and rotate   DE about D by the marked angle. Mark ∠ABC and rotate   DE about E by the marked angle. Label the intersection F. 2. Drag A, B, and C to different locations. What do you notice about the two triangles? 3. Write a conjecture about △ABC and △DEF. 4. Test your conjecture by measuring the sides and angles of △ABC and △DEF. 4-4 Triangle Congruence: SSS and SAS 249 249 �� 4-5 Use with Lesson 4-5 Activity 1 Predict Other Triangle Congruence Relationships Geometry software can help you investigate whether certain combinations of triangle parts will make only one triangle. If a combination makes only one triangle, then this arrangement can be used to prove two triangles congruent. TEKS G.9.B Congruence and the geometry of size: formulate … conjectures about the properties and attributes of polygons … based on explorations. Also G.2.A, G.3.B, G.10.B 1 Construct ∠CAB measuring 45° and ∠EDF measuring 110°. 2 Move ∠EDF so that DE overlays AC intersect, label the BA. DF and Where point G. Measure ∠DGA. 3 Move ∠CAB to the left and right without changing the measures of the angles. Observe what happens to the size of ∠DGA. 4 Measure the distance from A to D. Try to change the shape of the triangle without changing AD and the measures of ∠A and ∠D. Try This 1. Repeat Activity 1 using angle measures of your choice. Are your results the same? Explain. 2. Do the results change if one of the given angles measures 90°? 3. What theorem proves that the
measure of ∠DGA in Step 2 will always be the same? 4. In Step 3 of the activity, the angle measures in △ADG stayed the same as the size of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only one triangle? Explain. 5. Repeat Step 4 of the activity but measure the length of ̶̶ AG instead of ̶̶ AD. Are your results the same? Does this lead to a new congruence postulate or theorem? 6. If you are given two angles of a triangle, what additional piece of information is needed so that only one triangle is made? Make a conjecture based on your findings in Step 5. 250 250 Chapter 4 Triangle Congruence Activity 2 1 Construct ̶̶ YZ with a length of 6.5 cm. 2 Using ̶̶ YZ as a side, construct ∠XYZ measuring 43°. 3 Draw a circle at Z with a radius of 5 cm. Construct ̶̶̶ ZW, a radius of circle Z. 4 Move W around circle Z. Observe the possible shapes of △YZW. Try This 7. In Step 4 of the activity, how many different triangles were possible? Does Side-Side-Angle make only one triangle? 8. Repeat Activity 2 using an angle measure of 90° in Step 2 and a circle with a radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 9. Repeat the activity again using a measure of 90° in Step 2 and a circle with a radius of 4.25 cm in Step 3. Classify the resulting triangle by its angle measures. 10. Based on your results, complete the following conjecture. In a Side-Side-Angle combination, if the corresponding nonincluded angles are triangle is possible.?, then only one ̶̶̶̶ 4- 5 Technology Lab 251 251 4-5 Triangle Congruence: ASA, AAS, and HL TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.1.A, G.1.B, G.2.A, G.3.B, G.3.C, G.3.E, G.9.B Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS
, and HL. Vocabulary included side Why use this? Bearings are used to convey direction, helping people find their way to specific locations. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43° E, you face south and then turn 43° to the east. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. ̶̶ PQ is the included side of ∠P and ∠Q. Postulate 4-5-1 Angle-Side-Angle (ASA) Congruence POSTULATE HYPOTHESIS CONCLUSION If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. △ABC ≅ △DEF E X A M P L E 1 Problem-Solving Application Organizers of an orienteering race are planning a course with checkpoints A, B, and C. Does the table give enough information to determine the location of the checkpoints? Understand the Problem Bearing Distance A to B N 55° E 7.6 km B to C N 26° W C to A S 20° W The answer is whether the information in the table can be used to find the position of checkpoints A, B, and C. List the important information: The bearing from A to B is N 55° E. From B to C is N 26° W, and from C to A is S 20° W. The distance from A to B is 7.6 km. 252 252 Chapter 4 Triangle Congruence ��1 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of △ABC. Solve m∠CAB = 55° - 20° = 35° m∠CBA = 180° - (26° + 55°) = 99° You know the measures of ∠CAB and ∠CBA and the length of the included side ̶̶ AB. Therefore by ASA, a unique triangle ABC is determined. Look Back One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the
checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. E X A M P L E 2 Applying ASA Congruence Determine if you can use ASA to prove △UVX ≅ △WVX. Explain. ∠UXV ≅ ∠WXV as given. Since ∠WVX is a right angle that forms a linear pair with ∠UVX, ∠WVX ≅ ∠UVX. Also by the Reflexive Property. Therefore △UVX ≅ △WVX by ASA. ̶̶ VX ≅ ̶̶ VX 2. Determine if you can use ASA to prove △NKL ≅ △LMN. Explain. Construction Congruent Triangles Using ASA Use a straightedge to draw a segment and two angles, or copy the given segment and angles.     ̶̶ CD congruent to Construct the given segment. Construct ∠C congruent to one of the angles. Construct ∠D congruent to the other angle. Label the intersection of the rays as E. △CDE 4-5 Triangle Congruence: ASA, AAS, and HL 253 253 2��34�� You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). Theorem 4-5-2 Angle-Angle-Side (AAS) Congruence THEOREM HYPOTHESIS CONCLUSION If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. △GHJ ≅ △KLM PROOF PROOF Angle-Angle-Side Congruence ̶̶̶ LM ̶̶ HJ ≅ Given: ∠G ≅ ∠K, ∠J ≅ ∠M, Prove: △GHJ ≅ △KLM Proof: Statements Reasons 1. ∠G ≅ ∠K, ∠J ≅ ∠M 1. Given 2. ∠H ≅ ∠L ̶̶̶ LM
̶̶ HJ ≅ 3. 2. Third  Thm. 3. Given 4. △GHJ ≅ △KLM 4. ASA Steps 1, 3, and 2 E X A M P L E 3 Using AAS to Prove Triangles Congruent ̶̶ AB ǁ Use AAS to prove the triangles congruent. ̶̶ BC ≅ Given: Prove: △ABC ≅ △EDC Proof: ̶̶ ED, ̶̶ DC 3. Use AAS to prove the triangles congruent. ̶̶ JL bisects ∠KLM. ∠K ≅ ∠M Given: Prove: △JKL ≅ △JML There are four theorems for right triangles that are not used for acute or obtuse triangles. They are Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA), and Hypotenuse-Leg (HL). You will prove LL, HA, and LA in Exercises 21, 23, and 33. 254 254 Chapter 4 Triangle Congruence �� Theorem 4-5-3 Hypotenuse-Leg (HL) Congruence THEOREM HYPOTHESIS CONCLUSION If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. △ABC ≅ △DEF You will prove the Hypotenuse-Leg Theorem in Lesson 4-8. E X A M P L E 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A △VWX and △YXW According to the diagram, △VWX and △YXW are right triangles that share ̶̶̶ ̶̶̶ WX ≅ WX by the Reflexive hypotenuse ̶̶ ̶̶̶ WV ≅ Property. It is given that XY, therefore △VWX ≅ △YXW by HL. ̶̶̶ WX. B △VWZ and �
�YXZ This conclusion cannot be proved by HL. According to the diagram, △VWZ and △YXZ are right triangles, ̶̶̶ WZ and is congruent to hypotenuse ̶̶ XY. You do not know that hypotenuse ̶̶ XZ. ̶̶̶ WV ≅ 4. Determine if you can use the HL Congruence Theorem to prove △ABC ≅ △DCB. If not, tell what else you need to know. THINK AND DISCUSS 1. Could you use AAS to prove that these two triangles are congruent? Explain. 2. The arrangement of the letters in ASA matches the arrangement of what parts of congruent triangles? Include a sketch to support your answer. 3. GET ORGANIZED Copy and complete the graphic organizer. In each column, write a description of the method and then sketch two triangles, marking the appropriate congruent parts. 4-5 Triangle Congruence: ASA, AAS, and HL 255 255 �� 4-5 Exercises Exercises KEYWORD: MG7 4-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A triangle contains ∠ABC and ∠ACB with ̶̶ BC “closed in” between them. How would this help you remember the definition of included side. 252 Surveying Use the table for Exercises 2 and 3. A landscape designer surveyed the boundaries of a triangular park. She made the following table for the dimemsions of the land. A to B B to C C to A Bearing E S 25° E N 62° W Distance 115 ft?? 2. Draw the plot of land described by the table. Label the measures of the angles in the triangle. 3. Does the table have enough information to determine the locations of points A, B, and C? Explain Determine if you can use ASA to prove the triangles congruent. Explain. p. 253 4. △VRS and △VTS, given that ̶̶ VS bisects ∠RST and ∠RVT 5. △DEH and △FGH. Use AAS to prove the triangles congruent. p. 254 Given: ∠R and ∠P are right angles.
̶̶ QR ǁ ̶̶ SP Prove: △QPS ≅ △SRQ Proof. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. △ABC and △CDA 8. △XYV and △ZYV 256 256 Chapter 4 Triangle Congruence CBA115 ft ge07sec_04l05003aa�� Independent Practice For See Exercises Example 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Surveying Use the table for Exercises 9 and 10. From two different observation towers a fire is sighted. The locations of the towers are given in the following table. X to Y X to F Y to F Bearing E N 53° E N 16° W Distance 6 km?? 9. Draw the diagram formed by observation tower X, observation tower Y, and the fire F. Label the measures of the angles. 10. Is there enough information given in the table to pinpoint the location of the fire? Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Math History 11. 11. △MKJ and △MKL 12. △RST and △TUR 13. 13. Given: ̶̶ AB ≅ Prove: △ABC ≅ △DEF ̶̶ DE, ∠C ≅ ∠F Proof: Euclid wrote the mathematical text The Elements around 2300 years ago. It may be the second most reprinted book in history. Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 14. △GHJ and △JKG 15. △ABE and △DCE, given that E is the midpoint ̶̶ ̶̶ BC AD and of Multi-Step For each pair of triangles write a triangle congru
ence statement. Identify the transformation that moves one triangle to the position of the other triangle. 16. 17. 18. Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two triangles congruent. Draw a diagram that shows why this is true. 4-5 Triangle Congruence: ASA, AAS, and HL 257 257 �� 19. This problem will prepare you for the Multi-Step TAKS Prep on page 280. A carpenter built a truss to support the roof of a doghouse. ̶̶ a. The carpenter knows that MJ. Can the carpenter ̶̶ KJ ≅ conclude that △KJL ≅ △MJL? Why or why not? b. Suppose the carpenter also knows that ∠JLK is a right angle. Which theorem can be used to show that △KJL ≅ △MJL? 20. /////ERROR ANALYSIS///// Two proofs that △EFH ≅ △GHF are given. Which is incorrect? Explain the error. 21. Write a paragraph proof of the Leg-Leg (LL) Congruence Theorem. If the legs of one right triangle are congruent to the corresponding legs of another right triangle, the triangles are congruent. 22. Use AAS to prove the triangles congruent. ̶̶ AD ≅ Prove: △AED ≅ △CEB ̶̶ AD ǁ Given: ̶̶ BC, ̶̶ CB Proof: Statements Reasons ̶̶̶ AD ǁ ̶̶ BC 1. 2. ∠DAE ≅ ∠BCE 3. c. 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Vert.  Thm. 3. Given 4. f.? ̶̶̶̶ 23. Prove the Hypotenuse-Angle (HA) Theorem. ̶̶ ̶̶̶ JM ≅ KM ⊥ Given: Prove: △JKM ≅ △LKM
̶̶̶ LM, ∠JMK ≅ ∠LMK ̶̶ JL, 24. Write About It The legs of both right △DEF and right △RST are 3 cm and 4 cm. They each have a hypotenuse 5 cm in length. Describe two different ways you could prove that △DEF ≅ △RST. 25. Construction Use the method for constructing perpendicular lines to construct a right triangle. 26. What additional congruence statement is necessary to prove △XWY ≅ △XVZ by ASA? ∠XVZ ≅ ∠XWY ∠VUY ≅ ∠WUZ ̶̶ VZ ≅ ̶̶ XZ ≅ ̶̶̶ WY ̶̶ XY 258 258 Chapter 4 Triangle Congruence �� 27. Which postulate or theorem justifies the congruence statement △STU ≅ △VUT? ASA SSS HL SAS 28. Which of the following congruence statements is true? ∠A ≅ ∠B ̶̶ ̶̶ DE CE ≅ △AED ≅ △CEB △AED ≅ △BEC 29. In △RST, RT = 6y - 2. In △UVW, UW = 2y + 7. ∠R ≅ ∠U, and ∠S ≅ ∠V. What must be the value of y in order to prove that △RST ≅ △UVW? 1.25 2.25 9.0 11.5 30. Extended Response Draw a triangle. Construct a second triangle that has the same angle measures but is not congruent. Compare the lengths of each pair of corresponding sides. Consider the relationship between the lengths of the sides and the measures of the angles. Explain why Angle-Angle-Angle (AAA) is not a congruence principle. CHALLENGE AND EXTEND 31. Sports This bicycle frame includes △VSU and △VTU, which
lie in intersecting planes. From the given angle measures, can you conclude that △VSU ≅ △VTU? Explain. m∠VUS = (7y - 2) ° m∠VUT = (5 1 _ m∠USV = 5 2 _ y ° 3 m∠SVU = (3y - 6) ° m∠TVU = 2x ° ) x - 1 _ 2 2 m∠UTV = (4x + 8) ° ° � � � � 32. Given: △ABC is equilateral. C is the midpoint of ̶̶ DE. ∠DAC and ∠EBC are congruent and supplementary. Prove: △DAC ≅ △EBC �� 33. Write a two-column proof of the Leg-Angle (LA) Congruence Theorem. If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (Hint: There are two cases to consider.) 34. If two triangles are congruent by ASA, what theorem could you use to prove that the triangles are also congruent by AAS? Explain. SPIRAL REVIEW Identify the x- and y-intercepts. Use them to graph each line. (Previous course) 35. y = 3x - 6 36 37. y = -5x + 5 38. Find AB and BC if AC = 10. (Lesson 1-6) 39. Find m∠C. (Lesson 4-2) 4-5 Triangle Congruence: ASA, AAS, and HL 259 259 �� 4-6 Triangle Congruence: CPCTC TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning … Also G.3.E, G.7.A, G.10.B Objective Use CPCTC to prove parts of triangles are congruent. Vocabulary CPCTC Why learn this? You can use congruent triangles to estimate distances. CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles
congruent. E X A M P L E 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in the figure. If DE = 600 ft, what is AB? ̶̶ CB,because DC = CB = 500 ft. ∠D ≅ ∠B, because they are both right angles. ̶̶ DC ≅ ∠DCE ≅ ∠BCA, because vertical angles are congruent. Therefore △DCE ≅ △BCA by ASA or LA. By CPCTC, AB = ED = 600 ft. ̶̶ AB, so ̶̶ ED ≅ 1. A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? � �� E X A M P L E 2 Proving Corresponding Parts Congruent ̶̶ AB ≅ ̶̶ DC, ∠ABC ≅ ∠DCB Given: Prove: ∠A ≅ ∠D Proof: 2. Given: Prove: ̶̶ PR bisects ∠QPS and ∠QRS. ̶̶ PQ ≅ ̶̶ PS 260 260 Chapter 4 Triangle Congruence �� E X A M P L E 3 Using CPCTC in a Proof ̶̶ DF ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ ED ǁ ̶̶ DF, ̶̶ GF Given: Prove: Proof: Statements Reasons ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ DF ̶̶ DF 1. 2. 3. ∠EGD ≅ ∠FDG ̶̶̶ GD ≅ ̶̶̶ GD 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex. Prop. of ≅ 5. △EGD ≅ △FDG 5. SAS Steps 1, 3,
and 4 6. ∠EDG ≅ ∠FGD ̶̶ GF ̶̶ ED ǁ 7. 6. CPCTC 7. Converse of Alt. Int.  Thm. 3. Given: J is the midpoint of Prove: ̶̶ KL ǁ ̶̶̶ MN ̶̶̶ KM and ̶̶ NL. You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that the triangles are congruent, you can make conclusions about their corresponding parts. E X A M P L E 4 Using CPCTC in the Coordinate Plane Given: A (2, 3), B (5, -1), C (1, 0), D (-4, -1), E (0, 2), F (-1, -2) Prove: ∠ABC ≅ ∠DEF Step 1 Plot the points on a coordinate plane. Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. D = √  ( AB = √  (5 -2) 2 + (-1 - 3) 2 = √  9 + 16 = √  25 = 5 BC =  2 √ (1 - 5) 2 + (0 - (-1) ) = √  16 + 1 = √  17 AC = √  (1 - 2) 2 + (0 - 3) 2 = √  1 + 9 = √  10 DE =  2 √ + (2 - (-1) ) (0 - (-
4) ) 16 + 9 = √  25 = 5 = √  2 EF = √  (-1 - 0) 2 + (-2 - 2) 2 = √  1 + 16 = √  17 DF =  2 √ + (-2 - (-1) ) (-1 - (-4) ) 9 + 1 = √  10 = √  2 ̶̶ AB ≅ ̶̶ BC ≅ So and ∠ABC ≅ ∠DEF by CPCTC. ̶̶ EF, and ̶̶ DE, ̶̶ AC ≅ ̶̶ DF. Therefore △ABC ≅ △DEF by SSS, 4. Given: J (-1, -2), K (2, -1), L (-2, 0), R (2, 3), S (5, 2), T (1, 1) Prove: ∠JKL ≅ ∠RST 4-6 Triangle Congruence: CPCTC 261 261 �� THINK AND DISCUSS ̶̶̶ VW ≅ 1. In the figure, ̶̶ UV ≅ ̶̶ XY, ̶̶ YZ, and ∠V ≅ ∠Y. Explain why △UVW ≅ △XYZ. By CPCTC, which additional parts are congruent? 2. GET ORGANIZED Copy and complete the graphic organizer. Write all conclusions you can make using CPCTC. 4-6 Exercises Exercises GUIDED PRACTICE 1. Vocabulary You use CPCTC after proving triangles are congruent. Which parts of congruent triangles are referred to as corresponding parts. Engineering To find the height of p. 260 a windmill, a rancher places a marker at C and steps off the distance from C to B. Then the rancher walks the same distance from C in the opposite direction and places a marker at D. If DE
= 6.3 m, what is AB? KEYWORD: MG7 4-6 KEYWORD: MG7 Parent � � � � � �. 260 3. Given: X is the midpoint of ̶̶ ST. ̶̶ RX ⊥ ̶̶ ST Prove: ̶̶ RS ≅ ̶̶ RT Proof: 262 262 Chapter 4 Triangle Congruence ��. 261 4. Given: Prove: ̶̶ ̶̶ AC ≅ AD, ̶̶ AB bisects ∠CAD. ̶̶ CB ≅ ̶̶ DB Proof: Statements ̶̶ ̶̶ ̶̶̶ DB CB ≅ AD, ̶̶ AC ≅ 1. 2. b.? ̶̶̶̶ 3. △ACB ≅ △ADB 4. ∠CAB ≅ ∠DAB ̶̶ AB bisects ∠CAD 5. Reasons 1. a.? ̶̶̶̶ 2. Reflex. Prop. of ≅ 3. c. 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ Multi-Step Use the given set of points to prove each congruence statement. p. 261 5. E (-3, 3), F (-1, 3), G (-2, 0), J (0, -1), K (2, -1), L (1, 2) ; ∠EFG ≅ ∠JKL 6. A (2, 3), B (4, 1), C (1, -1), R (-1, 0), S (-3, -2), T (0, -4) ; ∠ACB ≅ ∠RTS Independent Practice For See Exercises Example 7 8–9 10–11 12–13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 7. Surveying To find the distance AB across a river, a surveyor first locates point C. He
measures the distance from C to B. Then he locates point D the same distance east of C. If DE = 420 ft, what is AB? 8. 8. Given: M is the midpoint of ̶̶ PQ and ̶̶ ̶̶ PS QR ≅ Prove: ̶̶ RS. 9. Given: ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ ̶̶̶ ZW Prove: ∠W ≅ ∠Y 10. Given: G is the midpoint of ̶̶ FH. 11. Given: ̶̶̶ LM bisects ∠JLK. Prove: M is the midpoint of ̶̶ KL ̶̶ JL ≅ ̶̶ JK. ̶̶ EF ≅ ̶̶ EH Prove: ∠1 ≅ ∠2 Multi-Step Use the given set of points to prove each congruence statement. 12. R (0, 0), S (2, 4), T (-1, 3), U (-1, 0), V (-3, -4), W (-4, -1) ; ∠RST ≅ ∠UVW 13. A (-1, 1), B (2, 3), C (2, -2), D (2, -3), E (-1, -5), F (-1, 0) ; ∠BAC ≅ ∠EDF 14. Given: △QRS is adjacent to △QTS. Prove: ̶̶ QS bisects ̶̶ RT. ̶̶ QS bisects ∠RQT. ∠R ≅ ∠T 15. Given: △ABE and △CDE with E the midpoint of ̶̶ AC and ̶̶ BD Prove: ̶̶ AB ǁ ̶̶ CD 4-6 Triangle Congruence: CPCTC 263 263 ��BC500 ftADE500 ftge07sec04l06004_A�� 16. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The front of a doghouse has the dimensions shown. a. How can you prove that △ADB ≅ △ADC? ̶̶ b. Prove that CD.
c. What is the length of ̶̶ BD and ̶̶ BD ≅ ̶̶ BC to the nearest tenth? �� Multi-Step Find the value of x. 17. 18. �� Use the diagram for Exercises 19–21. 19. Given: PS = RQ, m∠1 = m∠4 Prove: m∠3 = m∠2 20. Given: m∠1 = m∠2, m∠3 = m∠4 Prove: PS = RS 21. Given: PS = RQ, PQ = RS Prove: ̶̶ PQ ǁ ̶̶ RS 22. Critical Thinking Does the diagram contain enough information to allow you to conclude that ̶̶̶ ML? Explain. ̶̶ JK ǁ 23. Write About It Draw a diagram and explain how a surveyor can set up triangles to find the distance across a lake. Label each part of your diagram. List which sides or angles must be congruent. 24. Which of these will NOT be used as a reason in a proof of ̶̶ AD? ̶̶ AC ≅ SAS ASA CPCTC Reflexive Property 25. Given the points K (1, 2), L (0, -4), M (-2, -3), and N (-1, 3), which of these is true? ∠KNL ≅ ∠MNL ∠LNK ≅ ∠NLM 26. What is the value of y? ∠MLN ≅ ∠KLN ∠MNK ≅ ∠NKL 10 20 35 85 27. Which of these are NOT used to prove angles congruent? congruent triangles noncorresponding parts parallel lines perpendicular lines 264 264 Chapter 4 Triangle Congruence �� 28. Which set of coordinates represents the vertices of a triangle congruent to △RST? (Hint: Find the lengths of the sides of △RST.) (3, 4), (3, 0), (0, 0) (3, 3), (0, 4), (0, 0) (3, 1), (3, 3), (4, 6) (3, 0), (4
, 4), (0, 6) CHALLENGE AND EXTEND 29. All of the edges of a cube are congruent. All of the angles on each face of a cube are right angles. Use CPCTC to explain why any two diagonals on the faces of a cube (for example, must be congruent. ̶̶ AC and ̶̶ AF ) 30. Given: ̶̶ JK ≅ ̶̶̶ ML, ̶̶ JM ≅ ̶̶ KL Prove: ∠J ≅ ∠L (Hint: Draw an auxiliary line.) 31. Given: R is the midpoint of S is the midpoint of ̶̶ RS ⊥ ̶̶ AB. ̶̶ DC. ̶̶ AB, ∠ASD ≅ ∠BSC Prove: △ASD ≅ △BSC 32. △ABC is in plane M. △CDE is in plane P. Both planes have C in common and ∠A ≅ ∠E. What is the height AB to the nearest foot? �� SPIRAL REVIEW 33. Lina’s test scores in her history class are 90, 84, 93, 88, and 91. What is the minimum score Lina must make on her next test to have an average test score of 90? (Previous course) 34. One long-distance phone plan costs $3.95 per month plus $0.08 per minute of use. A second long-distance plan costs $0.10 per minute for the first 50 minutes used each month and then $0.15 per minute after that. Which plan is cheaper if you use an average of 75 long-distance minutes per month? (Previous course) A figure has vertices at (1, 3), (2, 2), (3, 2), and (4, 3). Identify the transformation of the figure that produces an image with each set of vertices. (Lesson 1-7) 35. (1, -3), (2, -2), (3, -2), (4, -3) 36. (-2, -1), (-1, -2), (0, -2), (1, -1) 37. Determine if you can use ASA to prove △ACB �
� △ECD. Explain. (Lesson 4-5) 4-6 Triangle Congruence: CPCTC 265 265 �� Quadratic Equations Algebra A quadratic equation is an equation that can be written in the form a x 2 + bx + c = 0. See Skills Bank page S66 Example Given: △ABC is isosceles with ̶̶ AB ≅ ̶̶ AC. Solve for x. Step 1 Set x 2 – 5x equal to 6 to get x 2 – 5x = 6. Step 2 Rewrite the quadratic equation by subtracting 6 from each side to get x 2 – 5x – 6 = 0. Step 3 Solve for x. Method 1: Factoring Method 2: Quadratic Formula x 2 - 5x - 6 = 0 (x - 6) (x + 1) = 0 Factor. x = -b ± √  b 2 - 4ac __ 2a x - 6 = 0 or x + 1 = 0 Set each factor equal to 0. x = - (-5) ± √  (-5) 2 - 4 (1) (-6) ___ 2 (1) Substitute 1 for a, -5 for b, and -6 for c. Simplify. Find the square root. Simplify. x = 6 or x = -1 Solve. x = 5 ± √  49 _ 2 5 ± 7 _ x = 2 or x = -2 _ x = 12 _ 2 2 x = 6 or x = -1 Step 4 Check each solution in the original equation. x 2 - 5x = 6 (6 ) 2 - 5 (6 ) 36 - 30 6 6 x 2 - 5x = 6 (-1) 2 - 5 (- ✓ Try This TAKS Grades 9–11 Obj. 5, 6 Solve for x in each isosceles triangle. ̶̶ 1. Given: FG ̶̶ FE ≅ 2. Given: ̶̶ JK ≅ ̶̶ JL 3. Given: ̶̶ YX ≅ ̶̶ YZ 4. Given: ̶̶ QP ≅ ̶̶ QR 266 266 Chapter 4 Triangle Congruence ��
�� 4-7 Introduction to Coordinate Proof TEK G.2.B Geometric structure: make conjectures about... polygons … and determine validity of the conjectures. Also G.3.B, G.9.B, G.10.B Objectives Position figures in the coordinate plane for use in coordinate proofs. Prove geometric concepts by using coordinate proof. Vocabulary coordinate proof Who uses this? The Bushmen in South Africa use the Global Positioning System to transmit data about endangered animals to conservationists. (See Exercise 24.) You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane • Use the origin as a vertex, keeping the figure in Quadrant I. • Center the figure at the origin. • Center a side of the figure at the origin. • Use one or both axes as sides of the figure. E X A M P L E 1 Positioning a Figure in the Coordinate Plane Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane. Method 1 You can center the longer side of the rectangle at the origin. Method 2 You can use the origin as a vertex of the rectangle. Depending on what you are using the figure to prove, one solution may be better than the other. For example, if you need to find the midpoint of the longer side, use the first solution. 1. Position a right triangle with leg lengths of 2 and 4 units in the coordinate plane. (Hint: Use the origin as the vertex of the right angle.) 4- 7 Introduction to Coordinate Proof 267 267 �� Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure. E X A M P L E 2 Writing a Proof Using Coordinate Geometry Write a coordinate proof. Given: Right △ABC has vertices A (0, 6), B (0, 0), and
C (4, 0). D is the ̶̶ AC. midpoint of Prove: The area of △DBC is one half the area of △ABC. Proof: △ABC is a right triangle with height AB and base BC. area of △ABC = 1 __ 2 bh = 1 __ 2 (4) (6) = 12 square units, 6 + 0 ____ 2 By the Midpoint Formula, the coordinates of D = ( 0 + 4 ____ 2 of △DBC, and the base is 4 units. area of △DBC = 1 __ 2 bh ) = (2, 3). The y-coordinate of D is the height = 1 __ 2 (4) (3) = 6 square units Since 6 = 1 __ 2 (12), the area of △DBC is one half the area of △ABC. 2. Use the information in Example 2 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. E X A M P L E 3 Assigning Coordinates to Vertices Position each figure in the coordinate plane and give the coordinates of each vertex. A a right triangle with leg B a rectangle with lengths a and b length c and width d Do not use both axes when positioning a figure unless you know the figure has a right angle. 3. Position a square with side length 4p in the coordinate plane and give the coordinates of each vertex. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables. 268 268 Chapter 4 Triangle Congruence �� E X A M P L E 4 Writing a Coordinate Proof Given: ∠B is a right angle in △ABC. D is the midpoint of Prove: The area of △DBC is one half the area of △ABC. ̶̶ AC. Step 1 Assign coordinates to each vertex
. The coordinates of A are (0, 2 j), the coordinates of B are (0, 0), and the coordinates of C are (2n, 0). Since you will use the Midpoint Formula to find the coordinates of D, use multiples of 2 for the leg lengths. Step 2 Position the figure in the coordinate plane. Step 3 Write a coordinate proof. Proof: △ABC is a right triangle with height 2j and base 2n. area of △ABC = 1 __ 2 bh = 1 __ 2 (2n) (2j) = 2nj square units Because the x- and y-axes intersect at right angles, they can be used to form the sides of a right triangle. By the Midpoint Formula, the coordinates of D = ( 0 + 2n 2j + 0 _____ _____ 2, 2 The height of △DBC is j units, and the base is 2n units. area of △DBC = 1 __ 2 bh ) = (n, j). = 1 __ 2 (2n) (j) = nj square units Since nj = 1 __ 2 (2nj), the area of △DBC is one half the area of △ABC. 4. Use the information in Example 4 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. THINK AND DISCUSS 1. When writing a coordinate proof why are variables used instead of numbers as coordinates for the vertices of a figure? 2. How does the way you position a figure in the coordinate plane affect your calculations in a coordinate proof? 3. Explain why it might be useful to assign 2p as a coordinate instead of just p. 4. GET ORGANIZED Copy and complete the graphic organizer. In each row, draw an example of each strategy that might be used when positioning a figure for a coordinate proof. 4- 7 Introduction to Coordinate Proof 269 269 �� 4-7 Exercises Exercises KEYWORD: MG7 4-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What is the relationship between coordinate geometry, coordinate plane, and coordinate proof? Position each figure in the coordinate
plane. p. 267 2. a rectangle with a length of 4 units and width of 1 unit 3. a right triangle with leg lengths of 1 unit and 3 units Write a proof using coordinate geometry. p. 268 4. Given: Right △PQR has coordinates P (0, 6), Q (8, 0), and R (0, 0). A is the midpoint of B is the midpoint of ̶̶ QR. ̶̶ PR. Prove: AB = 1 __ 2 PQ. 268 Position each figure in the coordinate plane and give the coordinates of each vertex. 5. a right triangle with leg lengths m and n 6. a rectangle with length a and width. 269 Independent Practice For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 Multi-Step Assign coordinates to each vertex and write a coordinate proof. 7. Given: ∠R is a right angle in △PQR. A is the midpoint of ̶̶ PR. B is the midpoint of ̶̶ QR. Prove: AB = 1 __ 2 PQ PRACTICE AND PROBLEM SOLVING Position each figure in the coordinate plane. 8. a square with side lengths of 2 units 9. a right triangle with leg lengths of 1 unit and 5 units Write a proof using coordinate geometry. 10. Given: Rectangle ABCD has coordinates A (0, 0), B (0, 10), C (6, 10), and D (6, 0). E is the midpoint of ̶̶ AB, and F is the midpoint of ̶̶ CD. Prove: EF = BC Position each figure in the coordinate plane and give the coordinates of each vertex. 11. a square with side length 2m 12. a rectangle with dimensions x and 3x Multi-Step Assign coordinates to each vertex and write a coordinate proof. ̶̶ CD. ̶̶ AB in rectangle ABCD. F is the midpoint of 13. Given: E is the midpoint of Prove: EF = AD 14. Critical Thinking Use variables to write the general form of the endpoints of a segment whose midpoint is (0, 0). 270 270 Chapter 4 Triangle Congruence �� Conservation The origin of the springbok�
�s name may come from its habit of pronking, or bouncing. When pronking, a springbok can leap up to 13 feet in the air. Springboks can run up to 53 miles per hour. 15. Recreation A hiking trail begins at E (0, 0). Bryan hikes from the start of the trail to a waterfall at W (3, 3) and then makes a 90° turn to a campsite at C (6, 0). a. Draw Bryan’s route in the coordinate plane. b. If one grid unit represents 1 mile, what is the total distance Bryan hiked? Round to the nearest tenth. Find the perimeter and area of each figure. 16. a right triangle with leg lengths of a and 2a units 17. a rectangle with dimensions s and t units Find the missing coordinates for each figure. 18. 19. 20. Conservation The Bushmen have sighted animals at the following coordinates: (-25, 31.5), (-23.2, 31.4), and (-24, 31.1). Prove that the distance between two of these locations is approximately twice the distance between two other. 21. Navigation Two ships depart from a port at P (20, 10). The first ship travels to a location at A (-30, 50), and the second ship travels to a location at B (70, -30). Each unit represents one nautical mile. Find the distance to the nearest nautical mile between the two ships. Verify that the port is at the midpoint between the two. Write a coordinate proof. 22. Given: Rectangle PQRS has coordinates P (0, 2), Q (3, 2), R (3, 0), and S (0, 0). ̶̶ PR and ̶̶ QS intersect at T (1.5, 1). Prove: The area of △RST is 1 __ 4 of the area of the rectangle _____ _____ 2, 2 23. Given ), with midpoint M ( Prove: AM = 1 __ 2 AB 24. Plot the points on a coordinate plane and connect them to form △KLM and △MPK. Write a coordinate proof. Given: K (-2, 1), L (-2, 3), M (1, 3), P (1, 1) Prove: △KLM ≅ △MPK 25. Write
About It When you place two sides of a figure on the coordinate axes, what are you assuming about the figure? 26. This problem will prepare you for the Multi-Step TAKS Prep on page 280. Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. a. Find BD and CE. b. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed A at the origin and and E, assuming that each unit of the coordinate plane represents one inch. ̶̶ AB on the y-axis. Find the coordinates of B, C, D, 4- 7 Introduction to Coordinate Proof 271 271 �� 27. The coordinates of the vertices of a right triangle are (0, 0), (4, 0), and (0, 2). Which is a true statement? The vertex of the right angle is at (4, 2). The midpoints of the two legs are at (2, 0) and (0, 1). The hypotenuse of the triangle is √  6 units. The shortest side of the triangle is positioned on the x-axis. 28. A rectangle has dimensions of 2g and 2f units. If one vertex is at the origin, which coordinates could NOT represent another vertex? (2g, 2f) (2f, 0) (2f, g) (-2f, 2g) 29. The coordinates of the vertices of a rectangle are (0, 0), (a, 0), (a, b), and (0, b). What is the perimeter of the rectangle? a + b ab 1 _ 2 ab 2a + 2b 30. A coordinate grid is placed over a map. City A is located at (-1, 2) and city C is located at (3, 5). If city C is at the midpoint between city A and city B, what are the coordinates of city B? (1, 3.5) (-5, -1) (7, 8) (2, 7) CHALLENGE AND EXTEND Find the missing coordinates for each figure. 31. 32. 33. The vertices of a right triangle are at (-2s, 2s), (0, 2s),
and (0, 0). What coordinates could be used so that a coordinate proof would be easier to complete? 34. Rectangle ABCD has dimensions of 2f and 2g units. The equation of the line containing ̶̶ g __ BD is y = x, and f ̶̶ g __ x + 2g. AC is y = - the equation of the line containing f Use algebra to show that the coordinates of E are (f, g). SPIRAL REVIEW Use the quadratic formula to solve for x. Round to the nearest hundredth if necessary. (Previous course) 35. 0 = 8 x 2 + 18x - 5 36. 0 = x 2 + 3x - 5 37. 0 = 3 x 2 - x - 10 Find each value. (Lesson 3-2) 38. x 39. y 40. Use A (-4, 3), B (-1, 3), C (-3, 1), D (0, -2), E (3, -2), and F (2, -4) to prove ∠ABC ≅ ∠EDF. (Lesson 4-6). 272 272 Chapter 4 Triangle Congruence �� 4-8 Isosceles and Equilateral Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons … and determine the validity of the conjectures.... Also G.3.C, G.10.B Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Vocabulary legs of an isosceles triangle vertex angle base base angles Who uses this? Astronomers use geometric methods. (See Example 1.) Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. ∠3 is the vertex angle. ∠1 and ∠2 are the base angles. Theorems Isosceles Triangle THEOREM HYPOTHESIS CONCLUSION 4-8-1 Isosceles Triangle Theorem If
two sides of a triangle are congruent, then the angles opposite the sides are congruent. 4-8-2 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. ∠B ≅ ∠C ̶̶ DE ≅ ̶̶ DF Theorem 4-8-1 is proven below. You will prove Theorem 4-8-2 in Exercise 35. PROOF PROOF ̶̶ AB ≅ Isosceles Triangle Theorem ̶̶ AC Given: Prove: ∠B ≅ ∠C Proof: Statements Reasons The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.” 1. Draw X, the mdpt. of 2. Draw the auxiliary line ̶̶ BC. ̶̶ AX. ̶̶ BX ≅ ̶̶ AB ≅ ̶̶ AX ≅ ̶̶ CX ̶̶ AC ̶̶ AX 3. 4. 5. 6. △ABX ≅ △ACX 7. ∠B ≅ ∠C 1. Every seg. has a unique mdpt. 2. Through two pts. there is exactly one line. 3. Def. of mdpt. 4. Given 5. Reflex. Prop. of ≅ 6. SSS Steps 3, 4, 5 7. CPCTC 4-8 Isosceles and Equilateral Triangles 273 273 �� E X A M P L E 1 Astronomy Application The distance from Earth to nearby stars can be measured using the parallax method, which requires observing the positions of a star 6 months apart. If the distance LM to a star in July is 4.0 × 10 13 km, explain why the distance LK to the star in January is the same. (Assume the distance from Earth to the Sun does not change.) Not drawn to scale m∠LKM = 180 - 90.4, so m∠LKM = 89.6°. Since ∠LKM ≅ ∠M, △LMK is isosceles by the Converse of the Isosceles Triangle Theorem. Thus LK = LM = 4.0 × 10 13 km. 1. If the distance from Earth to a star in
September is 4.2 × 10 13 km, what is the distance from Earth to the star in March? Explain. E X A M P L E 2 Finding the Measure of an Angle Find each angle measure. A m∠C m∠C = m∠B = x° m∠C + m∠B + m∠A = 180 x + x + 38 = 180 2x = 142 x = 71 Thus m∠C = 71°. B m∠S Isosc. △ Thm. △ Sum Thm. Substitute the given values. Simplify and subtract 38 from both sides. Divide both sides by 2. m∠S = m∠R 2x° = (x + 30) ° Isosc. △ Thm. Substitute the given values. x = 30 Subtract x from both sides. Thus m∠S = 2x° = 2 (30) = 60°. Find each angle measure. 2b. m∠N 2a. m∠H The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Corollary 4-8-3 Equilateral Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equilateral, then it is equiangular. (equilateral △ → equiangular △) ∠A ≅ ∠B ≅ ∠C You will prove Corollary 4-8-3 in Exercise 36. 274 274 Chapter 4 Triangle Congruence �� Corollary 4-8-4 Equiangular Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equiangular, then it is equilateral. (equiangular △ → equilateral △) ̶̶ DE ≅ ̶̶ DF ≅ ̶̶ EF E X A M P L E 3 Using Properties of Equilateral Triangles You will prove Corollary 4-8-4 in Exercise 37. Find each value. A x △ABC is equiangular. (3x + 15) ° = 60° 3x = 45 x = 15 B t △JKL is equilateral. 4t - 8 = 2t + 1 2t = 9 Equilateral △ →
equiangular △ The measure of each ∠ of an equiangular △ is 60°. Subtract 15 from both sides. Divide both sides by 3. Equiangular △ → equilateral △ Def. of equilateral △ Subtract 2t and add 8 to both sides. t = 4.5 Divide both sides by 2. 3. Use the diagram to find JL. E X A M P L E 4 Using Coordinate Proof A coordinate proof may be easier if you place one side of the triangle along the x-axis and locate a vertex at the origin or on the y-axis. Prove that the triangle whose vertices are the midpoints of the sides of an isosceles triangle is also isosceles. Given: △ABC is isosceles. X is the mdpt. of ̶̶ AC. Z is the mdpt. of ̶̶ AB. ̶̶ BC. Y is the mdpt. of Prove: △XYZ is isosceles. Proof: Draw a diagram and place the coordinates of △ABC and △XYZ as shown. By the Midpoint Formula, the coordinates of X are ( 2a + 0 ) = (a, b), _____ 2 the coordinates of Y are ( 2a + 4a ) = (3a, b), and the coordinates of Z ______ 2 are ( 4a + 0 _____ 2 By the Distance Formula, XZ = √ YZ = √  ̶̶ XZ ≅ Since XZ = YZ,  2 2 = √  + (0 - b) (2a - a) (2a - 3a) 2 + (0 - b) 2 = √  a 2 + b 2. ̶̶ YZ by definition. So △XYZ is isosceles. ) = (2a, 0). a 2 + b 2, and, 2b + 0 _____ 2, 2b + 0 _____ 2, 0 + 0 ____ 2 4. What if...? The coordinates of △
ABC are A (0, 2b), B (-2a, 0), and C (2a, 0). Prove △XYZ is isosceles. 4- 8 Isosceles and Equilateral Triangles 275 275 �� THINK AND DISCUSS 1. Explain why each of the angles in an equilateral triangle measures 60°. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, draw and mark a diagram for each type of triangle. 4-8 Exercises Exercises KEYWORD: MG7 4-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw isosceles △JKL with ∠K as the vertex angle. Name the legs, base, and base angles of the triangle. 274 2. Surveying To find the distance QR across a river, a surveyor locates three points Q, R, and S. QS = 41 m, and m∠S = 35°. The measure of exterior ∠PQS = 70°. Draw a diagram and explain how you can find QR Find each angle measure. p. 274 3. m∠ECD 4. m∠K 5. m∠X 6. m∠ Find each value. p. 275 7. y 8. x 9. BC 10. JK. 275 11. Given: △ABC is right isosceles. X is the midpoint of ̶̶ AC. ̶̶ AB ≅ ̶̶ BC Prove: △AXB is isosceles. 276 276 Chapter 4 Triangle Congruence �� Independent Practice For See Exercises Example 12 13–16 17–20 21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 12. Aviation A plane is flying parallel � �  AC. When the to the ground along plane is at A, an air-traffic controller in tower T measures the angle to the plane as 40°. After the plane has traveled 2.4 mi
to B, the angle to the plane is 80°. How can you find BT? � �� Find each angle measure. 13. m∠E 14. m∠TRU 15. m∠F �� 16. m∠A Find each value. 17. z 18. y 19. BC 20. XZ 21. Given: △ABC is isosceles. P is the midpoint ̶̶ AB. Q is the midpoint of ̶̶ AC. of ̶̶ AB ≅ ̶̶ PC ≅ ̶̶ AC ̶̶ QB Prove: Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 22. An equilateral triangle is an isosceles triangle. 23. The vertex angle of an isosceles triangle is congruent to the base angles. 24. An isosceles triangle is a right triangle. 25. An equilateral triangle and an obtuse triangle are congruent. 26. Critical Thinking Can a base angle of an isosceles triangle be an obtuse angle? Why or why not? 4- 8 Isosceles and Equilateral Triangles 277 277 �� 27. This problem will prepare you for the Multi-Step TAKS � Prep on page 280. The diagram shows the inside view of the support structure of the back of a doghouse. ̶̶ PS ≅ a. Find m∠SPT. b. Find m∠PQR and m∠PRQ. ̶̶ PT, m∠PST = 71°, and m∠QPS = m∠RPT = 18°. ̶̶ PQ ≅ ̶̶ PR, � � � � Multi-Step Find the measure of each numbered angle. 28. 29. �� 30. Write a coordinate proof. Given: ∠B is a right angle in isosceles right △ABC. X is the midpoint of ̶̶ AC. ̶̶ BA ≅ ̶̶ BC Prove: △AXB ≅ △CXB 31. Estimation Draw the
figure formed by (-2, 1), (5, 5), and (-1, -7). Estimate the measure of each angle and make a conjecture about the classification of the figure. Then use a protractor to measure each angle. Was your conjecture correct? Why or why not? 32. How many different isosceles triangles have a perimeter of 18 and sides whose lengths are natural numbers? Explain. Multi-Step Find the value of the variable in each diagram. 33. 34. 35. Prove the Converse of the Isosceles Triangle Theorem. Navigation 36. Complete the proof of Corollary 4-8-3. ̶̶ AC ≅ ̶̶ ̶̶ BC AB ≅ Given: Prove: ∠A ≅ ∠B ≅ ∠C ̶̶ ̶̶ AC, a. AB ≅? by the Isosceles Triangle Theorem. ̶̶̶̶ ̶̶ BC, ∠A ≅ ∠B by b. Proof: Since ̶̶ AC ≅ Since By the Transitive Property of ≅, ∠A ≅ ∠B ≅ ∠C.?. Therefore ∠A ≅ ∠C by c. ̶̶̶̶?. ̶̶̶̶ The taffrail log is dragged from the stern of a vessel to measure the speed or distance traveled during a voyage. The log consists of a rotator, recording device, and governor. 37. Prove Corollary 4-8-4. 38. Navigation The captain of a ship traveling along  AB sights an island C at an angle of 45°. The captain measures the distance the ship covers until it reaches B, where the angle to the island is 90°. Explain how to find the distance BC to the island. 39. Given: △ABC ≅ △CBA Prove: △ABC is isosceles. 40. Write About It Write the Isosceles Triangle Theorem and its converse as a biconditional. 278 278 Chapter 4 Triangle Congruence �� 41. Rewrite the paragraph proof of the Hypotenuse-Leg (HL) Congruence Theorem as a two-column proof. Given: △ABC and △DEF are right triangles. ∠C and ∠F are
right angles. ̶̶ ̶̶ DF, and AC ≅ Prove: △ABC ≅ △DEF ̶̶ AB ≅ ̶̶ DE.  EF. Mark G so that FG = CB. Thus Proof: On △DEF draw ̶̶ DF and ∠C and ∠F are right angles. ̶̶ AC ≅ lines. Thus ∠DFG is a right angle, and ∠DFG ≅ ∠C. △ABC ≅ △DGF by SAS. ̶̶̶ ̶̶ DG ≅ DE by the Transitive Property. By the Isosceles Triangle Theorem ∠G ≅ ∠E. ∠DFG ≅ ∠DFE since right angles are congruent. So △DGF ≅ △DEF by AAS. Therefore △ABC ≅ △DEF by the Transitive Property. ̶̶ AB by CPCTC. ̶̶ DE as given. ̶̶̶ DG ≅ ̶̶ DF ⊥ ̶̶ AB ≅ ̶̶ FG ≅ ̶̶ CB. From the diagram, ̶̶ EG by definition of perpendicular 42. Lorena is designing a window so that ∠R, ∠S, ∠T, and ̶̶ VU ≅ ̶̶ VT, and m∠UVT = 20°. ∠U are right angles, What is m∠RUV? 10° 70° 20° 80° 43. Which of these values of y makes △ABC isosceles 15 1 _ 2 44. Gridded Response The vertex angle of an isosceles triangle measures (6t - 9) °, and one of the base angles measures (4t) °. Find t. CHALLENGE AND EXTEND 45. In the figure, ̶̶ JK ≅ Prove m∠MKL must also be x°. ̶̶ JL, and ̶̶̶ KM ≅ ̶̶ KL. Let m∠J = x°. 46. An equilateral △ABC is placed on a coordinate plane. Each side length measures 2a. B is at the origin, and C is at (2a, 0). Find the coordinates of A. 47. An isosceles triangle has coordinates A (0,
0) and B (a, b). What are all possible coordinates of the third vertex? SPIRAL REVIEW Find the solutions for each equation. (Previous course) 48. x 2 + 5x + 4 = 0 49. x 2 - 4x + 3 = 0 50. x 2 - 2x + 1 = 0 Find the slope of the line that passes through each pair of points. (Lesson 3-5) 51. (2, -1) and (0, 5) 52. (-5, -10) and (20, -10) 53. (4, 7) and (10, 11) 54. Position a square with a perimeter of 4s in the coordinate plane and give the coordinates of each vertex. (Lesson 4-7) 4- 8 Isosceles and Equilateral Triangles 279 279 �� SECTION 4B Proving Triangles Congruent Gone to the Dogs You are planning to build a doghouse for your dog. The pitched roof of the doghouse will be supported by four trusses. Each truss will be an isosceles triangle with the dimensions shown. To determine the materials you need to purchase and how you will construct the trusses, you must first plan carefully. 1. You want to be sure that all four trusses are exactly the same size and shape. Explain how you could measure three lengths on each truss to ensure this. Which postulate or theorem are you using? 2. Prove that the two triangular halves of the truss are congruent. 3. What can you say about ̶̶ DB? Why is this true? and Use this to help you find the ̶̶ AC, and lengths of ̶̶ DB, ̶̶ AD, ̶̶ BC. ̶̶ AD 4. You want to make careful plans on a coordinate plane before you begin your construction of the trusses. Each unit of the coordinate plane represents 1 inch. How could you assign coordinates to vertices A, B, and C? 5. m∠ACB = 106°. What is the measure of each of the acute angles in the truss? Explain how you found your answer. 6. You can buy the wood for the trusses at the building supply store for $0.80 a foot. The store sells the wood in 6-foot lengths only. How much will you have to spend to get
enough wood for the 4 trusses of the doghouse? (Hint: You need to use the Pythagorean Theorem to find the two unknown side lengths of each truss.) 280 280 Chapter 4 Triangle Congruence �� Quiz for Lessons 4-4 Through 4-8 4-4 Triangle Congruence: SSS and SAS SECTION 4B 1. The figure shows one tower and the cables of a suspension bridge. ̶̶ AC ≅ ̶̶ BC, use SAS to explain why △ACD ≅ △BCD. Given that 2. Given: ̶̶ JK bisects ∠MJN. ̶̶ MJ ≅ ̶̶ NJ Prove: △MJK ≅ △NJK 4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 3. △RSU and △TUS 4. △ABC and △DCB Observers in two lighthouses K and L spot a ship S. 5. Draw a diagram of the triangle formed by the lighthouses and the ship. Label each measure. K to L K to S L to S Bearing E N 58° E N 77° W 6. Is there enough data in the table to pinpoint Distance 12 km?? the location of the ship? Why? 4-6 Triangle Congruence: CPCTC 7. Given: ̶̶ CD ǁ ̶̶ BE, Prove: ∠D ≅ ∠B ̶̶ DE ǁ ̶̶ CB 4-7 Introduction to Coordinate Proof 8. Position a square with side lengths of 9 units in the coordinate plane 9. Assign coordinates to each vertex and write a coordinate proof. ̶̶ Given: ABCD is a rectangle with M as the midpoint of AB. N is the midpoint of Prove: The area of △AMN is 1 __ 8 the area of rectangle ABCD. ̶̶ AD. 4-8 Isosceles and Equilateral Triangles Find each value. 10. m∠C 11. ST 12. Given: Isosceles △JKL has coordinates J (0, 0), K (2a, 2b), and L (4a, 0)
. M is the midpoint of ̶̶ JK, and N is the midpoint of ̶̶ KL. Prove: △KMN is isosceles. Ready to Go On? 281 281 �� EXTENSION EXTENSION Proving Constructions Valid TEK G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.3.B Objective Use congruent triangles to prove constructions valid. When performing a compass and straight edge construction, the compass setting remains the same width until you change it. This fact allows you to construct a segment congruent to a given segment. You can assume that two distances constructed with the same compass setting are congruent. The steps in the construction of a figure can be justified by combining the assumptions of compass and straightedge constructions and the postulates and theorems that are used for proving triangles congruent. You have learned that there exists exactly one midpoint on any line segment. The proof below justifies the construction of a midpoint. E X A M P L E 1 Proving the Construction of a Midpoint Given: diagram showing the steps in the construction Prove: M is the midpoint of ̶̶ AB. To construct a midpoint, see the construction of a perpendicular bisector on p. 172. Proof: Statements Reasons 1. Draw ̶̶ AC, ̶̶ BC, ̶̶̶ AD, and ̶̶ BD. ̶̶ AC ≅ ̶̶ CD ≅ ̶̶ BC ≅ ̶̶ CD 2. 3. ̶̶̶ AD ≅ ̶̶ BD 4. △ACD ≅ △BCD 5. ∠ACD ≅ ∠BCD ̶̶̶ CM ≅ ̶̶̶ CM 6. 7. △ ACM ≅ △BCM ̶̶̶ AM ≅ ̶̶̶ BM 8. 9. M is the midpt. of ̶̶ AB. 1. Through any two pts. there is exactly one line. 2. Same compass setting used 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of ≅ 7. SAS Steps 2, 5, 6 8. CPCTC 9. Def. of mdpt. 1. Given:
above diagram Prove:   CD is the perpendicular bisector of ̶̶ AB. 282 282 Chapter 4 Triangle Congruence �� E X A M P L E 2 Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: ∠A ≅ ∠D To review the construction of an angle congruent to another angle, see page 22. Proof: Since there is a straight line through any two points, you can draw ̶̶ EF. The same compass setting was used to construct ̶̶ AB, ̶̶ DE. The same compass setting was used ̶̶ DF ≅ ̶̶ AC, ̶̶ DF, ̶̶ BC ≅ ̶̶ EF. Therefore △BAC ≅ △EDF by SSS, ̶̶ BC and ̶̶ DE, so and to construct and ∠A ≅ ∠D by CPCTC. ̶̶ AC ≅ ̶̶ BC and ̶̶ AB ≅ ̶̶ EF, so 2. Prove the construction for bisecting an angle. (See page 23.) EXTENSION Exercises Exercises Use each diagram to prove the construction valid. 1. parallel lines 2. a perpendicular through a point not (See page 163 and page 170.) on the line (See page 179.) 3. constructing a triangle using SAS 4. constructing a triangle using ASA (See page 243.) (See page 253.) Extension 283 283 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute triangle.............. 216 CPCTC..................... 260 isosceles triangle........... 217 auxiliary line............... 223 equiangular triangle........ 216 legs of an isosceles triangle.. 273 base....................... 273 equilateral triangle......... 217 obtuse triangle............. 216 base angle...
............... 273 exterior.................... 225 remote interior angle....... 225 congruent polygons......... 231 exterior angle.............. 225 right triangle............... 216 coordinate proof............ 267 included angle.............. 242 scalene triangle............. 217 corollary................... 224 included side............... 252 triangle rigidity............. 242 corresponding angles....... 231 interior.................... 225 vertex angle................ 273 corresponding sides......... 231 interior angle............... 225 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a triangle with at least two congruent sides. ̶̶̶̶ 2. A name given to matching angles of congruent triangles is?. ̶̶̶̶ 3. A(n)? is the common side of two consecutive angles in a polygon. ̶̶̶̶ 4-1 Classifying Triangles (pp. 216–221) TEKS G.1.A E X A M P L E EXERCISES ■ Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. 4-2 Angle Relationships in Triangles (pp. 223–230) TEKS G.1.A, G.2.B E X A M P L E ■ Find m∠S. EXERCISES Find m∠N. 6. 12x = 3x + 42 + 6x 12x = 9x + 42 3x =
42 x = 14 m∠S = 6 (14) = 84° 284 284 Chapter 4 Triangle Congruence 7. In△LMN, m∠L = 8x °, m∠M = (2x + 1) °, and m∠N = (6x - 1) °. �� 4-3 Congruent Triangles (pp. 231–237) TEKS G.2.B, G.10.B E X A M P L E EXERCISES ■ Given: △DEF ≅ △JKL. Identify all pairs of Given: △PQR ≅ △XYZ. Identify the congruent congruent corresponding parts. Then find the value of x. ̶̶ DE ≅ ∠F ≅ ∠L, The congruent pairs follow: ∠D ≅ ∠J, ∠E ≅ ∠K, ̶̶ EF ≅ Since m∠E = m∠K, 90 = 8x - 22. After 22 is added to both sides, 112 = 8x. So x = 14. ̶̶ KL, and ̶̶ DF ≅ ̶̶ JK, ̶̶ JL. corresponding parts. 8. ̶̶ PR ≅? ̶̶̶̶ 9. ∠Y ≅? ̶̶̶ Given: △ABC ≅ △CDA Find each value. 10. x 11. CD 4-4 Triangle Congruence: SSS and SAS (pp. 242–249) TEKS G.2.A, G.3.B, G.3.E, G.10.B ■ Given: E X A M P L E S ̶̶ ̶̶ UT, and RS ≅ ̶̶ ̶̶ VS ≅ VT. V is the midpoint of ̶̶ RU. Prove: △RSV ≅ △UTV Proof: Statements Reasons EXERCISES 12. Given: ̶̶ AB ≅ ̶̶ DB ≅ Prove: △ADB ≅ △DAE ̶̶ DE, ̶̶ AE 13. Given: ̶̶ GJ bisects ̶̶ FH bisects and Prove: △FGK ≅ △HJK ̶̶
FH, ̶̶ GJ. ̶̶ RS ≅ ̶̶ VS ≅ ̶̶ UT ̶̶ VT 1. 2. 3. V is the mdpt. of ̶̶ UV ̶̶ RV ≅ 4. ̶̶ RU. 1. Given 2. Given 3. Given 4. Def. of mdpt. 14. Show that △ABC ≅ △XYZ when x = -6. 5. △RSV ≅ △UTV 5. SSS Steps 1, 2, 4 ■ Show that △ADB ≅ △CDB when s = 5. AB = s 2 - 4s AD = 14 - 2s 15. Show that △LMN ≅ △PQR when y = 25. = 5 2 - 4 (5 ) = 5 ̶̶ BD by the Reflexive Property. = 14 - 2 (5 ) = 4 ̶̶ CB. So △ADB ≅ △CDB by SSS. ̶̶ BD ≅ ̶̶ AB ≅ and ̶̶ AD ≅ ̶̶ CD Study Guide: Review 285 285 �� 4-5 Triangle Congruence: ASA, AAS, and HL (pp. 252–259) TEKS G.1.A, G.1.B, G.2.A, E X A M P L E S ■ Given: B is the midpoint of ̶̶ AE. ∠A ≅ ∠E, ∠ABC ≅ ∠EBD Prove: △ABC ≅ △EBD EXERCISES 16. Given: C is the midpoint ̶̶ AG. of ̶̶ HA ǁ ̶̶ GB Prove: △HAC ≅ △BGC G.3.B, G.3.C, G.3.E, G.9.B, G.10.B Proof: Statements Reasons 1. ∠A ≅ ∠E 2. ∠ABC ≅ ∠EBD 3. B is the mdpt. of ̶̶ EB ̶̶ AB ≅ 4. ̶̶ AE. 1. Given 2. Given 3. Given 4. Def

�� MN ⊥   PQ for M (2, 1), N (-3, 0), P (x, 4), and Q (3, y). Find a set of possible values for x and y. SPIRAL REVIEW Find the x- and y-intercepts of the line that contains each pair of points. (Previous course) 34. (-5, 0) and (0, -5) 35. (0, 1) and (2, -7) 36. (1, -3) and (3, 3) Use the given paragraph proof to write a two-column proof. (Lesson 2-7) 37. Given: ∠1 is supplementary to ∠3. Prove: ∠2 ≅ ∠3 Proof: It is given that ∠1 is supplementary to ∠3. ∠1 and ∠2 are a linear pair by the definition of a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 are supplementary. Thus ∠2 ≅ ∠3 by the Congruent Supplements Theorem. Given that m∠2 = 75°, tell whether each statement is true or false. Justify your answer with a postulate or theorem. (Lesson 3-2) 38. ∠1 ≅ ∠8 39. ∠2 ≅ ∠6 40. ∠3 ≅ ∠5 3- 5 Slopes of Lines 187 187 �� 3-6 Use with Lesson 3-6 Explore Parallel and Perpendicular Lines A graphing calculator can help you explore graphs of parallel and perpendicular lines. To graph a line on a calculator, you can enter the equation of the line in slope-intercept form. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. For example, the line y = 2x + 3 has a slope of 2 and crosses the y-axis at (0, 3). TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines,.... KEYWORD: MG7 Lab3 Activity 1 1 On a graphing calculator, graph the lines y = 3

x – 4, y = –3x – 4, and y = 3x + 1. Which lines appear to be parallel? What do you notice about the slopes of the parallel lines? 2 Graph y = 2x. Experiment with other equations to find a line that appears parallel to y = 2x. If necessary, graph y = 2x on graph paper and construct a parallel line. What is the slope of this new line? 3 Graph y = - 1__ 2x + 3. Try to graph a line that appears parallel to y = - 1 __ 2 x + 3. What is the slope of this new line? Try This 1. Create two new equations of lines that you think will be parallel. Graph these to confirm your conjecture. 2. Graph two lines that you think are parallel. Change the window settings on the calculator. Do the lines still appear parallel? Describe your results. 3. Try changing the y-intercepts of one of the parallel lines. Does this change whether the lines appear to be parallel? 188 188 Chapter 3 Parallel and Perpendicular Lines On a graphing calculator, perpendicular lines may not appear to be perpendicular on the screen. This is because the unit distances on the x-axis and y-axis can have different lengths. To make sure that the lines appear perpendicular on the screen, use a square window, which shows the x-axis and y-axis as having equal unit distances. One way to get a square window is to use the Zoom feature. On the Zoom menu, the ZDecimal and ZSquare commands change the window to a square window. The ZStandard command does not produce a square window. Activity 2 1 Graph the lines y = x and y = -x in a square window. Do the lines appear to be perpendicular? 2 Graph y = 3x - 2 in a square window. Experiment with other equations to find a line that appears perpendicular to y = 3x - 2. If necessary, graph y = 3x - 2 on graph paper and construct a perpendicular line. What is the slope of this new line? 3 Graph y = 2 __ 3 x in a square window. Try to graph a line that appears perpendicular to y = 2 __ 3 x. What is the slope of this new line? Try This 4. Create two new equations of lines that you think will be perpendicular. Graph these in a square window to confirm your conjecture. 5. Graph two lines that you think are perpendicular. Change the window settings on the calculator.

Do the lines still appear perpendicular? Describe your results. 6. Try changing the y-intercepts of one of the perpendicular lines. Does this change whether the lines appear to be perpendicular? 3- 6 Technology Lab 189 189 3-6 Lines in the Coordinate Plane TEKS G.7.B Dimensionality and the geometry of location: use... equations of lines to investigate... parallel lines, perpendicular lines,.... Also G.3.C, G.3.E, G.7.A, G.7.C Objectives Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding. Vocabulary point-slope form slope-intercept form Why learn this? The cost of some health club plans includes a one-time enrollment fee and a monthly fee. You can use the equations of lines to determine which plan is best for you. (See Example 4.) The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line © Forms of the Equation of a Line FORM EXAMPLE The point-slope form of a line is y - y 1 = m (x - x 1 ), where m is the slope and ( x 1, y 1 ) is a given point on the line. y - 3 = 2 (x - 4 ) m = 23, 4 ) The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. y = 3x + 6 m = 3, b = 6 The equation of a vertical line is x = a, where a is the x-intercept. The equation of a horizontal line is y = b, where b is the y-intercept. x = 5 y = 2 You will prove the slope-intercept form of a line in Exercise 48. PROOF PROOF Point-Slope Form of a Line Given: The slope of a line through points ( x 1, y 1 ) and ( x 2, y 2 ) is m = Prove: The equation of the line through ( x 1, y 1 ) with slope m is x - x 1 ). Proof: Let (x, y) be any point on

the linex - x 1 ) m = (x - x 1 ) m (x - x 1 ) = ( Slope formula Substitute (x, y) for ( x 2, y 2 ). Multiply both sides by (x - x 1 ). Simplify. y - y 1 = m (x - x 1 ) Sym. Prop. of = 190 190 Chapter 3 Parallel and Perpendicular Lines E X A M P L E 1 Writing Equations of Lines Write the equation of each line in the given form. A the line with slope 3 through (2, 1) in point-slope form y - y 1 = m (x - 2) Point-slope form Substitute 3 for m, 2 for x 1, and 1 for y 1. B the line through (0, 4) and (-1, 2) in slope-intercept form = -2 _ -1 - 0 y = mx + b 4 = 2 (0) + b 4 = b y = 2x + 4 Find the slope. Slope-intercept form Substitute 2 for m, 0 for x, and 4 for y to find b. Simplify. Write in slope-intercept form using m = 2 and b = 4. A line with y-intercept b contains the point (0, b). A line with x-intercept a contains the point (a, 0). C the line with x-intercept 2 and y-intercept 3 in point-slope form x - 2) 2 y = - 3 _ (x - 2) 2 Use the points (2, 0) and (0, 3) to find the slope. Point-slope form 3 _ for m, 2 for x 1, and 0 for y 1. Substitute - 2 Simplify. Write the equation of each line in the given form. 1a. the line with slope 0 through (4, 6) in slope-intercept form 1b. the line through (-3, 2) and (1, 2) in point-slope form E X A M P L E 2 Graphing Lines Graph each line The equation is given in slope-intercept form, with a slope of 3 __ 2 and a y-intercept of 3. Plot the point (0, 3) and then rise 3 and run 2 to find another point. Draw the line containing the two points. B y + 3 = -2 (

x - 1) The equation is given in point-slope form, with a slope of -2 = -2 ___ 1 through the point (1, -3). Plot the point (1, -3) and then rise -2 and run 1 to find another point. Draw the line containing the two points. 3- 6 Lines in the Coordinate Plane 191 191 �� Graph the line. C x = 3 The equation is given in the form for a vertical line with an x-intercept of 3. The equation tells you that the x-coordinate of every point on the line is 3. Draw the vertical line through (3, 0). Graph each line. 2a. y = 2x - 3 2b. y - 1 = - 2 _ (x + 2) 3 2c. y = -4 A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. Pairs of Lines Parallel Lines Intersecting Lines Coinciding Lines y = 5x + 8 y = 5x - 4 Same slope different y-intercept y = 2x - 5 y = 4x + 3 Different slopes y = 2x - 4 y = 2x - 4 Same slope same y-intercept E X A M P L E 3 Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. A y = 2x + 3, y = 2x - 1 Both lines have a slope of 2, and the y-intercepts are different. So the lines are parallel. B y = 3x - 5, 6x - 2y = 10 Solve the second equation for y to find the slope-intercept form. 6x - 2y = 10 -2y = -6x + 10 y = 3x - 5 Both lines have a slope of 3 and a y-intercept of -5, so they coincide. C 3x + 2y = 7, 3y = 4x + 7 Solve both equations for y to find the slope-intercept form. 3x + 2y = 7 3y = 4x + The slope is 4 _. 3 2y = -3x +. The slope is - 2 The lines have different slopes, so they intersect. 3. Deter

mine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide. 192 192 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Problem-Solving Application Audrey is trying to decide between two health club plans. After how many months would both plans’ total costs be the same? Plan A Plan B Enrollment Fee $140 $60 Monthly Fee $35 $55 Understand the Problem The answer is the number of months after which the costs of the two plans would be the same. Plan A costs $140 for enrollment and $35 per month. Plan B costs $60 for enrollment and $55 per month. Make a Plan Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations. Solve Plan A: y = 35x + 140 Plan B: y = 55x + 60 0 = -20x + 80 Subtract the second x = 4 y = 35 (4) + 140 = 280 equation from the first. Solve for x. Substitute 4 for x in the first equation. The lines cross at (4, 280). Both plans cost $280 after 4 months. Look Back Check your answer for each plan in the original problem. For 4 months, plan A costs $140 plus $35 (4) = $140 + $140 = $280. Plan B costs $60 + $55 (4) = $60 + $220 = $280, so the plans cost the same. Use the information above to answer the following. 4. What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan? THINK AND DISCUSS 1. Explain how to use the slopes and y-intercepts to determine if two lines are parallel. 2. Describe the relationship between the slopes of perpendicular lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3- 6 Lines in the Coordinate Plane 193 193 1234�� 3-6 Exercises Exercises KEYWORD: MG7 3-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1

. Vocabulary How can you recognize the slope-intercept form of an equation Write the equation of each line in the given form. p. 191 2. the line through (4, 7) and (-2, 1) in slope-intercept form 3. the line through (-4, 2) with slope 3 _ 4 4. the line with x-intercept 4 and y-intercept -2 in slope-intercept form in point-slope form Graph each line. p. 191. 192 5. y = -3x + 4 6. y + 4 = 2 _ (x - 6) 3 Determine whether the lines are parallel, intersect, or coincide. 7. x = 5 8. y = -3x + 4, y = -3x + 1 x + 2 _ 10. y = 1 _ 3 3, 3y = x + 2 9. 6x - 12y = -24, 3y = 2x + 18 11. 4x + 2y = 10, y = -2x + 15. 193 12. Transportation A speeding ticket in Conroe costs $115 for the first 10 mi/h over the speed limit and $1 for each additional mi/h. In Lakeville, a ticket costs $50 for the first 10 mi/h over the speed limit and $10 for each additional mi/h. If the speed limit is 55 mi/h, at what speed will the tickets cost approximately the same? Homework Help See For Exercises Example 13–15 16–18 19–22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Write the equation of each line in the given form. 13. the line through (0, -2) and (4, 6) in point-slope form 14. the line through (5, 2) and (-2, 2) in slope-intercept form 15. the line through (6, -4) with slope 2 _ 3 in point-slope form Graph each line. 16. y - 7 = x + 4 17. y = 1 _ 2 x - 2 18. y = 2 Determine whether the lines are parallel, intersect, or coincide. 19. y = x - 7, y = -x + 3 21. x + 2y = 6 20. y = 5 _ 2 x + 4,

2y = 5x - 4 22. 7x + 2y = 10, 3y = 4x - 5 23. Business Chris is comparing two sales positions that he has been offered. The first pays a weekly salary of $375 plus a 20% commission. The second pays a weekly salary of $325 plus a 25% commission. How much must he make in sales per week for the two jobs to pay the same? Write the equation of each line in slope-intercept form. Then graph the line. 24. through (-6, 2) and (3, 6) 26. through (5, -2) with slope 2 _ 3 27. x-intercept 4, y-intercept -3 25. horizontal line through (2, 3) Write the equation of each line in point-slope form. Then graph the line. 28. slope - 1 _ 2 30. through (5, -1) with slope -1 29. slope 3 _ 4 31. through (4, 6) and (-2, -5), x-intercept -2, y-intercept 2 194 194 Chapter 3 Parallel and Perpendicular Lines 32. ////ERROR ANALYSIS///// Write the equation of the line with slope -2 through the point (-4, 3) in slope-intercept form. Which equation is incorrect? Explain. Determine whether the lines are perpendicular. 33. y = 3x - 5, y = -3x + 1 x + 5, y = 3 _ 35, 34. y = -x + 1, y = x + 2 36. y = -2x + 4 Multi-Step Given the equation of the line and point P not on the line, find the equation of a line parallel to the given line and a line perpendicular to the given line through the given point. 37. y = 3x + 7, P (2, 3) 38. y = -2x - 5, P (-1, 4) 39. 4x + 3y = 8, P (4, -2) 40. 2x - 5y = 7, P (-2, 4) Food Multi-Step Use slope to determine if each triangle is a right triangle. If so, which angle is the right angle? 41. A (-5, 3), B (0, -2), C (5, 3) 42. D (1, 0), E (2, 7), F (

5, 1) 43. G (3, 4), H (-3, 4), J (1, -2) 44. K (-2, 4), L (2, 1), M (1, 8) In 2004, the world’s largest pizza was baked in Italy. The diameter of the pizza was 5.19 m (about 17 ft) and it weighed 124 kg (about 273 lb). 45. Food A restaurant charges $8 for a large cheese pizza plus $1.50 per topping. Another restaurant charges $11 for a large cheese pizza plus $0.75 per topping. How many toppings does a pizza have that costs the same at both restaurants? 46. Estimation Estimate the solution of the system of equations represented by the lines in the graph. Write the equation of the perpendicular bisector of the segment with the given endpoints. 47. (2, 5) and (4, 9) 48. (1, 1) and (3, 1) 49. (1, 3) and (-1, 4) 50. (-3, 2) and (-3, -10) 51. Line ℓ has equation y = - 1 __ 2 x + 4, and point P has coordinates (3, 5). a. Find the equation of line m that passes through P and is perpendicular to ℓ. b. Find the coordinates of the intersection of ℓ and m. c. What is the distance from P to ℓ? 52. Line p has equation y = x + 3, and line q has equation y = x - 1. a. Find the equation of a line r that is perpendicular to p and q. b. Find the coordinates of the intersection of p and r and the coordinates of the intersection of q and r. c. Find the distance between lines p and q. 3- 6 Lines in the Coordinate Plane 195 195 �� 53. This problem will prepare you for the Multi-Step TAKS Prep on page 200. For a car moving at 60 mi/h, the equation d = 88t gives the distance in feet d that the car travels in t seconds. a. Graph the line d = 88t. b

. On the same graph you made for part a, graph the line d = 300. What does the intersection of the two lines represent? c. Use the graph to estimate the number of seconds it takes the car to travel 300 ft. 54. Prove the slope-intercept form of a line, given the point-slope form. Given: The equation of the line through ( x 1, y 1 ) with slope m is y - y 1 = m (x - x 1 ). Prove: The equation of the line through (0, b) with slope m is y = mx + b. Plan: Substitute (0, b) for ( x 1, y 1 ) in the equation y - y 1 = m (x - x 1 ) and simplify. 55. Data Collection Use a graphing calculator and a motion detector to do the following: Walk in front of the motion detector at a constant speed, and write the equation of the resulting graph. 56. Critical Thinking A line contains the points (-4, 6) and (2, 2). Write a convincing argument that the line crosses the x-axis at (5, 0). Include a graph to verify your argument. 57. Write About It Determine whether the lines are parallel. Use slope to explain your answer. 58. Which graph best represents a solution to this system of equations? ⎧ -3x + y = 7 ⎨ 2x + y = -3 ⎩ 196 196 Chapter 3 Parallel and Perpendicular Lines �� 59. Which line is parallel to the line with the equation y = -2x + 5?   AB through A (2, 3) and B (1, 1 4x + 2y = 10 x + 1 _ 2 y = 1 60. Which equation best describes the graph shown = 3x - � � � � � � � � 61. Which line includes the points (-4, 2) and (6, -3)? y = 2x - 4 y = 2x CHALLENGE AND EXTEND 62. A right triangle is formed by the x-axis, the y-axis, and the line y = -2x + 5. Find the length of the hypotenuse. 63. If the length of the hypotenuse of a right triangle is 17 units and the legs lie along the x-axis and y-axis

, find a possible equation that describes the line that contains the hypotenuse. 64. Find the equations of three lines that form a triangle with a hypotenuse of 13 units. 65. Multi-Step Are the points (-2, -4), (5, -2) and (2, -3) collinear? Explain the method you used to determine your answer. 66. For the line y = x + 1 and the point P (3, 2), let d represent the distance from P to a point (x, y) on the line. a. Write an expression for d 2 in terms of x and y. Substitute the expression x + 1 for y and simplify. b. How could you use this expression to find the shortest distance from P to the line? Compare your result to the distance along a perpendicular line. SPIRAL REVIEW 67. The cost of renting DVDs from an online company is $5.00 per month plus $2.50 for each DVD rented. Write an equation for the total cost c of renting d DVDs from the company in one month. Graph the equation. How many DVDs did Sean rent from the company if his total bill for one month was $20.00? (Previous course) Use the coordinate plane for Exercises 68–70. Find the coordinates of the midpoint of each segment. (Lesson 1-6) ̶̶ AB ̶̶ BC ̶̶ AC 70. 68. 69. Use the slope formula to find the slope of each segment. (Lesson 3-5) ̶̶ AB ̶̶ BC ̶̶ AC 73. 71. 72. � � � � �� 3- 6 Lines in the Coordinate Plane 197 197 Scatter Plots and Lines of Best Fit Data Analysis Recall that a line has an infinite number of points on it. You can compute the slope of a line if you can identify two distinct points on the line. See Skills Bank page S79 Example 1 The table shows several possible measures of an angle and its supplement. Graph the points in the table. Then draw the line that best represents the data and write the equation of the line. Step 1 Use the table to write ordered pairs (x, 180 - x) and then plot the points. (30, 150), (60, 120), (90, 90), (120, 60), (150, 30) Step 2 Draw a line that

passes through all the points. x 30 60 90 120 150 y = 180 - x 150 120 90 60 30 Step 3 Choose two points from the line, such as (30, 150) and (120, 60). m = Use them to find the slope = 60 - 150 _ 120 - 30 = -90 _ 90 = -1 Slope formula Substitute (30, 150) for ( x 1, y 1 ) and (120, 60) for ( x 2, y 2 ). Simplify. Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) Point-slope form y - 150 = -1 (x - 30) Substitute (30, 150) for ( x 1, y 1 ) and -1 for m. y = -x + 180 Simplify. 198 198 Chapter 3 Parallel and Perpendicular Lines �� If you can draw a line through all the points in a set of data, the relationship is linear. If the points are close to a line, you can approximate the relationship with a line of best fit. Example 2 A physical therapist evaluates a client’s progress by measuring the angle of motion of an injured joint. The table shows the angle of motion of a client’s wrist over six weeks. Estimate the equation of the line of best fit. Step 1 Use the table to write ordered pairs and then plot the points. (1, 30), (2, 36), (3, 46), (4, 48), (5, 54), (6, 62) Step 2 Use a ruler to estimate a line of best fit. Try to get the edge of the ruler closest to all the points on the line. Week Angle Measure 1 2 3 4 5 6 30 36 46 48 54 62 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Step 3 A line passing through (2, 36) and (

6, 62) seems to be closest to all the points. Draw this line. Use the points (2, 36) and (6, 62) to find the slope of the line = 62 - 36 _ _ 6 - 2 = 6.5 Substitute (2, 36) for ( x 1, y 1 ) and (6, 62) for ( x 2, y 2 ). Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) y - 36 = 6.5 (x - 2) Point-slope form Substitute (2, 36) for ( x 1, y 1 ) and 6.5 for m. y = 6.5x + 23 Simplify. Try This TAKS Grades 9–11 Obj. 2, 3, 9 Estimate the equation of the line of best fit for each relationship. 1. 2. the relationship between an angle and its complement 3. Data Collection Use a graphing calculator and a motion detector to do the following: Set the equipment so that the graph shows distance on the y-axis and time on the x-axis. Walk in front of the motion dector while varying your speed slightly and use the resulting graph. On Track for TAKS 199 199 �� SECTION 3B Coordinate Geometry Red Light, Green Light When a driver approaches an intersection and sees a yellow traffic light, she must decide if she can make it through the intersection before the light turns red. Traffic engineers use graphs and equations to study this situation. 1. Traffic engineers can set the duration of the yellow lights on Lincoln Road for any length of time t up to 10 seconds. For each value of t, there is a critical distance d. If a car moving at the speed limit is more than d feet from the light when it turns yellow, the driver will have to stop. If the car is less than d feet from the light, the driver can continue through the intersection. The graph shows the relationship between t and d. Find the speed limit on Lincoln Road in miles per hour. (Hint : 22 ft/s = 15 mi/h) 2. Traffic engineers use the equation d = 22 __ 15 st to determine the critical distance for various durations of a yellow light. In the equation, s is the speed limit. The speed limit on Porter Street is 45 mi/h. Write the equation

of the critical distance for a yellow light on Porter Street and then graph the line. Does this line intersect the line for Lincoln Road? If so, where? Is the line for Porter Street steeper or flatter than the line for Lincoln Road? Explain how you know. 200 200 Chapter 3 Parallel and Perpendicular Lines �� SECTION 3B Quiz for Lesson 3-5 Through 3-6 3-5 Slopes of Lines Use the slope formula to determine the slope of each line. 1.  AC 2.   CD 3.  AB 4.   BD Find the slope of the line through the given points. 6. F (-1, 4) and G (5, -1) 8. K (4, 2) and L (-3, 2) 5. M (2, 3) and N (0, 7) 7. P (4, 0) and Q (1, -3) 9. Sonia is walking 4 miles home from school. She leaves at 4:00 P.M., and gets home at 4:45 P.M. Graph the line that represents Sonia’s distance from school at a given time. Find and interpret the slope of the line. Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither. 10.  EF and   GH for E (-2, 3), F (6, 1), G (6, 4), and H (2, 5) JK and   LM for J (4, 3), K (5, -1), L (-2, 4), and M (3, -5) 11.  12.  NP and   QR for N (5, -3), P (0, 4), Q (-3, -2), and R (4, 3) 13

.  ST and   VW for S (0, 3), T (0, 7), V (2, 3), and W (5, 3) 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 14. the line through (3, 8) and (-3, 4) in slope-intercept form 15. the line through (-5, 4) with slope 2 _ 3 16. the line with y-intercept 2 through the point (4, 1) in slope-intercept form in point-slope form Graph each line. 17. y = -2x + 5 18. y + 3 = 1 _ (x - 4) 4 19. x = 3 Write the equation of each line. 20. 21. 22. Determine whether the lines are parallel, intersect, or coincide. 23. y = -2x + 5 y = -2x - 5 24. 3x + 2y = 25. y = 4x -5 3x + 4y = 7 Ready to Go On? 201 201 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary alternate exterior angles..... 147 parallel planes.............. 146 same-side interior angles.... 147 alternate interior angles..... 147 perpendicular bisector...... 172 skew lines.................. 146 corresponding angles....... 147 perpendicular lines......... 146 slope.........................182 distance from a point-slope form........... 190 slope-intercept form........ 190 point to a line............ 172 parallel lines............... 146 rise........................ 182 transversal.....

............ 147 run........................ 182 Complete the sentences below with vocabulary words from the list above. 1. Angles on opposite sides of a transversal and between the lines it intersects are?. ̶̶̶̶ 2. Lines that are in different planes are?. ̶̶̶̶ 3. A(n) 4. The? is a line that intersects two coplanar lines at two points. ̶̶̶̶? is used to write the equation of a line with a given slope that passes ̶̶̶̶ through a given point. 5. The slope of a line is the ratio of the? to the ̶̶̶̶?. ̶̶̶̶ 3-1 Lines and Angles (pp. 146–151) E X A M P L E S EXERCISES Identify each of the following. Identify each of the following. ■ a pair of parallel segments ̶̶ AB ǁ ̶̶ CD ■ a pair of parallel planes plane ABC ǁ plane EFG ■ a pair of perpendicular segments ̶̶ AB ⊥ ̶̶ AE ■ a pair of skew segments ̶̶ AB and ̶̶ FG are skew. 6. a pair of skew segments 7. a pair of parallel segments 8. a pair of perpendicular segments 9. a pair of parallel planes 202 202 Chapter 3 Parallel and Perpendicular Lines �� Identify the transversal and classify each angle pair. ■ ∠4 and ∠6 p, corresponding angles ■ ∠1 and ∠2 q, alternate interior angles ■ ∠3 and ∠4 p, alternate exterior angles ■ ∠6 and ∠7 r, same-side interior angles Identify the transversal and classify each angle pair. 10. ∠5 and ∠2 11. ∠6 and ∠3 12. ∠2 and ∠4 13. ∠1 and ∠2 3-2 Angles Formed by Parallel Lines and Transversals (pp. 155–161) TEKS G.2.A, E X A M P L E S Find each angle measure. ■ m∠TUV EXERCISES Find each angle measure. 14. m∠WYZ G.3.C, G.3

.E, G.9.A By the Same-Side Interior Angles Theorem, (6x + 10) + (4x + 20) = 180. 15. m∠KLM x = 15 Solve for x. Substitute the value for x into the expression for m∠TUV. m∠TUV = 4 (15) + 20 = 80° ■ m∠ABC 16. m∠DEF 17. m∠QRS By the Corresponding Angles Postulate, 8x + 28 = 10x + 4. x = 12 Solve for x. Substitute the value for x into the expression for one of the obtuse angles. 10 (12) + 4 = 124° ∠ABC is supplementary to the 124° angle, so m∠ABC = 180 - 124 = 56°. Study Guide Review 203 203 �� 3-3 Proving Lines Parallel (pp. 162–169) TEKS G.1.A, G.3.C, G.3.E, G.9.A EXERCISES Use the given information and theorems and postulates you have learned to show that c ǁ d. 18. m∠4 = 58°, m∠6 = 58° 19. m∠1 = (23x + 38) °, m∠5 = (17x + 56) °, x = 3 20. m∠6 = (12x + 6) °, m∠3 = (21x + 9) °, x = 5 21. m∠1 = 99°, m∠7 = (13x + 8) °, Use the given information and theorems and postulates you have learned to show that p ǁ q. ■ m∠2 + m∠3 = 180° ∠2 and ∠3 are supplementary, so p ǁ q by the Converse of the Same-Side Interior Angles Theorem. ■ ∠8 ≅ ∠6 ∠8 ≅ ∠6, so p ǁ q by the Converse of the Corresponding Angles Postulate. ■ m∠1 = (7x - 3) °, m∠5 = 5x +

15, x = 9 m∠1 = 60°, and m∠5 = 60°. So ∠1 ≅ ∠5. p ǁ q by the Converse of the Alternate Exterior Angles Theorem. 3-4 Perpendicular Lines (pp. 172–178) TEKS G.1.A, G.2.A, G.3.C, G.3.E, G.9.A EXERCISES 22. Name the shortest segment from point K to ̶̶ LN. 23. Write and solve an inequality for x. 24. Given: Prove: ̶̶ AD ǁ ̶̶ AB ǁ ̶̶ BC, ̶̶ CD ̶̶ AD ⊥ ̶̶ AB, ̶̶ DC ⊥ ̶̶ BC E X A M P L E S ■ Name the shortest segment from ̶̶ WY. point X to ̶̶ XZ ■ Write and solve an inequality for x. x + 3 > 3 x > 0 Subtract 3 from both sides. ■ Given: m ⊥ p, ∠1 and ∠2 are complementary. Prove: p ǁ q Proof: It is given that m ⊥ p. ∠1 and ∠2 are complementary, so m∠1 + m∠2 = 90°. Thus m ⊥ q. Two lines perpendicular to the same line are parallel, so p ǁ q. 204 204 Chapter 3 Parallel and Perpendicular Lines �� 3-5 Slopes of Lines (pp. 182–187) TEKS G.7.A, G.7.B, G.7.C E X A M P L E S ■ Use the slope formula to determine the slope of the line. EXERCISES Use the slope formula to determine the slope of each line. 25. 26. slope of   WX = - (-3) _ 2 - (-4) = 6 _ 6 = 1 slope of   AB = ■ Use slopes to determine whether    AB and    CD are parallel, perpendicular, or neither for A (-1, 5), B (-3, 4), C

(3, -1), and D (4, -33 - (-1) -3 - (-1) = -2 _ _ 4 - 3 1 The slopes are opposite reciprocals, so the slope of   CD = = -2 Use slopes to determine if the lines are parallel, perpendicular, or neither. 27. EF and GH for E (8, 2), F (-3, 4), G (6, 1), and H (-4, 3) 28. JK and LM for J (4, 3), K (-4, -2), L (5, 6), and M (-3, 1) 29. ST and UV for S (-4, 5), T (2, 3), U (3, 1), and V (4, 4) lines are perpendicular. 3-6 Lines in the Coordinate Plane (pp. 190–197) TEKS G.3.C, G.3.E, G.7.A, G.7.B, G.7.C E X A M P L E S EXERCISES ■ Write the equation of the line through (5, -2) with slope 3 __ in slope-intercept form. 5 y - (-2) = 3 _ (x - 5 Solve for y. Simplify. x - 3 x - 5 Point-slope form ■ Determine whether the lines y = 4x + 6 and 8x - 2y = 4 are parallel, intersect, or coincide. Solve the second equation for y to find the slope-intercept form. 8x - 2y = 4 y = 4x - 2 Both the lines have a slope of 4 and have different y-intercepts, so they are parallel. Write the equation of each line in the given form. 30. the line through (6, 1) and (-3, 5) in slope-intercept form 31. the line through (-3, -4) with slope 2 _ 3 in slope-intercept form 32. the line with x-intercept 1 and y-intercept -2 in point-slope form Determine whether

the lines are parallel, intersect, or coincide. 33. -3x + 2y = 5, 6x - 4y = 8 34. y = 4x - 3, 5x + 2y = 1 35. y = 2x + 1, 2x - y = -1 Study Guide Review 205 205 �� Identify each of the following. 1. a pair of parallel planes 2. a pair of parallel segments 3. a pair of skew segments Find each angle measure. � � � � � � � � �� 4. 5. 6. Use the given information and the theorems and postulates you have learned to show f ǁ g. 7. m∠4 = (16x + 20) °, m∠5 = (12x + 32) °, x = 3 8. m∠3 = (18x + 6) °, m∠5 = (21x + 18) °, x = 4 Write a two-column proof. 9. Given: ∠1 ≅ ∠2, n ⊥ ℓ Prove: n ⊥ m Use the slope formula to determine the slope of each line. 10. 11. 12. 13. Greg is on a 32-mile bicycle trail from Elroy, Wisconsin, to Sparta, Wisconsin. He leaves Elroy at 9:30 A.M. and arrives in Sparta at 2:00 P.M. Graph the line that represents Greg’s distance from Elroy at a given time. Find and interpret the slope of the line. 14. Graph   QR and   ST for Q (3, 3), R (6, -5), S (-4, 6), and T (-1, -2). Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15. Write the equation of the line through (-2, -5) with slope - 3 _ 4 16. Determine whether the lines 6x + y = 3 and 2x + 3y = 1 are parallel, intersect, in point-slope form. or coincide. 206 206 Chapter 3 Parallel and Perpendicular Lines �� FOC

US ON ACT When you take the ACT Mathematics Test, you receive a separate subscore for each of the following areas: • Pre-Algebra/Elementary Algebra, • Intermediate Algebra/Coordinate Geometry, and • Plane Geometry/Trigonometry. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. 1. Which of the following is an equation of the line that passes through the point (2, -3) and is parallel to the line 4x - 5y = 1? (A) -4x + 5y = -23 (B) -5x - 4y = 2 (C) -2x - 5y = 11 (D) -4x - 5y = 7 (E) -5x + 4y = -22 2. In the figure below, line t crosses parallel lines ℓ and m. Which of the following statements are true? I. ∠1 and ∠6 are alternate interior angles. II. ∠2 ≅ ∠4 III. ∠2 ≅ ∠8 (F) I only (G) II only (H) III only (J) I and II only (K) II and III only Find out what percent of questions are from each area and concentrate on content that represents the greatest percent of questions. 3. In the standard (x, y) coordinate plane, the line that passes through (1, -7) and (-8, 5) is perpendicular to the line that passes through (3, 6) and (-1, b). What is the value of b? (A) 2 (B) 3 (C) 7 (D) 9 (E) 10 4. Lines m and n are cut by a transversal so that ∠2 and ∠5 are corresponding angles. If m∠2 = (x + 18) ° and m∠5 = (2x - 28) °, which value of x makes lines m and n parallel? (F) 3 1 _ 3 (G) 33 1 _ 3 (H) 46 (J) 63 1 _ 3 (K) 72 5. What is the distance between point G (4, 2) and the line through the points E (1, -2) and F (7, -2)? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 College Entrance

Exam Practice 207 207 �� Any Question Type: Interpret Coordinate Graphs When test items refer to a coordinate plane, it is important to interpret the coordinate graphs correctly. It is also important to understand the relationship between an equation and its graph. Multiple Choice Which statement best describes the graph of the following equations? y = 2x + 6 2y = 4x + 6 The lines are parallel. The lines are perpendicular. The lines coincide. The lines have the same y-intercept. It may help to graph the lines or visualize the graph in order to answer the question. The lines appear to be parallel. Write the equations of both lines in slope-intercept form. y = 2x + 6 y = 2x + 3 The slope is 2 and the y-intercept is 6. The slope is 2 and the y-intercept is 3. The lines have the same slope and different y-intercepts, so they are parallel. The answer is A. Gridded Response What is the rate of change of the following graph? Remember that the rate of change of the graph of a line is its slope. Choose two points on the line and use their coordinates to calculate the slope of the line. Use the points (5, 0) and (0, -2). m = - The slope is 2 _ 5 = 0.4. Enter 0.4 in the answer grid. 208 208 Chapter 3 Parallel and Perpendicular Lines �� A quick look at the graph of a line can tell you whether the slope is positive or negative. This may help you eliminate answer choices. Item C Multiple Choice Which equation describes the line through the point (4, 2) that is perpendicular to the line 3x - y = 7? Read each test item and answer the questions that follow. Item A Multiple Choice The line segment on the graph shows the altitude of a hot air balloon during a landing. Which statement best describes the slope of the line segment? The balloon descends about 5 feet per 8 seconds. The balloon descends about 8 feet per 5 seconds. The balloon descends about 1 foot per 2 seconds. The balloon descends about 2 feet per second. 1. What are the coordinates of two points on the graph? 2. How does the scale of the graph affect the appearance of the slope? 3. How is the slope of the line related to the rate of descent? Item B Gridded Response What is the

y-intercept of the line through (5, 2) that is parallel to the line x - 4y = 8? y = 3x - 10 y = -3x + 14. Graph the line represented by 3x - y = 7. What is its slope? 8. Is the slope of a line perpendicular to the line represented by 3x - y = 7 positive or negative? Based on your answer, can you eliminate any answer choices? 9. What is the slope of a line perpendicular to the line represented by 3x - y = 7? Item D Multiple Choice Which graph best represents a solution to the following system of equations? -2x + y = 6 3x + y = -2 10. What is the slope of the line represented by -2x + y = 6? Based on your answer, can you eliminate any answer choices? 4. Graph the line x - 4y = 8. What is its slope? 11. What is the slope of the line represented 5. Write an equation in point-slope form for the line through (5, 2) that is parallel to x - 4y = 8. 6. How would you use the equation in point- slope form to find the y-intercept of the line? by 3x + y = -2? Based on your answer, can you eliminate any answer choices? TAKS Tackler 209 209 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–3 Multiple Choice Use the diagram below for Items 1 and 2. 1. What type of angle pair are ∠JKM and ∠KMN? Corresponding angles Alternate exterior angles Same-side interior angles Alternate interior angles 2. What is m∠KML? 57° 80° 102° 125° 3. What is a possible value of x in the diagram? 2 3 4 5 4. A graphic artist used a computer illustration program to draw a line connecting points with coordinates (3, -1) and (4, 6). She needs to draw a second line parallel to the first line. What slope should the second line have. Which term describes a pair of vertical angles that are also supplementary? Acute Obtuse Right Straight 6. What is the equation of the line that passes through the points (-1, 8) and (4, -2)? y =

-2x + = 2x + 10 7. Given the points R (-5, 3), S (-5, 4), T (-3, 4), and U (-3, 1), which line is perpendicular to   TU?   RS   RT   ST   SU 8. Which of following is true if   XY and   UV are skew?   XY and   UV are coplanar. X, Y, and U are noncollinear.   XY ǁ   UV   XY ⊥   UV �� Make sure that you answer the question that is asked. Some problems require more than one step. You must perform all of the steps to get the correct answer. 9. Point C is the midpoint of B (7, 2). What is the length of nearest tenth. ̶̶ AB for A (1, -2) and ̶̶ AC? Round to the 3.0 3.6 5.0 7.2 Use the diagram below for Items 10 and 11. 10.  AD bisects ∠CAE, and  AE bisects ∠CAF. If m∠DAF = 120°, what is m∠DAE? 40° 60° 80° 100° 210 210 Chapter 3 Parallel and Perpendicular Lines A F 11. What is the intersection of  AD?  AF and ̶̶ FD ∠DAF �� 12. Which statement is true by the Transitive Property of Equality? If x + 3 = y, then y = x + 3. If k = 6, then 2k = 12. If a = b and b = 8, then a = 8. If m = n, then m + 7

= n + 7. 13. Which condition guarantees that r || s? STANDARDIZED TEST PREP Short Response 21. Given ℓ ǁ m with transversal t, explain why ∠1 and ∠8 are supplementary. ∠1 ≅ ∠2 ∠2 ≅ ∠7 ∠2 ≅ ∠3 ∠1 ≅ ∠4 22. Read the following conditional statement. If two angles are vertical angles, then they are congruent. a. Write the converse of this conditional statement. b. Give a counterexample to show that the 14. What is the converse of the following statement? converse is false. If x = 2, then x + 3 = 5. If x ≠ 2, then x + 3 = 5. If x = 2, then x + 3 ≠ 5. If x + 3 ≠ 5, then x ≠ 2. If x + 3 = 5, then x = 2. Gridded Response 15. Two lines a and b are cut by a transversal so that ∠1 and ∠2 are same-side interior angles. If m∠1 = (2x + 30) ° and m∠2 = (4x - 75) °, what value of x proves that a ǁ b? 23. Assume that the following statements are true when the bases are loaded in a baseball game. If a batter hits the ball over the fence, then the batter hits a home run. A batter hits a home run if and only if the result is four runs scored. a. If a batter hits the ball over the fence when the bases are loaded, can you conclude that four runs were scored? Explain your answer. b. If a batter hits a home run when the bases are loaded, can you conclude that the batter hit the ball over the fence? Explain your answer. 16. What is the slope of the line that passes through (3, 7) and (-5, 1)? Extended Response 17. ∠1 and ∠2 form a linear pair. m∠1 = (4x + 18)° and m∠2 = (3x - 6)°. What is the value of x? 18. Points A, B, and C are collinear, and B is between A and C. AB = 16 and AC = 27. What is the distance BC? 19. Ms

. Nelson wants to put a chain-link fence around 3 sides of a square-shaped lawn. Chain-link fencing is sold in sections that are each 6 feet wide. If Ms. Nelson’s lawn has an area of 3600 square feet, how many sections of fencing will she need? 20. What is the next number in this pattern? 67, 76, 83, 88,… 24. A car passes through a tollbooth at 8:00 A.M. and begins traveling east at an average speed of 45 miles per hour. A second car passes through the same tollbooth an hour later and begins traveling east at an average speed of 60 miles per hour. a. Write an equation for each car that relates the number of hours x since 8:00 A.M. to the distance in miles y the car has traveled. Explain what the slope of each equation represents. b. Graph the system of equations on the coordinate plane. c. If neither car stops, at what time will the second car catch up to the first car? Explain how you determined your answer. Cumulative Assessment, Chapters 1–3 211 211 �� Triangle Congruence 4A Triangles and Congruence 4-1 Classifying Triangles Lab Develop the Triangle Sum Theorem 4-2 Angle Relationships in Triangles 4-3 Congruent Triangles 4B Proving Triangle Congruence Lab Explore SSS and SAS Triangle Congruence 4-4 Triangle Congruence: SSS and SAS Lab Predict Other Triangle Congruence Relationships 4-5 Triangle Congruence: ASA, AAS, and HL 4-6 Triangle Congruence: CPCTC 4-7 Introduction to Coordinate Proof 4-8 Isosceles and Equilateral Triangles Ext Proving Constructions Valid KEYWORD: MG7 ChProj The Bob Bullock Texas State History Museum opened in Austin in April 2001. 212 212 Chapter 4 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. a statement that is accepted as true without proof 2. congruent segments B. an angle that measures greater than 90° and less than 180° 3. obtuse angle C. a statement that you can prove 4. postulate 5. triangle D. segments that have the same length E. a three-sided polygon F. an angle that measures greater than 0° and less than 90° Measure Angles Use a protractor to

measure each angle. 6. 7. Use a protractor to draw an angle with each of the following measures. 8. 20° 10. 105° 9. 63° 11. 158° Solve Equations with Fractions x + 7 = 25 Solve. 12. 9_ 2 14. x - 1_ 5 = 12_ 5 13. 3x - 2_ 3 = 4_ 3 15. 2y = 5y - 21_ 2 Connect Words and Algebra Write an equation for each statement. 16. Tanya’s age t is three times Martin’s age m. 17. Twice the length of a segment x is 9 ft. 18. The sum of 53° and twice an angle measure y is 90°. 19. The price of a radio r is $25 less than the price of a CD player p. 20. Half the amount of liquid j in a jar is 5 oz more than the amount of liquid b in a bowl. Triangle Congruence 213 213 Key Vocabulary/Vocabulario acute triangle triángulo acutángulo congruent polygons polígonos congruentes corollary corolario equilateral triangle triángulo equilátero exterior angle ángulo externo interior angle ángulo interno isosceles triangle triángulo isósceles obtuse triangle triángulo obtusángulo right triangle triángulo rectángulo scalene triangle triángulo escaleno Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The Latin word acutus means “pointed” or “sharp.” Draw a triangle that looks pointed or sharp. Do you think this is an acute triangle? 2. Consider the everyday meaning of the word exterior. Where do you think an exterior angle of a triangle is located? 3. You already know the definition of an obtuse angle. Use this meaning to make a conjecture about an obtuse triangle. Geometry TEKS G.1.A Geometric structure* develop an awareness of the structure of a mathematical system … G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures … G.2.B Geometric structure* make

conjectures about angles, lines, polygons … and determine validity of the conjectures, choosing from a variety of approaches … 4-2 Geo. Lab Les. 4-1 ★ Les. 4-2 Les. 4-3 ★ 4-4 Geo. Lab 4-5 Tech. Lab Les. 4-4 Les. 4-5 Les. 4-6 Les. 4-7 Les. 4-8 Ext.5.A Geometric patterns* use … geometric patterns ★ to develop algebraic expressions representing geometric properties G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures ★ G.9.B Congruence and the geometry of ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties and attributes of polygons … based on explorations … G.10.B Congruence and the geometry of size* justify ★ ★ ★ ★ ★ ★ ★ ★ and apply triangle congruence relationships * Knowledge and skills are written out completely on pages TX28–TX35. 214 214 Chapter 4 Reading Strategy: Read Geometry Symbols In Geometry we often use symbols to communicate information. When studying each lesson, read both the symbols and the words slowly and carefully. Reading aloud can sometimes help you translate symbols into words. � �� Throughout this course, you will use these symbols and combinations of these symbols to represent various geometric statements. Symbol Combinations Translated into Words   ST ǁ   UV ̶̶̶ ̶̶ GH BC ⊥ p → q Line ST is parallel to line UV. Segment BC is perpendicular to segment GH. If p, then q. m∠QRS = 45° The measure of angle QRS is 45 degrees. ∠CDE ≅ ∠LMN Angle CDE is congruent to angle LMN. Try This Rewrite each statement using symbols. 1. the absolute value of 2 times pi 2. The measure of angle 2 is 125 degrees.

3. Segment XY is perpendicular to line BC. 4. If not p, then not q. Translate the symbols into words. 5. m∠FGH = m∠VWX 6. ZA ǁ TU 7. ∼p → q 8. ST bisects ∠TSU. Triangle Congruence 215 215 4-1 Classifying Triangles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning... Objectives Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Who uses this? Manufacturers use properties of triangles to calculate the amount of material needed to make triangular objects. (See Example 4.) A triangle is a steel percussion instrument in the shape of an equilateral triangle. Different-sized triangles produce different musical notes when struck with a metal rod. Recall that a triangle (△) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. ̶̶ BC, and ̶̶ AB, A, B, and C are the triangle’s vertices. ̶̶ AC are the sides of △ABC. Triangle Classification By Angle Measures Acute Triangle Equiangular Triangle Right Triangle Obtuse Triangle Three acute angles Three congruent acute angles One right angle One obtuse angle E X A M P L E 1 Classifying Triangles by Angle Measures Classify each triangle by its angle measures. A △EHG ∠EHG is a right angle. So △EHG is a right triangle. B △EFH ∠EFH and ∠HFG form a linear pair, so they are supplementary. Therefore m∠EFH + m∠HFG = 180°. By substitution, m∠EFH + 60° = 180°. So m∠EFH = 120°. △EFH is an obtuse triangle by definition. 1. Use the diagram to classify △FHG by its angle measures. 216 216 Chapter 4 Triangle Congruence �� Triangle Classification By Side Length

s Equilateral Triangle Isosceles Triangle Scalene Triangle Three congruent sides At least two congruent sides No congruent sides E X A M P L E 2 Classifying Triangles by Side Lengths Classify each triangle by its side lengths. When you look at a figure, you cannot assume segments are congruent based on their appearance. They must be marked as congruent. A △ABC From the figure, and △ABC is equilateral. ̶̶ AB ≅ ̶̶ AC. So AC = 15, B △ABD By the Segment Addition Postulate, BD = BC + CD = 15 + 5 = 20. Since no sides are congruent, △ABD is scalene. 2. Use the diagram to classify △ACD by its side lengths. E X A M P L E 3 Using Triangle Classification Find the side lengths of the triangle. Step 1 Find the value of x. ̶̶ ̶̶ KL JK ≅ JK = KL (4x - 1.3) = (x + 3.2) 3x = 4.5 x = 1.5 Given Def. of ≅ segs. Substitute (4x - 13) for JK and (x + 3.2) for KL. Add 1.3 and subtract x from both sides. Divide both sides by 3. Step 2 Substitute 1.5 into the expressions to find the side lengths. JK = 4x - 1.3 = 4 (1.5) - 1.3 = 4.7 KL = x + 3.2 = (1.5) + 3.2 = 4.7 JL = 5x - 0.2 = 5 (1.5) - 0.2 = 7.3 3. Find the side lengths of equilateral △FGH. 4-1 Classifying Triangles 217 217 �� E X A M P L E 4 Music Application �� A manufacturer produces musical triangles by bending pieces of steel into the shape of an equilateral triangle. The triangles are available in side lengths of 4 inches, 7 inches, and 10 inches. How many 4-inch triangles can the manufacturer produce from a 100 inch piece of steel? The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3 (4) =

12 in. To find the number of triangles that can be made from 100 inches. of steel, divide 100 by the amount of steel needed for one triangle. 100 ÷ 12 = 8 1 _ 3 triangles There is not enough steel to complete a ninth triangle. So the manufacturer can make 8 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many triangles can be formed from a 100 in. piece of steel. 4a. 7 in. 4b. 10 in. THINK AND DISCUSS 1. For △DEF, name the three pairs of consecutive sides and the vertex formed by each. 2. Sketch an example of an obtuse isosceles triangle, or explain why it is not possible to do so. 3. Is every acute triangle equiangular? Explain and support your answer with a sketch. 4. Use the Pythagorean Theorem to explain why you cannot draw an equilateral right triangle. 5. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe each type of triangle. 218 218 Chapter 4 Triangle Congruence �� 4-1 Exercises Exercises KEYWORD: MG7 4-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, JK, KL, and JL are equal. How does this help you classify △JKL by its side lengths? 2. △XYZ is an obtuse triangle. What can you say about the types of angles in △XYZ Classify each triangle by its angle measures. p. 216 3. △DBC 4. △ABD 5. △ADC Classify each triangle by its side lengths. p. 217 6. △EGH 7. △EFH 8. △HFG Multi-Step Find the side lengths of each triangle. p. 217 9. 10. 218 11. Crafts A jeweler creates triangular earrings by bending pieces of silver wire. Each earring is an isosceles triangle with the dimensions shown. How many earrings can be made from a piece of wire that is 50 cm long? Independent Practice For See Exercises Example 12–14 15–17 18–20 21–22 1 2 3 4 TEKS TEKS T

AKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Classify each triangle by its angle measures. 12. △BEA 13. △DBC 14. △ABC Classify each triangle by its side lengths. 15. △PST 16. △RSP 17. △RPT Multi-Step Find the side lengths of each triangle. 18. 19. �� 20. Draw a triangle large enough to measure. Label the vertices X, Y, and Z. a. Name the three sides and three angles of the triangle. b. Use a ruler and protractor to classify the triangle by its side lengths and angle measures. 4-1 Classifying Triangles 219 219 �� Carpentry Use the following information for Exercises 21 and 22. A manufacturer makes trusses, or triangular supports, for the roofs of houses. Each truss is the shape of an isosceles triangle in which base ̶̶ ̶̶ PQ ≅ PR. The length of the ̶̶ QR is 4 __ 3 the length of each of the congruent sides. 21. The perimeter of each truss is 60 ft. Find each side length. 22. How many trusses can the manufacturer make from 150 feet of lumber? Draw an example of each type of triangle or explain why it is not possible. 23. isosceles right 24. equiangular obtuse 25. scalene right 26. equilateral acute 27. scalene equiangular 28. isosceles acute 29. An equilateral triangle has a perimeter of 105 in. What is the length of each side of the triangle? Architecture Classify each triangle by its angles and sides. 30. △ABC 31. △ACD Daniel Burnham designed and built the 22-story Flatiron Building in New York City in 1902. Source: www.greatbuildings.com 32. An isosceles triangle has a perimeter of 34 cm. The congruent sides measure (4x - 1) cm. The length of the third side is x cm. What is the value of x? 33. Architecture The base of the Flatiron Building is a triangle bordered by three streets: Broadway, Fifth Avenue, and East

Twenty-second Street. The Fifth Avenue side is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft. a. Find the two unknown side lengths. b. Classify the triangle by its side lengths. 34. Critical Thinking Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 35. An acute triangle is a scalene triangle. 36. A scalene triangle is an obtuse triangle. 37. An equiangular triangle is an isosceles triangle. 38. Write About It Write a formula for the side length s of an equilateral triangle, given the perimeter P. Explain how you derived the formula. 39. Construction Use the method for constructing congruent segments to construct an equilateral triangle. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. Marc folded a rectangular sheet of paper, ABCD, in half ̶̶ EF. He folded the resulting square diagonally and along then unfolded the paper to create the creases shown. a. Use the Pythagorean Theorem to find DE and CE. b. What is the m∠DEC? c. Classify △DEC by its side lengths and by its angle measures. 220 220 Chapter 4 Triangle Congruence �� 41. What is the side length of an equilateral triangle with a perimeter of 36 2 __ inches? 3 36 2 _ 3 18 1 _ 3 inches inches 12 1 _ 3 12 2 _ 9 inches inches 42. The vertices of △RST are R (3, 2), S (-2, 3), and T (-2, 1). Which of these best describes △RST? Isosceles Scalene Equilateral Right 43. Which of the following is NOT a correct classification of △LMN? Acute Isosceles Equiangular Right ̶̶ 44. Gridded Response △ABC is isosceles, and AB ≅ - x). What is the perimeter of △ABC? BC = ( 5 __ 2 ̶̶ AC. AB = ( 1 __ x + 1 __ 4 2 ), and CHALLENGE AND EXTEND 45. A triangle has

vertices with coordinates (0, 0), (a, 0), and (0, a), where a ≠ 0. Classify the triangle in two different ways. Explain your answer. 46. Write a two-column proof. Given: △ABC is equiangular. EF ǁ AC Prove: △EFB is equiangular. 47. Two sides of an equilateral triangle measure (y + 10) units and ( y 2 - 2) units. If the perimeter of the triangle is 21 units, what is the value of y? 48. Multi-Step The average length of the sides of △PQR is 24. How much longer then the average is the longest side? SPIRAL REVIEW Name the parent function of each function. (Previous course) 49. y = 5x 2 + 4 50. 2y = 3x + 4 51. y = 2 (x - 8) 2 + 6 Determine if each biconditional is true. If false, give a counterexample. (Lesson 2-4) 52. Two lines are parallel if and only if they do not intersect. 53. A triangle is equiangular if and only if it has three congruent angles. 54. A number is a multiple of 20 if and only if the number ends in a 0. Determine whether each line is parallel to, is perpendicular to, or coincides with y = 4x. (Lesson 3-6) 55. y = 4x + 2 57. 1 _ 2 y = 2x 56. 4y = -x + 8 58. -2y = 1 _ x 2 4-1 Classifying Triangles 221 221 �� 4-2 Use with Lesson 4-2 Activity Develop the Triangle Sum Theorem In this lab, you will use patty paper to discover a relationship between the measures of the interior angles of a triangle. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.3.D G.4.A, G.5.A 1 Draw and label △ABC on a sheet of notebook paper. 2 On patty paper draw a line ℓ and label a point P on the line. ̶̶ AB is on line ℓ and P and B 3

Place the patty paper on top of the triangle you drew. Align the papers so that coincide. Trace ∠B. Rotate the triangle and trace ∠C adjacent to ∠B. Rotate the triangle again and trace ∠A adjacent to ∠C. The diagram shows your final step. Try This 1. What do you notice about the three angles of the triangle that you traced? 2. Repeat the activity two more times using two different triangles. Do you get the same results each time? 3. Write an equation describing the relationship among the measures of the angles of △ABC. 4. Use inductive reasoning to write a conjecture about the sum of the measures of the angles of a triangle. 222 222 Chapter 4 Triangle Congruence 4-2 Angle Relationships in Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons …. Also G.1.A Objectives Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Vocabulary auxiliary line corollary interior exterior interior angle exterior angle remote interior angle Who uses this? Surveyors use triangles to make measurements and create boundaries. (See Example 1.) Triangulation is a method used in surveying. Land is divided into adjacent triangles. By measuring the sides and angles of one triangle and applying properties of triangles, surveyors can gather information about adjacent triangles. This engraving shows the county surveyor and commissioners laying out the town of Baltimore in 1730. Theorem 4-2-1 Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. m∠A + m∠B + m∠C = 180° The proof of the Triangle Sum Theorem uses an auxiliary line. An auxiliary line is a line that is added to a figure to aid in a proof. PROOF PROOF Triangle Sum Theorem Given: △ABC Prove: m∠1 + m∠2 + m∠3 = 180° Proof: Whenever you draw an auxiliary line, you must be able to justify its existence. Give this as the reason: Through any two points there is exactly one line. 4-2 Angle Relationships in Triangles 223 223 ��

�� E X A M P L E 1 Surveying Application The map of France commonly used in the 1600s was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures. A m∠NKM m∠KMN + m∠MNK + m∠NKM = 180° △ Sum Thm. 88 + 48 + m∠NKM = 180 136 + m∠NKM = 180 m∠NKM = 44° Substitute 88 for m∠KMN and 48 for m∠MNK. Simplify. Subtract 136 from both sides. B m∠JLK Step 1 Find m∠JKL. m∠NKM + m∠MKJ + m∠JKL = 180° Lin. Pair Thm. & ∠ Add. Post. 44 + 104 + m∠JKL = 180 148 + m∠JKL = 180 m∠JKL = 32° Substitute 44 for m∠NKM and 104 for m∠MKJ. Simplify. Subtract 148 from both sides. Step 2 Use substitution and then solve for m∠JLK. m∠JLK + m∠JKL + m∠KJL = 180° △ Sum Thm. m∠JLK + 32 + 70 = 180 Substitute 32 for m∠JKL and m∠JLK + 102 = 180 m∠JLK = 78° 70 for m∠KJL. Simplify. Subtract 102 from both sides. 1. Use the diagram to find m∠MJK. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Corollaries COROLLARY HYPOTHESIS CONCLUSION 4-2-2 The acute angles of a right triangle are complementary. 4-2-3 The measure of each angle of an equiangular triangle is 60°. ∠D and ∠E are complementary. m∠D + m∠E = 90° m∠A = m∠B = m∠C = 60° You will prove Corollaries 4-

2-2 and 4-2-3 in Exercises 24 and 25. 224 224 Chapter 4 Triangle Congruence 70°104°88°48°�� E X A M P L E 2 Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 22.9°. What is the measure of the other acute angle? Let the acute angles be ∠M and ∠N, with m∠M = 22.9°. m∠M + m∠N = 90 22.9 + m∠N = 90 Acute  of rt. △ are comp. Substitute 22.9 for m∠M. m∠N = 67.1° Subtract 22.9 from both sides. The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 2a. 63.7° 2b. x ° ° 2c. 48 2 _ 5 The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. ∠4 is an exterior angle. Its remote interior angles are ∠1 and ∠2. Theorem 4-2-4 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m∠4 = m∠1 + m∠2 You will prove Theorem 4-2-4 in Exercise 28. E X A M P L E 3 Applying the Exterior Angle Theorem Find m∠J. m∠J + m∠H = m∠FGH 5x + 17 + 6x - 1 = 126 11x + 16 = 126 11x = 110 x = 10 Ext. ∠ Thm. Substitute 5x + 17 for m∠J, 6x - 1 for m∠H, and 126 for m∠FGH. Simplify. Subtract 16 from both sides. Divide both sides by 11. m∠J = 5x + 17 = 5 (10) + 17 = 67° 3.

Find m∠ACD. 4-2 Angle Relationships in Triangles 225 225 �� Theorem 4-2-5 Third Angles Theorem THEOREM HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. ∠N ≅ ∠T You will prove Theorem 4-2-5 in Exercise 27. E X A M P L E 4 Applying the Third Angles Theorem Find m∠C and m∠F. ∠C ≅ ∠F m∠C = m∠F y 2 = 3y 2 - 72 Third  Thm. Def. of ≅ . Substitute y 2 for m∠C and 3 y 2 - 72 for m∠F. Subtract 3 y 2 from both sides. Divide both sides by -2. -2y 2 = -72 y 2 = 36 So m∠C = 36°. Since m∠F = m∠C, m∠F = 36°. 4. Find m∠P and m∠T. THINK AND DISCUSS 1. Use the Triangle Sum Theorem to explain why the supplement of one of the angles of a triangle equals in measure the sum of the other two angles of the triangle. Support your answer with a sketch. 2. Sketch a triangle and draw all of its exterior angles. How many exterior angles are there at each vertex of the triangle? How many total exterior angles does the triangle have? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write each theorem in words and then draw a diagram to represent it. 226 226 Chapter 4 Triangle Congruence �� 4-2 Exercises Exercises KEYWORD: MG7 4-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. To remember the meaning of remote interior angle, think of a television remote control. What is another way to remember the term remote? 2. An exterior angle is drawn at vertex

E of △DEF. What are its remote interior angles? 3. What do you call segments, rays, or lines that are added to a given diagram. 224 Astronomy Use the following information for Exercises 4 and 5. An asterism is a group of stars that is easier to recognize than a constellation. One popular asterism is the Summer Triangle, which is composed of the stars Deneb, Altair, and Vega. 4. What is the value of y? 5. What is the measure of each angle in the Summer Triangle. 225 The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 6. 20.8° 7 Find each angle measure. p. 225 9. m∠M ° 8. 24 2 _ 3 10. m∠L 11. In △ABC, m∠A = 65°, and the measure of an exterior angle at C is 117°. Find m∠B and the m∠BCA 12. m∠C and m∠F 13. m∠S and m∠U p. 226 14. For △ABC and △XYZ, m∠A = m∠X and m∠B = m∠Y. Find the measures of ∠C and ∠Z if m∠C = 4x + 7 and m∠Z = 3 (x + 5). 4-2 Angle Relationships in Triangles 227 227 �� Independent Practice For See Exercises Example 15 16–18 19–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 15. Navigation A sailor on ship A measures the angle between ship B and the pier and finds that it is 39°. A sailor on ship B measures the angle between ship A and the pier and finds that it is 57°. What is the measure of the angle between ships A and B? The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? ° 16. 76 1 _ 4 17. 2x° 18. 56.8° Find each

angle measure. 19. m∠XYZ 20. m∠C 21. m∠N and m∠P 22. m∠Q and m∠S 23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures.) 24. Complete the proof of Corollary 4-2-2. Given: △DEF with right ∠F Prove: ∠D and ∠E are complementary. Proof: Statements Reasons 1. △DEF with rt. ∠F 2. b.? ̶̶̶̶ 3. m∠D + m∠E + m∠F = 180° 4. m∠D + m∠E + 90° = 180° 5. e.? ̶̶̶̶ 6. ∠D and ∠E are comp. 1. a.? ̶̶̶̶ 2. Def. of rt. ∠ 3. c.? ̶̶̶̶? ̶̶̶̶ 5. Subtr. Prop. 4. d. 6. f.? ̶̶̶̶ 25. Prove Corollary 4-2-3 using two different methods of proof. Given: △ABC is equiangular. Prove: m∠A = m∠B = m∠C = 60° 26. Multi-Step The measure of one acute angle in a right triangle is 1 1 __ 4 times the measure of the other acute angle. What is the measure of the larger acute angle? 27. Write a two-column proof of the Third Angles Theorem. 228 228 Chapter 4 Triangle Congruence Ship AShip BPier57º39º�� 28. Prove the Exterior Angle Theorem. Given: △ABC with exterior angle ∠ACD Prove: m∠ACD = m∠A + m∠B (Hint: ∠BCA and ∠DCA form a linear pair.) Find each angle measure. 29. ∠UXW 31. ∠WZX 30. ∠UWY 32. ∠XYZ 33. Critical Thinking

What is the measure of any exterior angle of an equiangular triangle? What is the sum of the exterior angle measures? 34. Find m∠SRQ, given that ∠P ≅ ∠U, ∠Q ≅ ∠T, and m∠RST = 37.5°. 35. Multi-Step In a right triangle, one acute angle measure is 4 times the other acute angle measure. What is the measure of the smaller angle? 36. Aviation To study the forces of lift and drag, the Wright brothers built a glider, attached two ropes to it, and flew it like a kite. They modeled the two wind forces as the legs of a right triangle. a. What part of a right triangle is formed by each rope? b. Use the Triangle Sum Theorem to write an equation relating the angle measures in the right triangle. c. Simplify the equation from part b. What is the relationship between x and y? d. Use the Exterior Angle Theorem to write an expression for z in terms of x. e. If x = 37°, use your results from parts c and d to find y and z. 37. Estimation Draw a triangle and two exterior angles at each vertex. Estimate the measure of each angle. How are the exterior angles at each vertex related? Explain. 38. Given: Prove: ̶̶ AB ⊥ ̶̶ AD ǁ ̶̶ BD, ̶̶ CB ̶̶ BD ⊥ ̶̶ DC, ∠A ≅ ∠C 39. Write About It A triangle has angle measures of 115°, 40°, and 25°. Explain how to find the measures of the triangle’s exterior angles. Support your answer with a sketch. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. One of the steps in making an origami crane involves folding a square sheet of paper into the shape shown. a. ∠DCE is a right angle. ̶̶ FC bisects ∠DCE, and ̶̶ BC bisects ∠FCE. Find m∠FCB. b. Use the Triangle Sum Theorem to find m∠CBE. 4-2 Angle Relationships in Triangles 229 229 ��DragLiftRopezºyºxºge07sec04ll02005aABoehm��

41. What is the value of x? 19 52 42. Find the value of s. 23 28 57 71 34 56 43. ∠A and ∠B are the remote interior angles of ∠BCD in △ABC. Which of these equations must be true? m∠A - 180° = m∠B m∠A = 90° - m∠B m∠BCD = m∠BCA - m∠A m∠B = m∠BCD - m∠A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117°. Its remote interior angles measure ( 2y 2 + 7) ° and (61 - y 2 ) °. Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two same-side interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two same-side interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30°. What is the probability that the triangle has at least two congruent angles? 49. In △ABC, m∠B is 5° less than 1 1 __ 2 times m∠A. m∠C is 5° less than 2 1 __ 2 times m∠A. What is m∠A in degrees? SPIRAL REVIEW Make a table to show the value of each function when x is -2, 0, 1, and 4. (Previous course) 52. f (x) = (x - 3) 2 + 5 50. f (x) = 3x - 4 51. f (x) = x 2 + 1 53. Find the length of ̶̶̶ NQ. Name the theorem or postulate that justifies your answer. (Lesson 2-7) Classify each triangle by its side lengths. (Lesson 4-1) 54. △ACD 55. △BCD

56. △ABD 57. What if…? If CA = 8, What is the effect on the classification of △ACD? 230 230 Chapter 4 Triangle Congruence �� 4-3 Congruent Triangles TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.B Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Who uses this? Machinists used triangles to construct a model of the International Space Station’s support structure. Vocabulary corresponding angles corresponding sides congruent polygons Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent. Thus triangles that are the same size and shape are congruent. Properties of Congruent Polygons DIAGRAM CORRESPONDING ANGLES CORRESPONDING SIDES Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. △ABC ≅ △DEF polygon PQRS ≅ polygon WXYZ ∠A ≅ ∠D ∠B ≅ ∠E ∠C ≅ ∠F ∠P ≅ ∠W ∠Q ≅ ∠X ∠R ≅ ∠Y ∠S ≅ ∠ Z ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ AC ≅ ̶̶ DE ̶̶ EF ̶̶ DF ̶̶ PQ ≅ ̶̶ QR ≅ ̶̶ RS ≅ ̶̶ PS ≅ ̶̶̶ WX ̶̶ XY ̶̶ YZ ̶̶ WZ To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts. E X A M P L E 1 Naming Congruent Corresponding Parts △RST and △

XYZ represent the triangles of the space station’s support structure. If △RST ≅ △XYZ, identify all pairs of congruent corresponding parts. Angles: ∠R ≅ ∠X, ∠S ≅ ∠Y, ∠T ≅ ∠Z ̶̶ ST ≅ Sides: ̶̶ RT ≅ ̶̶ RS ≅ ̶̶ XY, ̶̶ YZ, ̶̶ XZ 1. If polygon LMNP ≅ polygon EFGH, identify all pairs of corresponding congruent parts. 4-3 Congruent Triangles 231 231 �� E X A M P L E 2 Using Corresponding Parts of Congruent Triangles Given: △EFH ≅ △GFH When you write a statement such as △ABC ≅ △DEF, you are also stating which parts are congruent. A Find the value of x. ∠FHE and ∠FHG are rt. . Def. of ⊥ lines ∠FHE ≅ ∠FHG m∠FHE = m∠FHG (6x - 12) ° = 90° 6x = 102 x = 7 B Find m∠GFH. Rt. ∠ ≅ Thm. Def. of ≅  Substitute values for m∠FHE and m∠FHG. Add 12 to both sides. Divide both sides by 6. m∠EFH + m∠FHE + m∠E = 180° △ Sum Thm. m∠EFH + 90 + 21.6 = 180 Substitute values for m∠FHE m∠EFH + 111.6 = 180 m∠EFH = 68.4 and m∠E. Simplify. Subtract 111.6 from both sides. ∠GFH ≅ ∠EFH m∠GFH = m∠EFH m∠GFH = 68.4° Corr.  of ≅  are ≅. Def. of ≅  Trans. Prop. of = Given: △ABC ≅ △DEF 2a. Find the value of x. 2b. Find m∠F. E X A M P L E 3 Proving Triangles Congruent

Given: ∠P and ∠M are right angles. R is the midpoint of ̶̶ ̶̶ NR PQ ≅ ̶̶ PM. ̶̶̶ MN, ̶̶ QR ≅ Prove: △PQR ≅ △MNR Proof: Statements Reasons 1. ∠P and ∠M are rt.  1. Given 2. ∠P ≅ ∠M 3. ∠PRQ ≅ ∠MRN 4. ∠Q ≅ ∠N ̶̶̶ PM. 6. 5. R is the mdpt. of ̶̶̶ MR ̶̶̶ MN ; ̶̶ PR ≅ ̶̶ PQ ≅ ̶̶ QR ≅ 7. ̶̶ NR 2. Rt. ∠ ≅ Thm. 3. Vert.  Thm. 4. Third  Thm. 5. Given 6. Def. of mdpt. 7. Given 8. △PQR ≅ △MNR 8. Def. of ≅  3. Given: ̶̶ AD bisects ̶̶ BE bisects ̶̶ AB ≅ Prove: △ABC ≅ △DEC ̶̶ BE. ̶̶ AD. ̶̶ DE, ∠A ≅ ∠D 232 232 Chapter 4 Triangle Congruence �� Overlapping Triangles “With overlapping triangles, it helps me to redraw the triangles separately. That way I can mark what I know about one triangle without getting confused by the other one.” Cecelia Medina Lamar High School E X A M P L E 4 Engineering Application Engineering The Rattler is one of the tallest wood-tracked roller coasters in the world. The ride sits on the side of a cliff at Six Flags Fiesta Texas. It is 5080 ft long and 179 ft high. The coaster travels at a speed of 65 mi/h. The bars that give structural support to a roller coaster form triangles. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. ̶̶ ML ⊥ Given: ̶̶ JL ≅ Prove: △JKL ≅ △MLK Proof: ̶̶ KL, ∠KLJ ≅ �

�LKM, ̶̶ MK ̶̶ JK ⊥ ̶̶ JK ≅ ̶̶ KL, ̶̶ ML, ̶̶ JK ⊥ ̶̶ KL, 1. Statements ̶̶̶ ML ⊥ ̶̶ KL Reasons 1. Given 2. ∠JKL and ∠MLK are rt. . 2. Def. of ⊥ lines 3. ∠JKL ≅ ∠MLK 4. ∠KLJ ≅ ∠LKM 5. ∠KJL ≅ ∠LMK ̶̶ JK ≅ ̶̶ KL ≅ ̶̶̶ ML, ̶̶ LK 6. 7. ̶̶ JL ≅ ̶̶̶ MK 8. △JKL ≅ △MLK 3. Rt. ∠ ≅ Thm. 4. Given 5. Third  Thm. 6. Given 7. Reflex. Prop. of ≅ 8. Def. of ≅  4. Use the diagram to prove the following. ̶̶ JK ≅ ̶̶̶ MK bisects ̶̶ JL bisects ̶̶ JL. Given: Prove: △JKN ≅ △LMN ̶̶̶ MK. ̶̶̶ ML, ̶̶ JK ǁ ̶̶̶ ML THINK AND DISCUSS 1. A roof truss is a triangular structure that supports a roof. How can you be sure that two roof trusses are the same size and shape? 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, name the congruent corresponding parts. 4-3 Congruent Triangles 233 233 �� 4-3 Exercises Exercises KEYWORD: MG7 4-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An everyday meaning of corresponding is “matching.” How can this help you find the corresponding parts of two triangles? 2. If △ABC ≅ △RST, what angle corresponds to ∠S. 231 Given: △RST ≅ △LMN

. Identify the congruent corresponding parts. ̶̶ RS ≅ ̶̶ TS ≅ 3. 6.? ̶̶̶̶? ̶̶̶̶ ̶̶ LN ≅ 4. 7. ∠L ≅? ̶̶̶̶? ̶̶̶̶ 5. ∠S ≅ 8. ∠N ≅? ̶̶̶̶? ̶̶̶̶ Given: △FGH ≅ △JKL. Find each value. p. 232 9. KL 10. 232. 233 11. Given: E is the midpoint of ̶̶ AB ≅ ̶̶ CD, ̶̶ AB ǁ ̶̶ CD ̶̶ AC and ̶̶ BD. Prove: △ABE ≅ △CDE Proof: Statements Reasons 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 5. Def. of mdpt. 4. d. 3. c. 6. f. 7. g.? ̶̶̶̶? ̶̶̶̶ ̶̶ AB ǁ ̶̶ CD 1. ̶̶ AB ≅ ̶̶ CD 3. 2. ∠ABE ≅ ∠CDE, ∠BAE ≅ ∠DCE 4. E is the mdpt. of ̶̶ AC and ̶̶ BD. 5. e.? ̶̶̶̶ 6. ∠AEB ≅ ∠CED 7. △ABE ≅ △CDE 12. Engineering The McDonald Observatory has four research telescopes and is a leading center for astronomical study. Prove that the triangles that make up the observatory dome are congruent. Given: ̶̶ ̶̶ ̶̶ SU ≅ SR, ST ≅ ∠UST ≅ ∠RST, and ∠U ≅ ∠R ̶̶ TU ≅ ̶̶ TR, Prove: △RTS ≅ △UTS 234 234 Chapter 4 Triangle Congruence �� Independent Practice Given: Polygon CDEF ≅ polygon KLMN. Identify the congruent corresponding parts. PRACTICE AND PROBLEM SOLVING 13. ̶̶ DE ≅ 15

. ∠F ≅? ̶̶̶̶? ̶̶̶̶ 14. ̶̶ KN ≅ 16. ∠L ≅? ̶̶̶̶? ̶̶̶̶ For See Exercises Example 13–16 17–18 19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 Given: △ABD ≅ △CBD. Find each value. 17. m∠C 18. y 19. Given: ̶̶̶ MP bisects ∠NMR. P is the midpoint of ̶̶ NR. ̶̶̶ MR, ∠N ≅ ∠R ̶̶̶ MN ≅ Prove: △MNP ≅ △MRP Proof: Statements Reasons 1. ∠N ≅ ∠R ̶̶̶ MP bisects ∠NMR. 2. 3. c. 4. d.? ̶̶̶̶? ̶̶̶̶ 5. P is the mdpt. of ̶̶ NR. 6. f.? ̶̶̶̶ ̶̶̶ MN ≅ ̶̶̶ MP ≅ ̶̶̶ MR ̶̶̶ MP 7. 8. 9. △MNP ≅ △MRP 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Def. of ∠ bisector 4. Third  Thm. 5. e.? ̶̶̶̶ 6. Def. of mdpt. 7. g. 8. h.? ̶̶̶̶? ̶̶̶̶ 9. Def. of ≅  20. Hobbies In a garden, triangular flower beds are separated by straight rows of grass as shown. Given: ∠ADC and ∠BCD are right angles. ̶̶ AD ≅ ̶̶ BC ̶̶ ̶̶ AC ≅ BD, ∠DAC ≅ ∠CBD Prove: △ADC ≅ △BCD 21. For two triangles, the following corresponding parts are given: ̶̶ ̶̶ ̶̶ PH, KH, GS ≅ ∠S ≅ ∠P, ∠G ≅ ∠K, and ∠R ≅ ∠H. Write three

different congruence statements. ̶̶ GR ≅ ̶̶ SR ≅ ̶̶ KP, 22. The two polygons in the diagram are congruent. Complete the following congruence statement for the polygons. polygon R? ≅ polygon V ̶̶̶̶? ̶̶̶̶ Write and solve an equation for each of the following. 23. △ABC ≅ △DEF. AB = 2x - 10, and DE = x + 20. Find the value of x and AB. 24. △JKL ≅ △MNP. m∠L = ( x 2 + 10) °, and m∠P = ( 2x 2 + 1) °. What is m∠L? 25. Polygon ABCD ≅ polygon PQRS. BC = 6x + 5, and QR = 5x + 7. Find the value of x and BC. 4-3 Congruent Triangles 235 235 ��ge07se_c04l03005aAEBDC�� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 238. ̶̶ JL and Many origami models begin with a square piece of paper, JKLM, that is folded along both diagonals to make the ̶̶̶ MK are perpendicular bisectors creases shown. of each other, and ∠NML ≅ ∠NKL. ̶̶ a. Explain how you know that KL and b. Prove △NML ≅ △NKL. ̶̶̶ ML are congruent. 27. Draw a diagram and then write a proof. ̶̶ BD ⊥ ̶̶ AC. D is the midpoint of ̶̶ AC. ̶̶ AB ≅ ̶̶ CB, and ̶̶ BD bisects ∠ABC. Given: Prove: △ABD ≅ △CBD 28. Critical Thinking Draw two triangles that are not congruent but have an area of 4 cm 2 each. 29. /////ERROR ANALYSIS///// Given △MPQ ≅ △EDF. Two solutions for finding m∠E are shown. Which is incorrect? Explain the error. 30. Write About It Given the diagram of the triangles, is there enough information to prove that △

HKL is congruent to △YWX? Explain. 31. Which congruence statement correctly indicates that the two given triangles are congruent? △ABC ≅ △DEF △ABC ≅ △FED △ABC ≅ △EFD △ABC ≅ △FDE 32. △MNP ≅ △RST. What are the values of x and y? x = 26, y = 21 1 _ 3 x = 27, y = 20 x = 25, y = 20 2 _ 3, y = 16 2 _ x = 30 1 _ 3 3 33. △ABC ≅ △XYZ. m∠A = 47.1°, and m∠C = 13.8°. Find m∠Y. 13.8 42.9 76.2 119.1 34. △MNR ≅ △SPQ, NL = 18, SP = 33, SR = 10, RQ = 24, and QP = 30. What is the perimeter of △MNR? 79 85 87 97 236 236 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 35. Multi-Step Given that the perimeter of TUVW is 149 units, find the value of x. Is △TUV ≅ △TWV? Explain. 36. Multi-Step Polygon ABCD ≅ polygon EFGH. ∠A is a right angle. m∠E = ( y 2 - 10) °, and m∠H = ( 2y 2 - 132) °. Find m∠D. 37. Given: ̶̶ RS ≅ ̶̶ RT, ∠S ≅ ∠T Prove: △RST ≅ △RTS SPIRAL REVIEW Two number cubes are rolled. Find the probability of each outcome. (Previous course) 38. Both numbers rolled are even. 39. The sum of the numbers rolled is 5. Classify each angle by its measure. (Lesson 1-3) 40. m∠DOC = 40° 41. m∠BOA = 90° 42. m∠

COA = 140° Find each angle measure. (Lesson 4-2) 43. ∠Q 44. ∠P 45. ∠QRS KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra 1 and 2, Geometry, Precalculus Q: What kind of degree or certification will you receive? A: I will receive an associate’s degree in applied science. Then I will take an exam to be certified as an EMT or paramedic. Q: How do you use math in your hands-on training? A: I calculate dosages based on body weight and age. I also calculate drug doses in milligrams per kilogram per hour or set up an IV drip to deliver medications at the correct rate. Q: What are your future career plans? A: When I am certified, I can work for a private ambulance service or with a fire department. I could also work in a hospital, transporting critically ill patients by ambulance or helicopter. 4-3 Congruent Triangles 237 237 Jordan Carter Emergency Medical Services Program �� SECTION 4A Triangles and Congruence Origami Origami is the Japanese art of paper folding. The Japanese word origami literally means “fold paper.” This ancient art form relies on properties of geometry to produce fascinating and beautiful shapes. Each of the figures shows a step in making an origami swan from a square piece of paper. The final figure shows the creases of an origami swan that has been unfolded. Step 1 Step 2 � Step 3 � � � � � � � � � � � � Fold the paper in half diagonally and crease it. Turn it over. Fold corners A and C to the center line and crease. Turn it over. � Step 4 Step 5 � � �� Fold in half along the center crease so that and ̶̶ DF are together. ̶̶ DE Step 6 � �� Fold the narrow point upward at a 90° angle and crease. Push in the fold so that the neck is inside the body. Fold the tip downward and crease. Push in the fold so that the head is inside the neck. Fold up the flap to form the wing. � 1. Use the fact that ABCD is a square � 2. to classify △ABD by its side lengths

and by its angle measures. ̶̶ ̶̶ DB bisects ∠ABC and ∠ADC. DE bisects ∠ADB. Find the measures of the angles in △EDB. Explain how you found the measures. ̶̶ 3. Given that DB bisects ∠ABC and ̶̶ ∠EDF, DE ≅ prove that △EDB ≅ △ FDB. ̶̶ BF, and ̶̶ BE ≅ ̶̶ DF, � � � � � � 238 238 Chapter 4 Triangle Congruence SECTION 4A Quiz for Lessons 4-1 Through 4-3 4-1 Classifying Triangles Classify each triangle by its angle measures. 1. △ACD 2. △ABD 3. △ADE Classify each triangle by its side lengths. 4. △PQR 5. △PRS 6. △PQS 4-2 Angle Relationships in Triangles Find each angle measure. 7. m∠M 8. m∠ABC 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 37° and 55°. Find the measure of ∠RTP, the angle formed by the roof of the house and the roof of the patio. 4-3 Congruent Triangles Given: △JKL ≅ △DEF. Identify the congruent corresponding parts. 10. ̶̶ KL ≅? ̶̶̶̶ 11. ̶̶ DF ≅? ̶̶̶̶ 12. ∠K ≅? ̶̶̶̶ 13. ∠F ≅? ̶̶̶̶ Given: △PQR ≅ △STU. Find each value. 14. PQ 15. y 16. Given:   AB ǁ   CD, ̶̶ CD, ̶̶ AB ≅ ̶̶ ̶̶ DB ⊥ AC ⊥ Prove: △ACD ≅ △DBA ̶̶ CD, ̶̶ AB ̶̶ AC ≅ ̶̶ BD, Proof: Statements Reasons 1.   AB ǁ   CD 2. ∠BAD ≅ �

�CDA ̶̶ AC ⊥ ̶̶ CD, ̶̶ DB ⊥ ̶̶ AB 3. 4. ∠ACD and ∠DBA are rt.  5. e.? ̶̶̶̶̶? ̶̶̶̶̶ ̶̶ CD, 6. f. ̶̶ AB ≅ 7. ̶̶ AC ≅ ̶̶ BD 8. h.? ̶̶̶̶̶ 9. △ACD ≅ △DBA 1. a. 2. b. 3. c. 4. d.? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶ 5. Rt. ∠ ≅ Thm. 6. Third  Thm. 7. g.? ̶̶̶̶̶ 8. Reflex Prop. of ≅ 9. i.? ̶̶̶̶̶ Ready to Go On? 239 239 �� 4-4 Explore SSS and SAS Triangle Congruence Use with Lesson 4-4 In Lesson 4-3, you used the definition of congruent triangles to prove triangles congruent. To use the definition, you need to prove that all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. In this lab, you will discover some shortcuts for proving triangles congruent. Activity 1 TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.4.A G.10.B 1 Measure and cut six pieces from the straws: two that are 2 inches long, two that are 4 inches long, and two that are 5 inches long. 2 Cut two pieces of string that are each about 20 inches long. 3 Thread one piece of each size of straw onto a piece of string. Tie the ends of the string together so that the pieces of straw form a triangle. 4 Using the remaining pieces, try to make another triangle with the same side lengths that is not congruent to the first triangle. Try This 1. Repeat Activity 1 using side lengths of your choice. Are your results the same? 2. Do you think it is possible

to make two triangles that have the same side lengths but that are not congruent? Why or why not? 3. How does your answer to Problem 2 provide a shortcut for proving triangles congruent? 4. Complete the following conjecture based on your results. Two triangles are congruent if?. ̶̶̶̶̶̶̶̶̶̶̶̶̶ 240 240 Chapter 4 Triangle Congruence Activity 2 1 Measure and cut two pieces from the straws: one that is 4 inches long and one that is 5 inches long. 2 Use a protractor to help you bend a paper clip to form a 30° angle. 3 Place the pieces of straw on the sides of the 30° angle. The straws will form two sides of your triangle. 4 Without changing the angle formed by the paper clip, use a piece of straw to make a third side for your triangle, cutting it to fit as necessary. Use additional paper clips or string to hold the straws together in a triangle. Try This 5. Repeat Activity 2 using side lengths and an angle measure of your choice. Are your results the same? 6. Suppose you know two side lengths of a triangle and the measure of the angle between these sides. Can the length of the third side be any measure? Explain. 7. How does your answer to Problem 6 provide a shortcut for proving triangles congruent? 8. Use the two given sides and the given angle from Activity 2 to form a triangle that is not congruent to the triangle you formed. (Hint: One of the given sides does not have to be adjacent to the given angle.) 9. Complete the following conjecture based on your results. Two triangles are congruent if?. ̶̶̶̶̶̶̶̶̶̶̶̶̶ 4- 4 Geometry Lab 241 241 4-4 Triangle Congruence: SSS and SAS TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.A, G.3.B, G.3.E Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles

congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. △ABC ≅ △FDE E X A M P L E 1 Using SSS to Prove Triangle Congruence Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why △PQR ≅ △PSR. ̶̶ ̶̶ SR. By QR ≅ It is given that ̶̶ PR ≅ the Reflexive Property of Congruence, Therefore △PQR ≅ △PSR by SSS. ̶̶ PS and that ̶̶ PQ ≅ ̶̶ PR. 1. Use SSS to explain why △ABC ≅ △CDA. An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides ̶̶ AB and ̶̶ BC. 242 242 Chapter 4 Triangle Congruence �� It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. Postulate 4-4-2 Side-Angle-Side (SAS) Congruence POSTULATE HYPOTHESIS CONCLUSION If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. △ABC ≅ △EFD E X A M P L E 2 Engineering Application The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. The figure shows part of the support structure of the Statue of Liberty. Use SAS to explain

why △KPN ≅ △LPM. It is given that ̶̶ NP ≅ and that By the Vertical Angles Theorem, ∠KPN ≅ ∠LPM. Therefore △KPN ≅ △LPM by SAS. ̶̶ KP ≅ ̶̶̶ MP. ̶̶ LP 2. Use SAS to explain why △ABC ≅ △DBC. The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angle, you can construct one and only one triangle. Construction Congruent Triangles Using SAS Use a straightedge to draw two segments and one angle, or copy the given segments and angle.    ̶̶ AB congruent to one Construct of the segments. Construct ∠A congruent to the given angle. ̶̶ Construct AC congruent to the other segment. Draw to complete △ABC. ̶̶ CB 4-4 Triangle Congruence: SSS and SAS 243 243 KPMNLge07se_c04l04009a3rd pass4/25/5cmurphy�� E X A M P L E 3 Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. A △UVW ≅ △YXW, x = 3 ZY = x - 1 = 3 - 1 = 2 XZ = x = 3 XY = 3x - 5 = 3 (3) - 5 = 4 ̶̶ ̶̶̶ ̶̶ UV ≅ XZ, and VW ≅ So △UVW ≅ △YXZ by SSS. ̶̶ YX. ̶̶̶ UW ≅ ̶̶ YZ. B △DEF ≅ △JGH, y = 7 JG = 2y + 1 = 2 (7) + 1 = 15 GH = y 2 - 4y + 3 = (7) 2 - 4 (7) + 3 = 24 m∠G = 12y + 42 = 12 (7) + 42 = 126° ̶̶ ̶̶ ̶̶ DE ≅ EF ≅ JG. So △DEF ≅ △JGH by SAS. ̶̶̶ GH, and ∠E ≅ ∠G. 3. △ADB

≅ △CDB Proving Triangles Congruent ̶̶ EG ≅ ̶̶ Given: ℓ ǁ m, HF Prove: △EGF ≅ △HFG Proof: Statements Reasons ̶̶ EG ≅ ̶̶ HF 1. 2. ℓ ǁ m 3. ∠EGF ≅ ∠HFG ̶̶ FG ≅ ̶̶ FG 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex Prop. of ≅ 5. △EGF ≅ △HFG 5. SAS Steps 1, 3, 4 4. Given: Prove: △RQP ≅ △SQP  QP bisects ∠RQS. ̶̶ QR ≅ ̶̶ QS 244 244 Chapter 4 Triangle Congruence �� THINK AND DISCUSS 1. Describe three ways you could prove that △ABC ≅ △DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 Exercises Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In △RST which angle is the included angle of sides ̶̶ ST and ̶̶ TR? Use SSS to explain why the triangles in each pair are congruent. p. 242 2. △ABD ≅ △CDB 3. △MNP ≅ △MQP. Design This Texas flag consists of a blue, p. 243 perpendicular stripe with a white star in the center. The star consists of five triangles. GJ = LG = 20 in., and GK = GH = 13 in. Use SAS to explain why △JGK ≅ △LGH Show that the triangles are congruent for the given value of the variable. p. 244 5. △GHJ ≅ △IHJ, x = 4 6. △RST ≅ △TUR, x =

18 4-4 Triangle Congruence: SSS and SAS 245 245 ��. Given: ̶̶ JK ≅ ̶̶̶ ML, ∠JKL ≅ ∠MLK p. 244 Prove: △JKL ≅ △MLK Proof: Statements Reasons ̶̶ JK ≅ ̶̶̶ ML 1. 2. b. ̶̶ KL ≅? ̶̶̶̶ ̶̶ LK 3. 4. △JKL ≅ △MLK 1. a.? ̶̶̶̶ 2. Given 3. c. 4. d.? ̶̶̶̶? ̶̶̶̶ Independent Practice Use SSS to explain why the triangles in each pair are congruent. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 8. △BCD ≅ △EDC 9. △GJK ≅ △GJL 10. Theater The lights shining on a stage appear to form two congruent right triangles. Given △ECB ≅ △DBC. ̶̶ DB, use SAS to explain why ̶̶ EC ≅ Show that the triangles are congruent for the given value of the variable. 11. △MNP ≅ △QNP, y = 3 12. △XYZ ≅ △STU, t = 5 13. Given: B is the midpoint of ̶̶ DC. ̶̶ AB ⊥ ̶̶ DC Prove: △ABD ≅ △ABC Proof: Statements ̶̶ DC. 1. B is the mdpt. of 2. b. 3. c.? ̶̶̶̶? ̶̶̶̶ 4. ∠ABD and ∠ABC are rt. . 5. ∠ABD ≅ ∠ABC 6. f.? ̶̶̶̶ 7. △ABD ≅ △ABC 246 246 Chapter 4 Triangle Congruence Reasons 1. a.? ̶̶̶̶ 2. Def. of mdpt

. 3. Given 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶ 6. Reflex. Prop. of ≅ 7. g.? ̶̶̶̶ �� Which postulate, if any, can be used to prove the triangles congruent? 14. 16. 15. 17. 18. Explain what additional information, if any, you would need to prove △ABC ≅ △DEC by each postulate. b. SAS a. SSS Multi-Step Graph each triangle. Then use the Distance Formula and the SSS Postulate to determine whether the triangles are congruent. 19. △QRS and △TUV 20. △ABC and △DEF Q (-2, 0), R (1, -2), S (-3, -2) T (5, 1), U (3, -2), V (3, 2) A (2, 3), B (3, -1), C (7, 2) D (-3, 1), E (1, 2), F (-3, 5) 21. Given: ∠ZVY ≅ ∠WYV, ∠ZVW ≅ ∠WYZ, ̶̶ ̶̶̶ YZ VW ≅ Prove: △ZVY ≅ △WYV Proof: Statements Reasons 1. ∠ZVY ≅ ∠WYV, ∠ZVW ≅ WYZ 2. m∠ZVY = m∠WYV, m∠ZVW = m∠WYZ 3. m∠ZVY + m∠ZVW = m∠WYV + m∠WYZ 4. c.? ̶̶̶̶ 5. ∠WVY ≅ ∠ZYV ̶̶̶ VW ≅ ̶̶ YZ 6. 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Add. Prop. of = 4. ∠ Add. Post. 5. d. 6. e.? ̶̶̶̶? ̶̶̶̶ 7. f.? ̶̶̶̶ 8. �

�ZVY ≅ △WYV 7. Reflex. Prop. of ≅ 8. g.? ̶̶̶̶ 22. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that ∠A and ∠D are right angles. Explain how you could make just two additional measurements on each truss to ensure that the trusses are congruent. b. You verify that the trusses are congruent and find ̶̶ EF to the that AB = AC = 2.5 ft. Find the length of nearest tenth. Explain. 4-4 Triangle Congruence: SSS and SAS 247 247 �� 23. Critical Thinking Draw two isosceles triangles that are not congruent but that have a perimeter of 15 cm each. Ecology 24. △ABC ≅ △ADC for what value of x? Explain why the SSS Postulate can be used to prove the two triangles congruent. 25. Ecology A wing deflector is a triangular structure made of logs that is filled with large rocks and placed in a stream to guide the current or prevent erosion. Wing deflectors are often used in pairs. Suppose an engineer wants to build two wing deflectors. The logs that form the sides of each wing deflector are perpendicular. How can the engineer make sure that the two wing deflectors are congruent? Wing deflectors are designed to reduce the width-to-depth ratio of a stream. Reducing the width increases the velocity of the stream. 26. Write About It If you use the same two sides and included angle to repeat the construction of a triangle, are your two constructed triangles congruent? Explain. 27. Construction Use three segments (SSS) to construct a scalene triangle. Suppose you then use the same segments in a different order to construct a second triangle. Will the result be the same? Explain. 28. Which of the three triangles below can be proven congruent by SSS or SAS? I and II II and III I and III I, II, and III 29. What is the perimeter of polygon ABCD? 29.9 cm 39.8 cm 49.8 cm 59.8 cm 30. Jacob wants to prove that △FGH ≅ �

�JKL using SAS. ̶̶ ̶̶ JL. What additional JK and ̶̶ FH ≅ ̶̶ He knows that FG ≅ piece of information does he need? ∠H ≅ ∠L ∠F ≅ ∠G ∠F ≅ ∠J ∠G ≅ ∠K 31. What must the value of x be in order to prove that △EFG ≅ △EHG by SSS? 1.5 4.25 4.67 5.5 248 248 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 32. Given:. ∠ADC and ∠BCD are ̶̶ AD ≅ supplementary. ̶̶ CB Prove: △ADB ≅ △CBD (Hint: Draw an auxiliary line.) 33. Given: ∠QPS ≅ ∠TPR, ̶̶ PQ ≅ ̶̶ PT, ̶̶ PR ≅ ̶̶ PS Prove: △PQR ≅ △PTS Algebra Use the following information for Exercises 34 and 35. Find the value of x. Then use SSS or SAS to write a paragraph proof showing that two of the triangles are congruent. 34. m∠FKJ = 2x° m∠KFJ = (3x + 10) ° KJ = 4x + 8 HJ = 6 (x - 4) 35. ̶̶ FJ bisects ∠KFH. m∠KFJ = (2x + 6) ° m∠HFJ = (3x - 21) ° FK = 8x - 45 FH = 6x + 9 SPIRAL REVIEW Solve and graph each inequality. (Previous course) 36. x _ 2 37. 2a + 4 > 3a - 8 ≤ 5 38. -6m - 1 ≤ -13 Solve each equation. Write a justification for each step. (Lesson 2-5) 40. a _ 4 39. 4x - 7 = 21 + 5 = -8 41. 6r = 4r + 10 Given: △EFG ≅ △GHE. Find each value. (Lesson 4-3) 42. x 43. m∠F

EG 44. m∠FGH Using Technology Use geometry software to complete the following. 1. Draw a triangle and label the vertices A, B, and C. Draw a point and label it D. Mark a vector from A to B and translate D by the marked vector. Label the image E. Draw   DE. Mark ∠BAC and rotate   DE about D by the marked angle. Mark ∠ABC and rotate   DE about E by the marked angle. Label the intersection F. 2. Drag A, B, and C to different locations. What do you notice about the two triangles? 3. Write a conjecture about △ABC and △DEF. 4. Test your conjecture by measuring the sides and angles of △ABC and △DEF. 4-4 Triangle Congruence: SSS and SAS 249 249 �� 4-5 Use with Lesson 4-5 Activity 1 Predict Other Triangle Congruence Relationships Geometry software can help you investigate whether certain combinations of triangle parts will make only one triangle. If a combination makes only one triangle, then this arrangement can be used to prove two triangles congruent. TEKS G.9.B Congruence and the geometry of size: formulate … conjectures about the properties and attributes of polygons … based on explorations. Also G.2.A, G.3.B, G.10.B 1 Construct ∠CAB measuring 45° and ∠EDF measuring 110°. 2 Move ∠EDF so that DE overlays AC intersect, label the BA. DF and Where point G. Measure ∠DGA. 3 Move ∠CAB to the left and right without changing the measures of the angles. Observe what happens to the size of ∠DGA. 4 Measure the distance from A to D. Try to change the shape of the triangle without changing AD and the measures of ∠A and ∠D. Try This 1. Repeat Activity 1 using angle measures of your choice. Are your results the same? Explain. 2. Do the results change if one of the given angles measures 90°? 3. What theorem proves that the

measure of ∠DGA in Step 2 will always be the same? 4. In Step 3 of the activity, the angle measures in △ADG stayed the same as the size of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only one triangle? Explain. 5. Repeat Step 4 of the activity but measure the length of ̶̶ AG instead of ̶̶ AD. Are your results the same? Does this lead to a new congruence postulate or theorem? 6. If you are given two angles of a triangle, what additional piece of information is needed so that only one triangle is made? Make a conjecture based on your findings in Step 5. 250 250 Chapter 4 Triangle Congruence Activity 2 1 Construct ̶̶ YZ with a length of 6.5 cm. 2 Using ̶̶ YZ as a side, construct ∠XYZ measuring 43°. 3 Draw a circle at Z with a radius of 5 cm. Construct ̶̶̶ ZW, a radius of circle Z. 4 Move W around circle Z. Observe the possible shapes of △YZW. Try This 7. In Step 4 of the activity, how many different triangles were possible? Does Side-Side-Angle make only one triangle? 8. Repeat Activity 2 using an angle measure of 90° in Step 2 and a circle with a radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 9. Repeat the activity again using a measure of 90° in Step 2 and a circle with a radius of 4.25 cm in Step 3. Classify the resulting triangle by its angle measures. 10. Based on your results, complete the following conjecture. In a Side-Side-Angle combination, if the corresponding nonincluded angles are triangle is possible.?, then only one ̶̶̶̶ 4- 5 Technology Lab 251 251 4-5 Triangle Congruence: ASA, AAS, and HL TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.1.A, G.1.B, G.2.A, G.3.B, G.3.C, G.3.E, G.9.B Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS

, and HL. Vocabulary included side Why use this? Bearings are used to convey direction, helping people find their way to specific locations. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43° E, you face south and then turn 43° to the east. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. ̶̶ PQ is the included side of ∠P and ∠Q. Postulate 4-5-1 Angle-Side-Angle (ASA) Congruence POSTULATE HYPOTHESIS CONCLUSION If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. △ABC ≅ △DEF E X A M P L E 1 Problem-Solving Application Organizers of an orienteering race are planning a course with checkpoints A, B, and C. Does the table give enough information to determine the location of the checkpoints? Understand the Problem Bearing Distance A to B N 55° E 7.6 km B to C N 26° W C to A S 20° W The answer is whether the information in the table can be used to find the position of checkpoints A, B, and C. List the important information: The bearing from A to B is N 55° E. From B to C is N 26° W, and from C to A is S 20° W. The distance from A to B is 7.6 km. 252 252 Chapter 4 Triangle Congruence ��1 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of △ABC. Solve m∠CAB = 55° - 20° = 35° m∠CBA = 180° - (26° + 55°) = 99° You know the measures of ∠CAB and ∠CBA and the length of the included side ̶̶ AB. Therefore by ASA, a unique triangle ABC is determined. Look Back One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the

checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. E X A M P L E 2 Applying ASA Congruence Determine if you can use ASA to prove △UVX ≅ △WVX. Explain. ∠UXV ≅ ∠WXV as given. Since ∠WVX is a right angle that forms a linear pair with ∠UVX, ∠WVX ≅ ∠UVX. Also by the Reflexive Property. Therefore △UVX ≅ △WVX by ASA. ̶̶ VX ≅ ̶̶ VX 2. Determine if you can use ASA to prove △NKL ≅ △LMN. Explain. Construction Congruent Triangles Using ASA Use a straightedge to draw a segment and two angles, or copy the given segment and angles.     ̶̶ CD congruent to Construct the given segment. Construct ∠C congruent to one of the angles. Construct ∠D congruent to the other angle. Label the intersection of the rays as E. △CDE 4-5 Triangle Congruence: ASA, AAS, and HL 253 253 2��34�� You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). Theorem 4-5-2 Angle-Angle-Side (AAS) Congruence THEOREM HYPOTHESIS CONCLUSION If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. △GHJ ≅ △KLM PROOF PROOF Angle-Angle-Side Congruence ̶̶̶ LM ̶̶ HJ ≅ Given: ∠G ≅ ∠K, ∠J ≅ ∠M, Prove: △GHJ ≅ △KLM Proof: Statements Reasons 1. ∠G ≅ ∠K, ∠J ≅ ∠M 1. Given 2. ∠H ≅ ∠L ̶̶̶ LM

̶̶ HJ ≅ 3. 2. Third  Thm. 3. Given 4. △GHJ ≅ △KLM 4. ASA Steps 1, 3, and 2 E X A M P L E 3 Using AAS to Prove Triangles Congruent ̶̶ AB ǁ Use AAS to prove the triangles congruent. ̶̶ BC ≅ Given: Prove: △ABC ≅ △EDC Proof: ̶̶ ED, ̶̶ DC 3. Use AAS to prove the triangles congruent. ̶̶ JL bisects ∠KLM. ∠K ≅ ∠M Given: Prove: △JKL ≅ △JML There are four theorems for right triangles that are not used for acute or obtuse triangles. They are Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA), and Hypotenuse-Leg (HL). You will prove LL, HA, and LA in Exercises 21, 23, and 33. 254 254 Chapter 4 Triangle Congruence �� Theorem 4-5-3 Hypotenuse-Leg (HL) Congruence THEOREM HYPOTHESIS CONCLUSION If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. △ABC ≅ △DEF You will prove the Hypotenuse-Leg Theorem in Lesson 4-8. E X A M P L E 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A △VWX and △YXW According to the diagram, △VWX and △YXW are right triangles that share ̶̶̶ ̶̶̶ WX ≅ WX by the Reflexive hypotenuse ̶̶ ̶̶̶ WV ≅ Property. It is given that XY, therefore △VWX ≅ △YXW by HL. ̶̶̶ WX. B △VWZ and �

�YXZ This conclusion cannot be proved by HL. According to the diagram, △VWZ and △YXZ are right triangles, ̶̶̶ WZ and is congruent to hypotenuse ̶̶ XY. You do not know that hypotenuse ̶̶ XZ. ̶̶̶ WV ≅ 4. Determine if you can use the HL Congruence Theorem to prove △ABC ≅ △DCB. If not, tell what else you need to know. THINK AND DISCUSS 1. Could you use AAS to prove that these two triangles are congruent? Explain. 2. The arrangement of the letters in ASA matches the arrangement of what parts of congruent triangles? Include a sketch to support your answer. 3. GET ORGANIZED Copy and complete the graphic organizer. In each column, write a description of the method and then sketch two triangles, marking the appropriate congruent parts. 4-5 Triangle Congruence: ASA, AAS, and HL 255 255 �� 4-5 Exercises Exercises KEYWORD: MG7 4-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A triangle contains ∠ABC and ∠ACB with ̶̶ BC “closed in” between them. How would this help you remember the definition of included side. 252 Surveying Use the table for Exercises 2 and 3. A landscape designer surveyed the boundaries of a triangular park. She made the following table for the dimemsions of the land. A to B B to C C to A Bearing E S 25° E N 62° W Distance 115 ft?? 2. Draw the plot of land described by the table. Label the measures of the angles in the triangle. 3. Does the table have enough information to determine the locations of points A, B, and C? Explain Determine if you can use ASA to prove the triangles congruent. Explain. p. 253 4. △VRS and △VTS, given that ̶̶ VS bisects ∠RST and ∠RVT 5. △DEH and △FGH. Use AAS to prove the triangles congruent. p. 254 Given: ∠R and ∠P are right angles.

̶̶ QR ǁ ̶̶ SP Prove: △QPS ≅ △SRQ Proof. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. △ABC and △CDA 8. △XYV and △ZYV 256 256 Chapter 4 Triangle Congruence CBA115 ft ge07sec_04l05003aa�� Independent Practice For See Exercises Example 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Surveying Use the table for Exercises 9 and 10. From two different observation towers a fire is sighted. The locations of the towers are given in the following table. X to Y X to F Y to F Bearing E N 53° E N 16° W Distance 6 km?? 9. Draw the diagram formed by observation tower X, observation tower Y, and the fire F. Label the measures of the angles. 10. Is there enough information given in the table to pinpoint the location of the fire? Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Math History 11. 11. △MKJ and △MKL 12. △RST and △TUR 13. 13. Given: ̶̶ AB ≅ Prove: △ABC ≅ △DEF ̶̶ DE, ∠C ≅ ∠F Proof: Euclid wrote the mathematical text The Elements around 2300 years ago. It may be the second most reprinted book in history. Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 14. △GHJ and △JKG 15. △ABE and △DCE, given that E is the midpoint ̶̶ ̶̶ BC AD and of Multi-Step For each pair of triangles write a triangle congru

ence statement. Identify the transformation that moves one triangle to the position of the other triangle. 16. 17. 18. Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two triangles congruent. Draw a diagram that shows why this is true. 4-5 Triangle Congruence: ASA, AAS, and HL 257 257 �� 19. This problem will prepare you for the Multi-Step TAKS Prep on page 280. A carpenter built a truss to support the roof of a doghouse. ̶̶ a. The carpenter knows that MJ. Can the carpenter ̶̶ KJ ≅ conclude that △KJL ≅ △MJL? Why or why not? b. Suppose the carpenter also knows that ∠JLK is a right angle. Which theorem can be used to show that △KJL ≅ △MJL? 20. /////ERROR ANALYSIS///// Two proofs that △EFH ≅ △GHF are given. Which is incorrect? Explain the error. 21. Write a paragraph proof of the Leg-Leg (LL) Congruence Theorem. If the legs of one right triangle are congruent to the corresponding legs of another right triangle, the triangles are congruent. 22. Use AAS to prove the triangles congruent. ̶̶ AD ≅ Prove: △AED ≅ △CEB ̶̶ AD ǁ Given: ̶̶ BC, ̶̶ CB Proof: Statements Reasons ̶̶̶ AD ǁ ̶̶ BC 1. 2. ∠DAE ≅ ∠BCE 3. c. 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ 1. a. 2. b.? ̶̶̶̶? ̶̶̶̶ 3. Vert.  Thm. 3. Given 4. f.? ̶̶̶̶ 23. Prove the Hypotenuse-Angle (HA) Theorem. ̶̶ ̶̶̶ JM ≅ KM ⊥ Given: Prove: △JKM ≅ △LKM

̶̶̶ LM, ∠JMK ≅ ∠LMK ̶̶ JL, 24. Write About It The legs of both right △DEF and right △RST are 3 cm and 4 cm. They each have a hypotenuse 5 cm in length. Describe two different ways you could prove that △DEF ≅ △RST. 25. Construction Use the method for constructing perpendicular lines to construct a right triangle. 26. What additional congruence statement is necessary to prove △XWY ≅ △XVZ by ASA? ∠XVZ ≅ ∠XWY ∠VUY ≅ ∠WUZ ̶̶ VZ ≅ ̶̶ XZ ≅ ̶̶̶ WY ̶̶ XY 258 258 Chapter 4 Triangle Congruence �� 27. Which postulate or theorem justifies the congruence statement △STU ≅ △VUT? ASA SSS HL SAS 28. Which of the following congruence statements is true? ∠A ≅ ∠B ̶̶ ̶̶ DE CE ≅ △AED ≅ △CEB △AED ≅ △BEC 29. In △RST, RT = 6y - 2. In △UVW, UW = 2y + 7. ∠R ≅ ∠U, and ∠S ≅ ∠V. What must be the value of y in order to prove that △RST ≅ △UVW? 1.25 2.25 9.0 11.5 30. Extended Response Draw a triangle. Construct a second triangle that has the same angle measures but is not congruent. Compare the lengths of each pair of corresponding sides. Consider the relationship between the lengths of the sides and the measures of the angles. Explain why Angle-Angle-Angle (AAA) is not a congruence principle. CHALLENGE AND EXTEND 31. Sports This bicycle frame includes △VSU and △VTU, which

lie in intersecting planes. From the given angle measures, can you conclude that △VSU ≅ △VTU? Explain. m∠VUS = (7y - 2) ° m∠VUT = (5 1 _ m∠USV = 5 2 _ y ° 3 m∠SVU = (3y - 6) ° m∠TVU = 2x ° ) x - 1 _ 2 2 m∠UTV = (4x + 8) ° ° � � � � 32. Given: △ABC is equilateral. C is the midpoint of ̶̶ DE. ∠DAC and ∠EBC are congruent and supplementary. Prove: △DAC ≅ △EBC �� 33. Write a two-column proof of the Leg-Angle (LA) Congruence Theorem. If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (Hint: There are two cases to consider.) 34. If two triangles are congruent by ASA, what theorem could you use to prove that the triangles are also congruent by AAS? Explain. SPIRAL REVIEW Identify the x- and y-intercepts. Use them to graph each line. (Previous course) 35. y = 3x - 6 36 37. y = -5x + 5 38. Find AB and BC if AC = 10. (Lesson 1-6) 39. Find m∠C. (Lesson 4-2) 4-5 Triangle Congruence: ASA, AAS, and HL 259 259 �� 4-6 Triangle Congruence: CPCTC TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning … Also G.3.E, G.7.A, G.10.B Objective Use CPCTC to prove parts of triangles are congruent. Vocabulary CPCTC Why learn this? You can use congruent triangles to estimate distances. CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles

congruent. E X A M P L E 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in the figure. If DE = 600 ft, what is AB? ̶̶ CB,because DC = CB = 500 ft. ∠D ≅ ∠B, because they are both right angles. ̶̶ DC ≅ ∠DCE ≅ ∠BCA, because vertical angles are congruent. Therefore △DCE ≅ △BCA by ASA or LA. By CPCTC, AB = ED = 600 ft. ̶̶ AB, so ̶̶ ED ≅ 1. A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? � �� E X A M P L E 2 Proving Corresponding Parts Congruent ̶̶ AB ≅ ̶̶ DC, ∠ABC ≅ ∠DCB Given: Prove: ∠A ≅ ∠D Proof: 2. Given: Prove: ̶̶ PR bisects ∠QPS and ∠QRS. ̶̶ PQ ≅ ̶̶ PS 260 260 Chapter 4 Triangle Congruence �� E X A M P L E 3 Using CPCTC in a Proof ̶̶ DF ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ ED ǁ ̶̶ DF, ̶̶ GF Given: Prove: Proof: Statements Reasons ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ DF ̶̶ DF 1. 2. 3. ∠EGD ≅ ∠FDG ̶̶̶ GD ≅ ̶̶̶ GD 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex. Prop. of ≅ 5. △EGD ≅ △FDG 5. SAS Steps 1, 3,

and 4 6. ∠EDG ≅ ∠FGD ̶̶ GF ̶̶ ED ǁ 7. 6. CPCTC 7. Converse of Alt. Int.  Thm. 3. Given: J is the midpoint of Prove: ̶̶ KL ǁ ̶̶̶ MN ̶̶̶ KM and ̶̶ NL. You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that the triangles are congruent, you can make conclusions about their corresponding parts. E X A M P L E 4 Using CPCTC in the Coordinate Plane Given: A (2, 3), B (5, -1), C (1, 0), D (-4, -1), E (0, 2), F (-1, -2) Prove: ∠ABC ≅ ∠DEF Step 1 Plot the points on a coordinate plane. Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. D = √  ( AB = √  (5 -2) 2 + (-1 - 3) 2 = √  9 + 16 = √  25 = 5 BC =  2 √ (1 - 5) 2 + (0 - (-1) ) = √  16 + 1 = √  17 AC = √  (1 - 2) 2 + (0 - 3) 2 = √  1 + 9 = √  10 DE =  2 √ + (2 - (-1) ) (0 - (-

4) ) 16 + 9 = √  25 = 5 = √  2 EF = √  (-1 - 0) 2 + (-2 - 2) 2 = √  1 + 16 = √  17 DF =  2 √ + (-2 - (-1) ) (-1 - (-4) ) 9 + 1 = √  10 = √  2 ̶̶ AB ≅ ̶̶ BC ≅ So and ∠ABC ≅ ∠DEF by CPCTC. ̶̶ EF, and ̶̶ DE, ̶̶ AC ≅ ̶̶ DF. Therefore △ABC ≅ △DEF by SSS, 4. Given: J (-1, -2), K (2, -1), L (-2, 0), R (2, 3), S (5, 2), T (1, 1) Prove: ∠JKL ≅ ∠RST 4-6 Triangle Congruence: CPCTC 261 261 �� THINK AND DISCUSS ̶̶̶ VW ≅ 1. In the figure, ̶̶ UV ≅ ̶̶ XY, ̶̶ YZ, and ∠V ≅ ∠Y. Explain why △UVW ≅ △XYZ. By CPCTC, which additional parts are congruent? 2. GET ORGANIZED Copy and complete the graphic organizer. Write all conclusions you can make using CPCTC. 4-6 Exercises Exercises GUIDED PRACTICE 1. Vocabulary You use CPCTC after proving triangles are congruent. Which parts of congruent triangles are referred to as corresponding parts. Engineering To find the height of p. 260 a windmill, a rancher places a marker at C and steps off the distance from C to B. Then the rancher walks the same distance from C in the opposite direction and places a marker at D. If DE

= 6.3 m, what is AB? KEYWORD: MG7 4-6 KEYWORD: MG7 Parent � � � � � �. 260 3. Given: X is the midpoint of ̶̶ ST. ̶̶ RX ⊥ ̶̶ ST Prove: ̶̶ RS ≅ ̶̶ RT Proof: 262 262 Chapter 4 Triangle Congruence ��. 261 4. Given: Prove: ̶̶ ̶̶ AC ≅ AD, ̶̶ AB bisects ∠CAD. ̶̶ CB ≅ ̶̶ DB Proof: Statements ̶̶ ̶̶ ̶̶̶ DB CB ≅ AD, ̶̶ AC ≅ 1. 2. b.? ̶̶̶̶ 3. △ACB ≅ △ADB 4. ∠CAB ≅ ∠DAB ̶̶ AB bisects ∠CAD 5. Reasons 1. a.? ̶̶̶̶ 2. Reflex. Prop. of ≅ 3. c. 4. d. 5. e.? ̶̶̶̶? ̶̶̶̶? ̶̶̶̶ Multi-Step Use the given set of points to prove each congruence statement. p. 261 5. E (-3, 3), F (-1, 3), G (-2, 0), J (0, -1), K (2, -1), L (1, 2) ; ∠EFG ≅ ∠JKL 6. A (2, 3), B (4, 1), C (1, -1), R (-1, 0), S (-3, -2), T (0, -4) ; ∠ACB ≅ ∠RTS Independent Practice For See Exercises Example 7 8–9 10–11 12–13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 7. Surveying To find the distance AB across a river, a surveyor first locates point C. He

measures the distance from C to B. Then he locates point D the same distance east of C. If DE = 420 ft, what is AB? 8. 8. Given: M is the midpoint of ̶̶ PQ and ̶̶ ̶̶ PS QR ≅ Prove: ̶̶ RS. 9. Given: ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ ̶̶̶ ZW Prove: ∠W ≅ ∠Y 10. Given: G is the midpoint of ̶̶ FH. 11. Given: ̶̶̶ LM bisects ∠JLK. Prove: M is the midpoint of ̶̶ KL ̶̶ JL ≅ ̶̶ JK. ̶̶ EF ≅ ̶̶ EH Prove: ∠1 ≅ ∠2 Multi-Step Use the given set of points to prove each congruence statement. 12. R (0, 0), S (2, 4), T (-1, 3), U (-1, 0), V (-3, -4), W (-4, -1) ; ∠RST ≅ ∠UVW 13. A (-1, 1), B (2, 3), C (2, -2), D (2, -3), E (-1, -5), F (-1, 0) ; ∠BAC ≅ ∠EDF 14. Given: △QRS is adjacent to △QTS. Prove: ̶̶ QS bisects ̶̶ RT. ̶̶ QS bisects ∠RQT. ∠R ≅ ∠T 15. Given: △ABE and △CDE with E the midpoint of ̶̶ AC and ̶̶ BD Prove: ̶̶ AB ǁ ̶̶ CD 4-6 Triangle Congruence: CPCTC 263 263 ��BC500 ftADE500 ftge07sec04l06004_A�� 16. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The front of a doghouse has the dimensions shown. a. How can you prove that △ADB ≅ △ADC? ̶̶ b. Prove that CD.

c. What is the length of ̶̶ BD and ̶̶ BD ≅ ̶̶ BC to the nearest tenth? �� Multi-Step Find the value of x. 17. 18. �� Use the diagram for Exercises 19–21. 19. Given: PS = RQ, m∠1 = m∠4 Prove: m∠3 = m∠2 20. Given: m∠1 = m∠2, m∠3 = m∠4 Prove: PS = RS 21. Given: PS = RQ, PQ = RS Prove: ̶̶ PQ ǁ ̶̶ RS 22. Critical Thinking Does the diagram contain enough information to allow you to conclude that ̶̶̶ ML? Explain. ̶̶ JK ǁ 23. Write About It Draw a diagram and explain how a surveyor can set up triangles to find the distance across a lake. Label each part of your diagram. List which sides or angles must be congruent. 24. Which of these will NOT be used as a reason in a proof of ̶̶ AD? ̶̶ AC ≅ SAS ASA CPCTC Reflexive Property 25. Given the points K (1, 2), L (0, -4), M (-2, -3), and N (-1, 3), which of these is true? ∠KNL ≅ ∠MNL ∠LNK ≅ ∠NLM 26. What is the value of y? ∠MLN ≅ ∠KLN ∠MNK ≅ ∠NKL 10 20 35 85 27. Which of these are NOT used to prove angles congruent? congruent triangles noncorresponding parts parallel lines perpendicular lines 264 264 Chapter 4 Triangle Congruence �� 28. Which set of coordinates represents the vertices of a triangle congruent to △RST? (Hint: Find the lengths of the sides of △RST.) (3, 4), (3, 0), (0, 0) (3, 3), (0, 4), (0, 0) (3, 1), (3, 3), (4, 6) (3, 0), (4

, 4), (0, 6) CHALLENGE AND EXTEND 29. All of the edges of a cube are congruent. All of the angles on each face of a cube are right angles. Use CPCTC to explain why any two diagonals on the faces of a cube (for example, must be congruent. ̶̶ AC and ̶̶ AF ) 30. Given: ̶̶ JK ≅ ̶̶̶ ML, ̶̶ JM ≅ ̶̶ KL Prove: ∠J ≅ ∠L (Hint: Draw an auxiliary line.) 31. Given: R is the midpoint of S is the midpoint of ̶̶ RS ⊥ ̶̶ AB. ̶̶ DC. ̶̶ AB, ∠ASD ≅ ∠BSC Prove: △ASD ≅ △BSC 32. △ABC is in plane M. △CDE is in plane P. Both planes have C in common and ∠A ≅ ∠E. What is the height AB to the nearest foot? �� SPIRAL REVIEW 33. Lina’s test scores in her history class are 90, 84, 93, 88, and 91. What is the minimum score Lina must make on her next test to have an average test score of 90? (Previous course) 34. One long-distance phone plan costs $3.95 per month plus $0.08 per minute of use. A second long-distance plan costs $0.10 per minute for the first 50 minutes used each month and then $0.15 per minute after that. Which plan is cheaper if you use an average of 75 long-distance minutes per month? (Previous course) A figure has vertices at (1, 3), (2, 2), (3, 2), and (4, 3). Identify the transformation of the figure that produces an image with each set of vertices. (Lesson 1-7) 35. (1, -3), (2, -2), (3, -2), (4, -3) 36. (-2, -1), (-1, -2), (0, -2), (1, -1) 37. Determine if you can use ASA to prove △ACB �

� △ECD. Explain. (Lesson 4-5) 4-6 Triangle Congruence: CPCTC 265 265 �� Quadratic Equations Algebra A quadratic equation is an equation that can be written in the form a x 2 + bx + c = 0. See Skills Bank page S66 Example Given: △ABC is isosceles with ̶̶ AB ≅ ̶̶ AC. Solve for x. Step 1 Set x 2 – 5x equal to 6 to get x 2 – 5x = 6. Step 2 Rewrite the quadratic equation by subtracting 6 from each side to get x 2 – 5x – 6 = 0. Step 3 Solve for x. Method 1: Factoring Method 2: Quadratic Formula x 2 - 5x - 6 = 0 (x - 6) (x + 1) = 0 Factor. x = -b ± √  b 2 - 4ac __ 2a x - 6 = 0 or x + 1 = 0 Set each factor equal to 0. x = - (-5) ± √  (-5) 2 - 4 (1) (-6) ___ 2 (1) Substitute 1 for a, -5 for b, and -6 for c. Simplify. Find the square root. Simplify. x = 6 or x = -1 Solve. x = 5 ± √  49 _ 2 5 ± 7 _ x = 2 or x = -2 _ x = 12 _ 2 2 x = 6 or x = -1 Step 4 Check each solution in the original equation. x 2 - 5x = 6 (6 ) 2 - 5 (6 ) 36 - 30 6 6 x 2 - 5x = 6 (-1) 2 - 5 (- ✓ Try This TAKS Grades 9–11 Obj. 5, 6 Solve for x in each isosceles triangle. ̶̶ 1. Given: FG ̶̶ FE ≅ 2. Given: ̶̶ JK ≅ ̶̶ JL 3. Given: ̶̶ YX ≅ ̶̶ YZ 4. Given: ̶̶ QP ≅ ̶̶ QR 266 266 Chapter 4 Triangle Congruence ��

�� 4-7 Introduction to Coordinate Proof TEK G.2.B Geometric structure: make conjectures about... polygons … and determine validity of the conjectures. Also G.3.B, G.9.B, G.10.B Objectives Position figures in the coordinate plane for use in coordinate proofs. Prove geometric concepts by using coordinate proof. Vocabulary coordinate proof Who uses this? The Bushmen in South Africa use the Global Positioning System to transmit data about endangered animals to conservationists. (See Exercise 24.) You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane • Use the origin as a vertex, keeping the figure in Quadrant I. • Center the figure at the origin. • Center a side of the figure at the origin. • Use one or both axes as sides of the figure. E X A M P L E 1 Positioning a Figure in the Coordinate Plane Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane. Method 1 You can center the longer side of the rectangle at the origin. Method 2 You can use the origin as a vertex of the rectangle. Depending on what you are using the figure to prove, one solution may be better than the other. For example, if you need to find the midpoint of the longer side, use the first solution. 1. Position a right triangle with leg lengths of 2 and 4 units in the coordinate plane. (Hint: Use the origin as the vertex of the right angle.) 4- 7 Introduction to Coordinate Proof 267 267 �� Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure. E X A M P L E 2 Writing a Proof Using Coordinate Geometry Write a coordinate proof. Given: Right △ABC has vertices A (0, 6), B (0, 0), and

C (4, 0). D is the ̶̶ AC. midpoint of Prove: The area of △DBC is one half the area of △ABC. Proof: △ABC is a right triangle with height AB and base BC. area of △ABC = 1 __ 2 bh = 1 __ 2 (4) (6) = 12 square units, 6 + 0 ____ 2 By the Midpoint Formula, the coordinates of D = ( 0 + 4 ____ 2 of △DBC, and the base is 4 units. area of △DBC = 1 __ 2 bh ) = (2, 3). The y-coordinate of D is the height = 1 __ 2 (4) (3) = 6 square units Since 6 = 1 __ 2 (12), the area of △DBC is one half the area of △ABC. 2. Use the information in Example 2 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. E X A M P L E 3 Assigning Coordinates to Vertices Position each figure in the coordinate plane and give the coordinates of each vertex. A a right triangle with leg B a rectangle with lengths a and b length c and width d Do not use both axes when positioning a figure unless you know the figure has a right angle. 3. Position a square with side length 4p in the coordinate plane and give the coordinates of each vertex. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables. 268 268 Chapter 4 Triangle Congruence �� E X A M P L E 4 Writing a Coordinate Proof Given: ∠B is a right angle in △ABC. D is the midpoint of Prove: The area of △DBC is one half the area of △ABC. ̶̶ AC. Step 1 Assign coordinates to each vertex

. The coordinates of A are (0, 2 j), the coordinates of B are (0, 0), and the coordinates of C are (2n, 0). Since you will use the Midpoint Formula to find the coordinates of D, use multiples of 2 for the leg lengths. Step 2 Position the figure in the coordinate plane. Step 3 Write a coordinate proof. Proof: △ABC is a right triangle with height 2j and base 2n. area of △ABC = 1 __ 2 bh = 1 __ 2 (2n) (2j) = 2nj square units Because the x- and y-axes intersect at right angles, they can be used to form the sides of a right triangle. By the Midpoint Formula, the coordinates of D = ( 0 + 2n 2j + 0 _____ _____ 2, 2 The height of △DBC is j units, and the base is 2n units. area of △DBC = 1 __ 2 bh ) = (n, j). = 1 __ 2 (2n) (j) = nj square units Since nj = 1 __ 2 (2nj), the area of △DBC is one half the area of △ABC. 4. Use the information in Example 4 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. THINK AND DISCUSS 1. When writing a coordinate proof why are variables used instead of numbers as coordinates for the vertices of a figure? 2. How does the way you position a figure in the coordinate plane affect your calculations in a coordinate proof? 3. Explain why it might be useful to assign 2p as a coordinate instead of just p. 4. GET ORGANIZED Copy and complete the graphic organizer. In each row, draw an example of each strategy that might be used when positioning a figure for a coordinate proof. 4- 7 Introduction to Coordinate Proof 269 269 �� 4-7 Exercises Exercises KEYWORD: MG7 4-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What is the relationship between coordinate geometry, coordinate plane, and coordinate proof? Position each figure in the coordinate

plane. p. 267 2. a rectangle with a length of 4 units and width of 1 unit 3. a right triangle with leg lengths of 1 unit and 3 units Write a proof using coordinate geometry. p. 268 4. Given: Right △PQR has coordinates P (0, 6), Q (8, 0), and R (0, 0). A is the midpoint of B is the midpoint of ̶̶ QR. ̶̶ PR. Prove: AB = 1 __ 2 PQ. 268 Position each figure in the coordinate plane and give the coordinates of each vertex. 5. a right triangle with leg lengths m and n 6. a rectangle with length a and width. 269 Independent Practice For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 Multi-Step Assign coordinates to each vertex and write a coordinate proof. 7. Given: ∠R is a right angle in △PQR. A is the midpoint of ̶̶ PR. B is the midpoint of ̶̶ QR. Prove: AB = 1 __ 2 PQ PRACTICE AND PROBLEM SOLVING Position each figure in the coordinate plane. 8. a square with side lengths of 2 units 9. a right triangle with leg lengths of 1 unit and 5 units Write a proof using coordinate geometry. 10. Given: Rectangle ABCD has coordinates A (0, 0), B (0, 10), C (6, 10), and D (6, 0). E is the midpoint of ̶̶ AB, and F is the midpoint of ̶̶ CD. Prove: EF = BC Position each figure in the coordinate plane and give the coordinates of each vertex. 11. a square with side length 2m 12. a rectangle with dimensions x and 3x Multi-Step Assign coordinates to each vertex and write a coordinate proof. ̶̶ CD. ̶̶ AB in rectangle ABCD. F is the midpoint of 13. Given: E is the midpoint of Prove: EF = AD 14. Critical Thinking Use variables to write the general form of the endpoints of a segment whose midpoint is (0, 0). 270 270 Chapter 4 Triangle Congruence �� Conservation The origin of the springbok�

�s name may come from its habit of pronking, or bouncing. When pronking, a springbok can leap up to 13 feet in the air. Springboks can run up to 53 miles per hour. 15. Recreation A hiking trail begins at E (0, 0). Bryan hikes from the start of the trail to a waterfall at W (3, 3) and then makes a 90° turn to a campsite at C (6, 0). a. Draw Bryan’s route in the coordinate plane. b. If one grid unit represents 1 mile, what is the total distance Bryan hiked? Round to the nearest tenth. Find the perimeter and area of each figure. 16. a right triangle with leg lengths of a and 2a units 17. a rectangle with dimensions s and t units Find the missing coordinates for each figure. 18. 19. 20. Conservation The Bushmen have sighted animals at the following coordinates: (-25, 31.5), (-23.2, 31.4), and (-24, 31.1). Prove that the distance between two of these locations is approximately twice the distance between two other. 21. Navigation Two ships depart from a port at P (20, 10). The first ship travels to a location at A (-30, 50), and the second ship travels to a location at B (70, -30). Each unit represents one nautical mile. Find the distance to the nearest nautical mile between the two ships. Verify that the port is at the midpoint between the two. Write a coordinate proof. 22. Given: Rectangle PQRS has coordinates P (0, 2), Q (3, 2), R (3, 0), and S (0, 0). ̶̶ PR and ̶̶ QS intersect at T (1.5, 1). Prove: The area of △RST is 1 __ 4 of the area of the rectangle _____ _____ 2, 2 23. Given ), with midpoint M ( Prove: AM = 1 __ 2 AB 24. Plot the points on a coordinate plane and connect them to form △KLM and △MPK. Write a coordinate proof. Given: K (-2, 1), L (-2, 3), M (1, 3), P (1, 1) Prove: △KLM ≅ △MPK 25. Write

About It When you place two sides of a figure on the coordinate axes, what are you assuming about the figure? 26. This problem will prepare you for the Multi-Step TAKS Prep on page 280. Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. a. Find BD and CE. b. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed A at the origin and and E, assuming that each unit of the coordinate plane represents one inch. ̶̶ AB on the y-axis. Find the coordinates of B, C, D, 4- 7 Introduction to Coordinate Proof 271 271 �� 27. The coordinates of the vertices of a right triangle are (0, 0), (4, 0), and (0, 2). Which is a true statement? The vertex of the right angle is at (4, 2). The midpoints of the two legs are at (2, 0) and (0, 1). The hypotenuse of the triangle is √  6 units. The shortest side of the triangle is positioned on the x-axis. 28. A rectangle has dimensions of 2g and 2f units. If one vertex is at the origin, which coordinates could NOT represent another vertex? (2g, 2f) (2f, 0) (2f, g) (-2f, 2g) 29. The coordinates of the vertices of a rectangle are (0, 0), (a, 0), (a, b), and (0, b). What is the perimeter of the rectangle? a + b ab 1 _ 2 ab 2a + 2b 30. A coordinate grid is placed over a map. City A is located at (-1, 2) and city C is located at (3, 5). If city C is at the midpoint between city A and city B, what are the coordinates of city B? (1, 3.5) (-5, -1) (7, 8) (2, 7) CHALLENGE AND EXTEND Find the missing coordinates for each figure. 31. 32. 33. The vertices of a right triangle are at (-2s, 2s), (0, 2s),

and (0, 0). What coordinates could be used so that a coordinate proof would be easier to complete? 34. Rectangle ABCD has dimensions of 2f and 2g units. The equation of the line containing ̶̶ g __ BD is y = x, and f ̶̶ g __ x + 2g. AC is y = - the equation of the line containing f Use algebra to show that the coordinates of E are (f, g). SPIRAL REVIEW Use the quadratic formula to solve for x. Round to the nearest hundredth if necessary. (Previous course) 35. 0 = 8 x 2 + 18x - 5 36. 0 = x 2 + 3x - 5 37. 0 = 3 x 2 - x - 10 Find each value. (Lesson 3-2) 38. x 39. y 40. Use A (-4, 3), B (-1, 3), C (-3, 1), D (0, -2), E (3, -2), and F (2, -4) to prove ∠ABC ≅ ∠EDF. (Lesson 4-6). 272 272 Chapter 4 Triangle Congruence �� 4-8 Isosceles and Equilateral Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons … and determine the validity of the conjectures.... Also G.3.C, G.10.B Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Vocabulary legs of an isosceles triangle vertex angle base base angles Who uses this? Astronomers use geometric methods. (See Example 1.) Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. ∠3 is the vertex angle. ∠1 and ∠2 are the base angles. Theorems Isosceles Triangle THEOREM HYPOTHESIS CONCLUSION 4-8-1 Isosceles Triangle Theorem If

two sides of a triangle are congruent, then the angles opposite the sides are congruent. 4-8-2 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. ∠B ≅ ∠C ̶̶ DE ≅ ̶̶ DF Theorem 4-8-1 is proven below. You will prove Theorem 4-8-2 in Exercise 35. PROOF PROOF ̶̶ AB ≅ Isosceles Triangle Theorem ̶̶ AC Given: Prove: ∠B ≅ ∠C Proof: Statements Reasons The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.” 1. Draw X, the mdpt. of 2. Draw the auxiliary line ̶̶ BC. ̶̶ AX. ̶̶ BX ≅ ̶̶ AB ≅ ̶̶ AX ≅ ̶̶ CX ̶̶ AC ̶̶ AX 3. 4. 5. 6. △ABX ≅ △ACX 7. ∠B ≅ ∠C 1. Every seg. has a unique mdpt. 2. Through two pts. there is exactly one line. 3. Def. of mdpt. 4. Given 5. Reflex. Prop. of ≅ 6. SSS Steps 3, 4, 5 7. CPCTC 4-8 Isosceles and Equilateral Triangles 273 273 �� E X A M P L E 1 Astronomy Application The distance from Earth to nearby stars can be measured using the parallax method, which requires observing the positions of a star 6 months apart. If the distance LM to a star in July is 4.0 × 10 13 km, explain why the distance LK to the star in January is the same. (Assume the distance from Earth to the Sun does not change.) Not drawn to scale m∠LKM = 180 - 90.4, so m∠LKM = 89.6°. Since ∠LKM ≅ ∠M, △LMK is isosceles by the Converse of the Isosceles Triangle Theorem. Thus LK = LM = 4.0 × 10 13 km. 1. If the distance from Earth to a star in

September is 4.2 × 10 13 km, what is the distance from Earth to the star in March? Explain. E X A M P L E 2 Finding the Measure of an Angle Find each angle measure. A m∠C m∠C = m∠B = x° m∠C + m∠B + m∠A = 180 x + x + 38 = 180 2x = 142 x = 71 Thus m∠C = 71°. B m∠S Isosc. △ Thm. △ Sum Thm. Substitute the given values. Simplify and subtract 38 from both sides. Divide both sides by 2. m∠S = m∠R 2x° = (x + 30) ° Isosc. △ Thm. Substitute the given values. x = 30 Subtract x from both sides. Thus m∠S = 2x° = 2 (30) = 60°. Find each angle measure. 2b. m∠N 2a. m∠H The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Corollary 4-8-3 Equilateral Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equilateral, then it is equiangular. (equilateral △ → equiangular △) ∠A ≅ ∠B ≅ ∠C You will prove Corollary 4-8-3 in Exercise 36. 274 274 Chapter 4 Triangle Congruence �� Corollary 4-8-4 Equiangular Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equiangular, then it is equilateral. (equiangular △ → equilateral △) ̶̶ DE ≅ ̶̶ DF ≅ ̶̶ EF E X A M P L E 3 Using Properties of Equilateral Triangles You will prove Corollary 4-8-4 in Exercise 37. Find each value. A x △ABC is equiangular. (3x + 15) ° = 60° 3x = 45 x = 15 B t △JKL is equilateral. 4t - 8 = 2t + 1 2t = 9 Equilateral △ →

equiangular △ The measure of each ∠ of an equiangular △ is 60°. Subtract 15 from both sides. Divide both sides by 3. Equiangular △ → equilateral △ Def. of equilateral △ Subtract 2t and add 8 to both sides. t = 4.5 Divide both sides by 2. 3. Use the diagram to find JL. E X A M P L E 4 Using Coordinate Proof A coordinate proof may be easier if you place one side of the triangle along the x-axis and locate a vertex at the origin or on the y-axis. Prove that the triangle whose vertices are the midpoints of the sides of an isosceles triangle is also isosceles. Given: △ABC is isosceles. X is the mdpt. of ̶̶ AC. Z is the mdpt. of ̶̶ AB. ̶̶ BC. Y is the mdpt. of Prove: △XYZ is isosceles. Proof: Draw a diagram and place the coordinates of △ABC and △XYZ as shown. By the Midpoint Formula, the coordinates of X are ( 2a + 0 ) = (a, b), _____ 2 the coordinates of Y are ( 2a + 4a ) = (3a, b), and the coordinates of Z ______ 2 are ( 4a + 0 _____ 2 By the Distance Formula, XZ = √ YZ = √  ̶̶ XZ ≅ Since XZ = YZ,  2 2 = √  + (0 - b) (2a - a) (2a - 3a) 2 + (0 - b) 2 = √  a 2 + b 2. ̶̶ YZ by definition. So △XYZ is isosceles. ) = (2a, 0). a 2 + b 2, and, 2b + 0 _____ 2, 2b + 0 _____ 2, 0 + 0 ____ 2 4. What if...? The coordinates of △

ABC are A (0, 2b), B (-2a, 0), and C (2a, 0). Prove △XYZ is isosceles. 4- 8 Isosceles and Equilateral Triangles 275 275 �� THINK AND DISCUSS 1. Explain why each of the angles in an equilateral triangle measures 60°. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, draw and mark a diagram for each type of triangle. 4-8 Exercises Exercises KEYWORD: MG7 4-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw isosceles △JKL with ∠K as the vertex angle. Name the legs, base, and base angles of the triangle. 274 2. Surveying To find the distance QR across a river, a surveyor locates three points Q, R, and S. QS = 41 m, and m∠S = 35°. The measure of exterior ∠PQS = 70°. Draw a diagram and explain how you can find QR Find each angle measure. p. 274 3. m∠ECD 4. m∠K 5. m∠X 6. m∠ Find each value. p. 275 7. y 8. x 9. BC 10. JK. 275 11. Given: △ABC is right isosceles. X is the midpoint of ̶̶ AC. ̶̶ AB ≅ ̶̶ BC Prove: △AXB is isosceles. 276 276 Chapter 4 Triangle Congruence �� Independent Practice For See Exercises Example 12 13–16 17–20 21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 12. Aviation A plane is flying parallel � �  AC. When the to the ground along plane is at A, an air-traffic controller in tower T measures the angle to the plane as 40°. After the plane has traveled 2.4 mi

to B, the angle to the plane is 80°. How can you find BT? � �� Find each angle measure. 13. m∠E 14. m∠TRU 15. m∠F �� 16. m∠A Find each value. 17. z 18. y 19. BC 20. XZ 21. Given: △ABC is isosceles. P is the midpoint ̶̶ AB. Q is the midpoint of ̶̶ AC. of ̶̶ AB ≅ ̶̶ PC ≅ ̶̶ AC ̶̶ QB Prove: Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 22. An equilateral triangle is an isosceles triangle. 23. The vertex angle of an isosceles triangle is congruent to the base angles. 24. An isosceles triangle is a right triangle. 25. An equilateral triangle and an obtuse triangle are congruent. 26. Critical Thinking Can a base angle of an isosceles triangle be an obtuse angle? Why or why not? 4- 8 Isosceles and Equilateral Triangles 277 277 �� 27. This problem will prepare you for the Multi-Step TAKS � Prep on page 280. The diagram shows the inside view of the support structure of the back of a doghouse. ̶̶ PS ≅ a. Find m∠SPT. b. Find m∠PQR and m∠PRQ. ̶̶ PT, m∠PST = 71°, and m∠QPS = m∠RPT = 18°. ̶̶ PQ ≅ ̶̶ PR, � � � � Multi-Step Find the measure of each numbered angle. 28. 29. �� 30. Write a coordinate proof. Given: ∠B is a right angle in isosceles right △ABC. X is the midpoint of ̶̶ AC. ̶̶ BA ≅ ̶̶ BC Prove: △AXB ≅ △CXB 31. Estimation Draw the

figure formed by (-2, 1), (5, 5), and (-1, -7). Estimate the measure of each angle and make a conjecture about the classification of the figure. Then use a protractor to measure each angle. Was your conjecture correct? Why or why not? 32. How many different isosceles triangles have a perimeter of 18 and sides whose lengths are natural numbers? Explain. Multi-Step Find the value of the variable in each diagram. 33. 34. 35. Prove the Converse of the Isosceles Triangle Theorem. Navigation 36. Complete the proof of Corollary 4-8-3. ̶̶ AC ≅ ̶̶ ̶̶ BC AB ≅ Given: Prove: ∠A ≅ ∠B ≅ ∠C ̶̶ ̶̶ AC, a. AB ≅? by the Isosceles Triangle Theorem. ̶̶̶̶ ̶̶ BC, ∠A ≅ ∠B by b. Proof: Since ̶̶ AC ≅ Since By the Transitive Property of ≅, ∠A ≅ ∠B ≅ ∠C.?. Therefore ∠A ≅ ∠C by c. ̶̶̶̶?. ̶̶̶̶ The taffrail log is dragged from the stern of a vessel to measure the speed or distance traveled during a voyage. The log consists of a rotator, recording device, and governor. 37. Prove Corollary 4-8-4. 38. Navigation The captain of a ship traveling along  AB sights an island C at an angle of 45°. The captain measures the distance the ship covers until it reaches B, where the angle to the island is 90°. Explain how to find the distance BC to the island. 39. Given: △ABC ≅ △CBA Prove: △ABC is isosceles. 40. Write About It Write the Isosceles Triangle Theorem and its converse as a biconditional. 278 278 Chapter 4 Triangle Congruence �� 41. Rewrite the paragraph proof of the Hypotenuse-Leg (HL) Congruence Theorem as a two-column proof. Given: △ABC and △DEF are right triangles. ∠C and ∠F are

right angles. ̶̶ ̶̶ DF, and AC ≅ Prove: △ABC ≅ △DEF ̶̶ AB ≅ ̶̶ DE.  EF. Mark G so that FG = CB. Thus Proof: On △DEF draw ̶̶ DF and ∠C and ∠F are right angles. ̶̶ AC ≅ lines. Thus ∠DFG is a right angle, and ∠DFG ≅ ∠C. △ABC ≅ △DGF by SAS. ̶̶̶ ̶̶ DG ≅ DE by the Transitive Property. By the Isosceles Triangle Theorem ∠G ≅ ∠E. ∠DFG ≅ ∠DFE since right angles are congruent. So △DGF ≅ △DEF by AAS. Therefore △ABC ≅ △DEF by the Transitive Property. ̶̶ AB by CPCTC. ̶̶ DE as given. ̶̶̶ DG ≅ ̶̶ DF ⊥ ̶̶ AB ≅ ̶̶ FG ≅ ̶̶ CB. From the diagram, ̶̶ EG by definition of perpendicular 42. Lorena is designing a window so that ∠R, ∠S, ∠T, and ̶̶ VU ≅ ̶̶ VT, and m∠UVT = 20°. ∠U are right angles, What is m∠RUV? 10° 70° 20° 80° 43. Which of these values of y makes △ABC isosceles 15 1 _ 2 44. Gridded Response The vertex angle of an isosceles triangle measures (6t - 9) °, and one of the base angles measures (4t) °. Find t. CHALLENGE AND EXTEND 45. In the figure, ̶̶ JK ≅ Prove m∠MKL must also be x°. ̶̶ JL, and ̶̶̶ KM ≅ ̶̶ KL. Let m∠J = x°. 46. An equilateral △ABC is placed on a coordinate plane. Each side length measures 2a. B is at the origin, and C is at (2a, 0). Find the coordinates of A. 47. An isosceles triangle has coordinates A (0,

0) and B (a, b). What are all possible coordinates of the third vertex? SPIRAL REVIEW Find the solutions for each equation. (Previous course) 48. x 2 + 5x + 4 = 0 49. x 2 - 4x + 3 = 0 50. x 2 - 2x + 1 = 0 Find the slope of the line that passes through each pair of points. (Lesson 3-5) 51. (2, -1) and (0, 5) 52. (-5, -10) and (20, -10) 53. (4, 7) and (10, 11) 54. Position a square with a perimeter of 4s in the coordinate plane and give the coordinates of each vertex. (Lesson 4-7) 4- 8 Isosceles and Equilateral Triangles 279 279 �� SECTION 4B Proving Triangles Congruent Gone to the Dogs You are planning to build a doghouse for your dog. The pitched roof of the doghouse will be supported by four trusses. Each truss will be an isosceles triangle with the dimensions shown. To determine the materials you need to purchase and how you will construct the trusses, you must first plan carefully. 1. You want to be sure that all four trusses are exactly the same size and shape. Explain how you could measure three lengths on each truss to ensure this. Which postulate or theorem are you using? 2. Prove that the two triangular halves of the truss are congruent. 3. What can you say about ̶̶ DB? Why is this true? and Use this to help you find the ̶̶ AC, and lengths of ̶̶ DB, ̶̶ AD, ̶̶ BC. ̶̶ AD 4. You want to make careful plans on a coordinate plane before you begin your construction of the trusses. Each unit of the coordinate plane represents 1 inch. How could you assign coordinates to vertices A, B, and C? 5. m∠ACB = 106°. What is the measure of each of the acute angles in the truss? Explain how you found your answer. 6. You can buy the wood for the trusses at the building supply store for $0.80 a foot. The store sells the wood in 6-foot lengths only. How much will you have to spend to get

enough wood for the 4 trusses of the doghouse? (Hint: You need to use the Pythagorean Theorem to find the two unknown side lengths of each truss.) 280 280 Chapter 4 Triangle Congruence �� Quiz for Lessons 4-4 Through 4-8 4-4 Triangle Congruence: SSS and SAS SECTION 4B 1. The figure shows one tower and the cables of a suspension bridge. ̶̶ AC ≅ ̶̶ BC, use SAS to explain why △ACD ≅ △BCD. Given that 2. Given: ̶̶ JK bisects ∠MJN. ̶̶ MJ ≅ ̶̶ NJ Prove: △MJK ≅ △NJK 4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 3. △RSU and △TUS 4. △ABC and △DCB Observers in two lighthouses K and L spot a ship S. 5. Draw a diagram of the triangle formed by the lighthouses and the ship. Label each measure. K to L K to S L to S Bearing E N 58° E N 77° W 6. Is there enough data in the table to pinpoint Distance 12 km?? the location of the ship? Why? 4-6 Triangle Congruence: CPCTC 7. Given: ̶̶ CD ǁ ̶̶ BE, Prove: ∠D ≅ ∠B ̶̶ DE ǁ ̶̶ CB 4-7 Introduction to Coordinate Proof 8. Position a square with side lengths of 9 units in the coordinate plane 9. Assign coordinates to each vertex and write a coordinate proof. ̶̶ Given: ABCD is a rectangle with M as the midpoint of AB. N is the midpoint of Prove: The area of △AMN is 1 __ 8 the area of rectangle ABCD. ̶̶ AD. 4-8 Isosceles and Equilateral Triangles Find each value. 10. m∠C 11. ST 12. Given: Isosceles △JKL has coordinates J (0, 0), K (2a, 2b), and L (4a, 0)

. M is the midpoint of ̶̶ JK, and N is the midpoint of ̶̶ KL. Prove: △KMN is isosceles. Ready to Go On? 281 281 �� EXTENSION EXTENSION Proving Constructions Valid TEK G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.3.B Objective Use congruent triangles to prove constructions valid. When performing a compass and straight edge construction, the compass setting remains the same width until you change it. This fact allows you to construct a segment congruent to a given segment. You can assume that two distances constructed with the same compass setting are congruent. The steps in the construction of a figure can be justified by combining the assumptions of compass and straightedge constructions and the postulates and theorems that are used for proving triangles congruent. You have learned that there exists exactly one midpoint on any line segment. The proof below justifies the construction of a midpoint. E X A M P L E 1 Proving the Construction of a Midpoint Given: diagram showing the steps in the construction Prove: M is the midpoint of ̶̶ AB. To construct a midpoint, see the construction of a perpendicular bisector on p. 172. Proof: Statements Reasons 1. Draw ̶̶ AC, ̶̶ BC, ̶̶̶ AD, and ̶̶ BD. ̶̶ AC ≅ ̶̶ CD ≅ ̶̶ BC ≅ ̶̶ CD 2. 3. ̶̶̶ AD ≅ ̶̶ BD 4. △ACD ≅ △BCD 5. ∠ACD ≅ ∠BCD ̶̶̶ CM ≅ ̶̶̶ CM 6. 7. △ ACM ≅ △BCM ̶̶̶ AM ≅ ̶̶̶ BM 8. 9. M is the midpt. of ̶̶ AB. 1. Through any two pts. there is exactly one line. 2. Same compass setting used 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of ≅ 7. SAS Steps 2, 5, 6 8. CPCTC 9. Def. of mdpt. 1. Given:

above diagram Prove:   CD is the perpendicular bisector of ̶̶ AB. 282 282 Chapter 4 Triangle Congruence �� E X A M P L E 2 Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: ∠A ≅ ∠D To review the construction of an angle congruent to another angle, see page 22. Proof: Since there is a straight line through any two points, you can draw ̶̶ EF. The same compass setting was used to construct ̶̶ AB, ̶̶ DE. The same compass setting was used ̶̶ DF ≅ ̶̶ AC, ̶̶ DF, ̶̶ BC ≅ ̶̶ EF. Therefore △BAC ≅ △EDF by SSS, ̶̶ BC and ̶̶ DE, so and to construct and ∠A ≅ ∠D by CPCTC. ̶̶ AC ≅ ̶̶ BC and ̶̶ AB ≅ ̶̶ EF, so 2. Prove the construction for bisecting an angle. (See page 23.) EXTENSION Exercises Exercises Use each diagram to prove the construction valid. 1. parallel lines 2. a perpendicular through a point not (See page 163 and page 170.) on the line (See page 179.) 3. constructing a triangle using SAS 4. constructing a triangle using ASA (See page 243.) (See page 253.) Extension 283 283 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute triangle.............. 216 CPCTC..................... 260 isosceles triangle........... 217 auxiliary line............... 223 equiangular triangle........ 216 legs of an isosceles triangle.. 273 base....................... 273 equilateral triangle......... 217 obtuse triangle............. 216 base angle...

............... 273 exterior.................... 225 remote interior angle....... 225 congruent polygons......... 231 exterior angle.............. 225 right triangle............... 216 coordinate proof............ 267 included angle.............. 242 scalene triangle............. 217 corollary................... 224 included side............... 252 triangle rigidity............. 242 corresponding angles....... 231 interior.................... 225 vertex angle................ 273 corresponding sides......... 231 interior angle............... 225 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a triangle with at least two congruent sides. ̶̶̶̶ 2. A name given to matching angles of congruent triangles is?. ̶̶̶̶ 3. A(n)? is the common side of two consecutive angles in a polygon. ̶̶̶̶ 4-1 Classifying Triangles (pp. 216–221) TEKS G.1.A E X A M P L E EXERCISES ■ Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. 4-2 Angle Relationships in Triangles (pp. 223–230) TEKS G.1.A, G.2.B E X A M P L E ■ Find m∠S. EXERCISES Find m∠N. 6. 12x = 3x + 42 + 6x 12x = 9x + 42 3x =

42 x = 14 m∠S = 6 (14) = 84° 284 284 Chapter 4 Triangle Congruence 7. In△LMN, m∠L = 8x °, m∠M = (2x + 1) °, and m∠N = (6x - 1) °. �� 4-3 Congruent Triangles (pp. 231–237) TEKS G.2.B, G.10.B E X A M P L E EXERCISES ■ Given: △DEF ≅ △JKL. Identify all pairs of Given: △PQR ≅ △XYZ. Identify the congruent congruent corresponding parts. Then find the value of x. ̶̶ DE ≅ ∠F ≅ ∠L, The congruent pairs follow: ∠D ≅ ∠J, ∠E ≅ ∠K, ̶̶ EF ≅ Since m∠E = m∠K, 90 = 8x - 22. After 22 is added to both sides, 112 = 8x. So x = 14. ̶̶ KL, and ̶̶ DF ≅ ̶̶ JK, ̶̶ JL. corresponding parts. 8. ̶̶ PR ≅? ̶̶̶̶ 9. ∠Y ≅? ̶̶̶ Given: △ABC ≅ △CDA Find each value. 10. x 11. CD 4-4 Triangle Congruence: SSS and SAS (pp. 242–249) TEKS G.2.A, G.3.B, G.3.E, G.10.B ■ Given: E X A M P L E S ̶̶ ̶̶ UT, and RS ≅ ̶̶ ̶̶ VS ≅ VT. V is the midpoint of ̶̶ RU. Prove: △RSV ≅ △UTV Proof: Statements Reasons EXERCISES 12. Given: ̶̶ AB ≅ ̶̶ DB ≅ Prove: △ADB ≅ △DAE ̶̶ DE, ̶̶ AE 13. Given: ̶̶ GJ bisects ̶̶ FH bisects and Prove: △FGK ≅ △HJK ̶̶

FH, ̶̶ GJ. ̶̶ RS ≅ ̶̶ VS ≅ ̶̶ UT ̶̶ VT 1. 2. 3. V is the mdpt. of ̶̶ UV ̶̶ RV ≅ 4. ̶̶ RU. 1. Given 2. Given 3. Given 4. Def. of mdpt. 14. Show that △ABC ≅ △XYZ when x = -6. 5. △RSV ≅ △UTV 5. SSS Steps 1, 2, 4 ■ Show that △ADB ≅ △CDB when s = 5. AB = s 2 - 4s AD = 14 - 2s 15. Show that △LMN ≅ △PQR when y = 25. = 5 2 - 4 (5 ) = 5 ̶̶ BD by the Reflexive Property. = 14 - 2 (5 ) = 4 ̶̶ CB. So △ADB ≅ △CDB by SSS. ̶̶ BD ≅ ̶̶ AB ≅ and ̶̶ AD ≅ ̶̶ CD Study Guide: Review 285 285 �� 4-5 Triangle Congruence: ASA, AAS, and HL (pp. 252–259) TEKS G.1.A, G.1.B, G.2.A, E X A M P L E S ■ Given: B is the midpoint of ̶̶ AE. ∠A ≅ ∠E, ∠ABC ≅ ∠EBD Prove: △ABC ≅ △EBD EXERCISES 16. Given: C is the midpoint ̶̶ AG. of ̶̶ HA ǁ ̶̶ GB Prove: △HAC ≅ △BGC G.3.B, G.3.C, G.3.E, G.9.B, G.10.B Proof: Statements Reasons 1. ∠A ≅ ∠E 2. ∠ABC ≅ ∠EBD 3. B is the mdpt. of ̶̶ EB ̶̶ AB ≅ 4. ̶̶ AE. 1. Given 2. Given 3. Given 4. Def