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10. Draw a two-point perspective view. 11. Determine whether the drawing represents the given object. Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 12. 13. 14. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 15. (2, 4, 9) and (3, 7, 2) 16. (0, 0, 0) and (4, 7, -4) 17. (5, 1, 0) and (0, 3, 4) Find the lateral area and surface area of each figure. Give exact answers, using π if necessary. 18. 19. 20. Lesson 10-2 Lesson 10-3 Lesson 10-4 21. The dimensions of a cylinder with r = 9 cm and h = 12 cm are multiplied by 1 _. 3 Describe the effect on the surface area. S22S22 TEKS TAKS Practice �� Lesson 10-5 Find the lateral area and surface area of each figure. Give exact answers, using π if necessary. 22. 23. 24. 25. The dimensions of a square pyramid with B = 64 in 2 and h = 7 in. are tripled. Describe the effect on the surface area. 26. The dimensions of a right cone with r = 14 in. and ℓ = 24 in. are multiplied by 1 _. 2 Describe the effect on the surface area. Lesson 10-6 Find the volume of each figure. Round to the nearest tenth. 27. 28. 29. 30. The dimensions of a prism with B = 14 cm 2 and h = 8 cm are doubled. Describe the effect on the volume. 31. The dimensions of a cylinder with r = 6 cm and h = 4 cm are multiplied by 2 _. 3 Describe the effect on the volume. Lesson 10-7 Find the volume of each figure. Round to the nearest tenth. 32. 33. 34. 35. The dimensions of a cone with r = 8 cm and ℓ = 17 cm are multiplied by 1 _. 2 Describe the effect on the volume. 36. The dimensions of a pyramid with B = 128 m m 2 and h = 56 mm are tripled. Describe the effect on
the volume. Lesson 10-8 Find the surface area and volume of each figure. Give your answers in terms of π. 37. 38. 39. 40. The radius of a sphere with r = 24 cm is multiplied by 1 _ 3 surface area and volume.. Describe the effect on the 41. The radius of a sphere with r = 15 mm is multiplied by 4. Describe the effect on the surface area and volume. TEKS TAKS Practice S23 S23 �� Chapter 11 Skills Practice Lesson 11-1 Identify each line or segment that intersects each circle. Identify each line or segment that intersects each circle. 1. 1. 2. Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 3. 4. The segments in each figure are tangent to the circle. Find each length. 5. PQ 6. WZ Lesson 11-2 Find each measure. Round to the nearest tenth, if necessary. 7. m ⁀ FB 8. PQ 9. ⊙T ≅ ⊙W. Find m∠VWX. 10. BD Lesson 11-3 Find the area of each sector or segment. Round to the nearest tenth. 11. 12. S24S24 TEKS TAKS Practice �� Find each arc length. Give your answers in terms of π and rounded to the nearest tenth. 13. 14. Lesson 11-4 Find each measure or value. Round to the nearest tenth, if necessary. 16. x 15. m∠ABD 17. x 18. angle measures of HJKL 19. m ⁀ DF Lesson 11-5 20. m∠JMK 21. m∠RTQ 22. x 23. m∠AFE 24. m ⁀ GL Lesson 11-6 Find the value of the variable. Round to the nearest tenth, if necessary. 25. 26. 27. Lesson 11-7 Write the equation of each circle. 28. ⊙A with center A (2, -3) and radius 6
29. ⊙B that passes through (3, 4) and has center B (-2, 1) Graph each equation. 30. (x + 3) 2 + (y - 4) 2 = 1 31. x 2 + (y + 4) 2 = 16 TEKS TAKS Practice S25 S25 �� Chapter 12 Skills Practice Lesson 12-1 Copy each figure and the line of reflection. Draw the reflection of the figure Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 1. 1. 2. Reflect the figure with the given vertices across the given line. 3. A (-4, 1), B (2, 4), C (3, -2) ; x-axis 4. D (3, 1), E (2, 4), F (-2, 2), G (2, -2) ; y = x Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 5. 6. Translate the figure with the given vertices along the given vector. 7. A (-2, 1), B (4, 3), C (2, -2) ; 〈2, 3〉 8. D (-1, 3), E (2, 4), F (3, 3), G (3, -2) ; 〈2, -2〉 Copy each figure and the angle of rotation. Draw the rotation of the figure about the point P by m∠A. 9. 10. Rotate the figure with the given vertices about the origin using the given angle of rotation. 11. A (2, 3), B (-2, 1), C (1, -1) ; 90° 12. D (-2, 3), E (2, 4), F (3, 1), G (-2, 2) ; 180° Draw the result of each composition of isometries. 13. Translate △ABC along  v and then reflect it across line ℓ. 14. Reflect △DEF across line m and then translate it along  w. Lesson 12-2 Less
on 12-3 Lesson 12-4 15. Copy the figure and draw two lines of reflection that produce an equivalent transformation. S26S26 TEKS TAKS Practice �� Lesson 12-5 Describe the symmetry of each figure. Copy the shape and draw all lines of symmetry. If there is rotational symmetry, give the angle and order. 16. 18. 17. Tell whether each figure has plane symmetry, symmetry about an axis, or neither. 19. 21. 20. Lesson 12-6 Copy the given figure and use it to create a tessellation. 22. 23. 24. Classify each tessellation as regular, semiregular, or neither. 25. 26. 27. Lesson 12-7 Copy each figure and center of dilation P. Draw the image of the figure under a dilation with the given scale factor. 28. scale factor: 3 29. scale factor: - 2 _ 3 Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 30. A (1, 3), B (1, 5), C (4, 3) ; scale factor 2 31. E (-2, 2), F (2, 4), G (4, -2) ; scale factor - 1 _ 2 TEKS TAKS Practice S27 S27 �� Chapter 1 Applications Practice Athletics Use the following information for Exercises 1–3. During gym class, a teacher notices the following. Decide if each resembles a point, segment, ray, or line. (Lesson 1-1) 1. Kyle starts running in a straight line. Suppose he does not stop running. 2. Agnes runs a quarter-mile in a straight line. 3. Jimmy stands perfectly still. Travel Use the following information for Exercises 4–6. The Perez family is driving from Austin, Texas, to Dallas, Texas. The city of Waco is the approximate midpoint between these two cities. It is 102 miles from Austin to Waco. (Lesson 1-2) 9. Entomology Because the insect is symmetrical, ∠1 ≅ ∠4 and ∠2 ≅ ∠3. Also, ∠1 and ∠2 are complementary, and ∠3 and ∠4 are complementary. If m∠1 = 48.5
°, find m∠2, m∠3, and m∠4. (Lesson 1-4) � � � � Architecture Use the following information for Exercises 10 and 11. The bricks used to make a building are one-fourth as tall as they are wide, and the bricks are 2.25 inches tall. (Lesson 1-5) 4. What is the total distance from Austin to 10. What is the area of the largest face of each Dallas? brick? 5. The approximate midpoint from Waco to Dallas is Milford. What is the distance from Austin to Milford? 11. A certain exterior wall is 33 bricks long and 20 bricks tall. What is the area of the wall in square inches? 6. The Perez family averages 64 miles per hour. About how long will the entire drive take? Probability Use the following information for Exercises 7 and 8. In a carnival game, each contestant spins the wheel and wins the prize indicated by the color. (Lesson 1-3) 12. Sports A football coach has his team run sprints diagonally across a football field. If the field is 120 yards long and 160 feet wide, what is the distance they run? Write your answer to the nearest hundredth of a foot. (Lesson 1-6) 13. Crafts The picture below shows half of a stenciled design. The full design should resemble a sun. Name two transformations that can be performed on the image so that the image and its preimage form a complete picture. Be as specific as possible, referring to L and P. (Lesson 1-7) 7. Using a protractor, measure each angle on the wheel. 8. Since there are 360° in a circle, the probability of the wheel landing on a given color is the number of degrees in the angle divided by 360°. Find the probability of the wheel landing on each prize. Express your answer as a fraction in lowest terms. S28S28 TEKS TAKS Practice �� Chapter 2 Applications Practice 1. Health Mike collected the following data about the heights of twelve students in his tenth-grade class. Use the table to make a conjecture about the heights of boys and girls in the tenth grade. (Lesson 2-1) Height (in.) of Tenth-Grade Students Boys Girls 70 67 71 64 68 64 67 65 70 68 67 66 2. Government Voter Turnout Year 1998
1996 Voters 12,530 Presidential elections are held every four years. Elections for senators are held every two years. So in years not divisible by 4, only Senate seats are up for election. The table shows voter turnout for a small town during recent election years. Make a conjecture based on the data. (Lesson 2-1) 15,210 14,380 8,750 7,370 2000 2002 2004 3. Biology Write the converse, inverse, and contrapositive of the conditional statement “If an animal is a fish, then it swims in salt water.” Find the truth value of each. (Lesson 2-2) 4. Gardening Write the converse, inverse, and contrapositive of the conditional statement “If a plant is watered, then it will grow.” Find the truth value of each. (Lesson 2-2) 5. Sports Determine if the conjecture is valid by the Law of Detachment. (Lesson 2-3) Given: If you participate in a triathlon, then you run, swim, and bike. Margie runs, swims, and bikes. Conjecture: Margie participates in a triathlon. 6. Health Students are required to have certain immunizations before attending school to prevent the spread of disease. Write the conditional statement and converse within the biconditional “Students can attend public school if and only if they have the required immunizations.” (Lesson 2-4) 7. Weather Hurricanes are assigned category numbers to describe the amount of flooding and wind damage they are likely to cause. Write the statement “If a hurricane has sustained winds of more than 155 miles per hour, then it is Category 5” as a biconditional statement. (Lesson 2-4) 8. Athletics The equation c = 5w + 25 relates the number of workouts w to the cost c of a weight training group. If Matthew plans to spend $200 on weight training, how many workouts can he participate in? Solve the equation for w and justify each step. (Lesson 2-5) 9. Nutrition Rick has allotted himself 200 Calories for his evening snack, which consists of a glass of milk and crackers. A glass of milk has 110 Calories, and each cracker has 15 Calories. The equation s = 110 + 15c relates the number of crackers c to the total number of Calories s in Rick’s evening snack. How many crackers can Rick
have? Solve the equation for c and justify each step. (Lesson 2-5) 10. Travel On a city map, the library, post office, and police station are collinear points in that order. The distance from the library to the post office is 2.3 miles. The distance from the post office to the police station is 5.1 miles. Which theorem can you use to conclude that the distance from the library to the police station is 7.4 miles? (Lesson 2-6) 11. Recreation Kyle is making a kite from the pattern below by cutting four triangles from different pieces of material. Write a paragraph proof to show that m∠3 = 90°. (Lesson 2-7) Given: ∠1 ≅ ∠2 Prove: m∠3 = 90° TEKS TAKS Practice S29 S29 �� Chapter 3 Applications Practice 1. Recreation A scuba diver leaves a flag on 5. Transportation The railroad ties in the the surface of the water to alert boaters of his location. Describe two parallel lines and a transversal in the flag. (Lesson 3-1) diagram are all parallel. m∠1 = 19x - 5 and m∠ 2 = 4x + 5y. Find x and y so that the ties are all perpendicular to the tracks. (Lesson 3-4) 2. Carpentry In the stairs shown, the horizontal treads and the vertical risers are all parallel. m∠1 = (14x + 6) ° and m∠2 = (19x - 24) °. Find x. (Lesson 3-2) 6. Art The sides of a picture frame are cut so that the opposite sides of the frame are parallel and the consecutive sides are perpendicular. Find the values of x and y in the diagram. (Lesson 3-4) 3. Transportation The train tracks shown cross the street lanes. The lanes of the street are parallel. Find x in the diagram. (Lesson 3-3) 7. Recreation At 1:00 P.M., a boat on a river passes a point that is 3 miles from a lodge. At 5:30 P.M., the boat passes a point that is 8 miles from the lodge. Graph the line that represents the boat’s distance from the lodge. Find and interpret the slope of the line. (Lesson 3-5) 8. Sports A marathon runner
runs 10 miles by 3:00 P.M. and 25 miles by 4:30 P.M. Graph the line that represents her distance run. Find and interpret the slope of the line. (Lesson 3-5) 9. Business A cab company charges $8 per ride plus $0.25 per mile. Another cab company charges $5 per ride plus $0.35 per mile. For how many miles will two cab rides cost the same amount? (Lesson 3-6) 4. Sports At a track meet, the starting blocks are placed along a line that is a transversal to the lanes. m∠1 = 12x - 8, m∠2 = 8x + 12, and x = 5. Show that the lines between the lanes are parallel. (Lesson 3-3) � � � 10. Food A pizza parlor is catering a school event. Pete’s Pizza charges $85 for the first 20 students and $5 for each additional student. Polly’s Pizza charges $125 for the first 20 students and $3 for each additional student. For how many students will the pizza parlors cost the same? (Lesson 3-6) S30S30 TEKS TAKS Practice �� Chapter 4 Applications Practice 1. Camping Three poles are used to create the frame for a tent. The front of the tent is an isosceles triangle with the base is 1.5 times the length of the sides. The perimeter of the triangle is 21 ft. Find each side length. (Lesson 4-1) ̶̶ BC. The length of ̶̶ AB ≅ 2. Geography The universities in Durham, Chapel Hill, and Raleigh, North Carolina form what is known as the Research Triangle. Use the map to find the measure of the angle whose vertex is at Durham. (Lesson 4-2) �� 3. Business Oil derricks are used as supports for oil drilling equipment. Use the diagram to prove the following. (Lesson 4-3) 6. Surveying To find the distance AB across a lake, first locate point C. Then measure the distance from C to B. Locate point D the same distance from C as B, but in the opposite direction. Then measure the distance from C to A and locate point E in a similar manner. What is the distance AB across the lake? (Less
on 4-6) 7. The first step in creating a Sierpinski triangle is to connect the midpoints of the sides of a triangle as shown. (Lesson 4-7) Given: ̶̶̶ ̶̶ ̶̶ AB ≅ HB ≅ HG, ∠GAB ≅ ∠BHG, ∠AGB ≅ ∠HBG ̶̶ AG Prove: △AGB ≅ △HBG 4. Sports A kite is made up of two pairs of congruent triangles. Use SAS to explain why △ABD ≅ △CBD. (Lesson 4-4) 5. Recreation A student is estimating the height of a water slide. From a certain distance, the angle from where he is standing to a point on the highest part of the slide is 35°. From a distance 200 m closer, the same angle is 45°. a. Draw a triangle with the point at the top of the slide as one vertex, and the points where the measurements were taken as the other vertices. b. Which postulate or theorem can be used to show that this triangle is uniquely determined? (Lesson 4-5) Given: Equilateral △ABC, D is the midpoint ̶̶ AB, E is the midpoint of ̶̶ AC, and F is of the midpoint of ̶̶ BC. Prove: The area of △DEF is 1 __ 4 the area of △ABC. 8. Recreation A boat is sailing parallel to the coastline along   XY. When the boat is at X, the measure of the angle from the lighthouse W to the boat is 30°. After the boat has traveled 5 miles to Y, the angle from the lighthouse to the boat is 60°. How can you find WY? (Lesson 4-8) TEKS TAKS Practice S31 S31 �� Chapter 5 Applications Practice 1. Building The guy wires ̶̶ AB and ̶̶ CB supporting a cell phone tower are congruent and are equally spaced from the base of the tower. How do these wires ensure that the cell phone tower is perpendicular to the ground? (Lesson 5-1) 2. Safety City planners want to relocate their town’s firehouse so that it is the same distance from the three main streets of the town
. Draw a sketch to show where the firehouse should be positioned. Justify your sketch. (Lesson 5-2) 3. Safety A lifeguard needs to watch three areas of a water park. Draw a sketch to show where she should stand to be the same distance from all the swimmers. Justify your sketch. (Lesson 5-2) 4. Art An artist is designing a sculpture composed of a pedestal with a triangular top. The vertices of the top are A (-4, 2), B (2, 4), and C (4, -3). Where should the artist attach the pedestal so that the triangle is balanced? (Lesson 5-3) 5. Measurement City engineers plan to build a bridge across the pond shown. What will be the length of the bridge, GH? (Lesson 5-4) � �� S32S32 TEKS TAKS Practice Engineering Use the following information for Exercises 6 and 7. Playground engineers are planning a sidewalk that will connect the swings, seesaw, and slide. (Lesson 5-5) 6. If the angle at the swings is the largest, which portion of the sidewalk will be the longest? 7. The distance from the swings to the seesaw is 37 ft. Can the lengths of the other sides be 40 ft and 50 ft? Explain. 8. Geography The cities of Allenville, Baytown, College City, and Dean Park are shown on the map. Baytown and Dean Park are each 30 miles from College City. Which city is closer to Allenville: Baytown or Dean Park? (Lesson 5-6) 9. Mark is late for school. He usually goes around the park so he can walk along the water. Today he decides to cut through the park. About how many feet does he save by going through the park? (Lesson 5-7) 10. Sports A baseball diamond is a square with a side length of 90 ft. What is the distance from first base to third base? (Lesson 5-8) 11. Recreation Haley, who is 5 ft tall, is flying a kite on 100 ft of string. How high is the kite? (Lesson 5-8) ��
�� Chapter 6 Applications Practice 1. Safety A stop sign is in the shape of a regular octagon. What is the value of x? (Lesson 6-1) Design Use the following information for Exercises 7–9. 2. Hobbies Nancy is planting a garden shaped like a regular pentagon. She bought metal edging to surround the garden and prevent weeds. What angle should the edging form at the vertices of the garden? (Lesson 6-1) Fishing Use the following information for Exercises 3–5. The hinges for the trays in a tackle box form parallelograms to ensure that the trays stay parallel to the base of the box. In ABCD, AB = 21 in., AE = 9 in., and m∠BCD = 125°. Find each measure. (Lesson 6-2) 3. DC 4. EC 5. m∠ADC 6. Design A glide rocker uses hinged parallelograms to move the chair back and forth. In ABCD, AB = DC, and AD = BC. ̶̶ BC, The sides of the parallelogram, rotate together to move the chair. Why is ABCD always a parallelogram? (Lesson 6-3) ̶̶ AD and When extended, the legs of a folding table must form a rectangle so the tabletop is parallel to the ground. Given that JK = 48 in. and KN = 36 in., find each length. (Lesson 6-4) 7. JM 8. JN 9. NM 10. Hobbies Elise is creating a decorative page for her scrapbook. She has a piece of ribbon that is 12 inches long. She wants to outline a rhombus with the ribbon. How can Elise cut the ribbon to ensure that the final shape is a rhombus? (Lesson 6-5) 11. Carpentry Luke is cutting a rectangular window frame. The dimensions of the window are to be 3 feet by 4 feet. What should the diagonal of the frame measure so that the window is rectangular? (Lesson 6-5) 12. Hobbies Addie is making a kite with diagonals of 32 inches and 18 inches. She wants to put a ribbon around the edge of the kite. She will add an 8-foot tail to the kite, made of the same ribbon. If ribbon can be purchased in packages of 3 yards, how many packages
should she buy for the entire project? (Lesson 6-6) 13. Carpentry Aaron is building a shadow box for his baseball memorabilia. The shadow box will be in the shape of a trapezoid, as shown below. The wood for the box costs $1.59 per foot. Estimate the cost of the lumber. (Lesson 6-6) TEKS TAKS Practice S33 S33 �� Chapter 7 Applications Practice 9. Geography Riverside Park has campsites available for rent. Lot A has 50 ft of street frontage and 80 ft of river frontage. Find the river frontage for lots B, C, and D. (Lesson 7-4) 10. Architecture An amphitheater is being built according to the design shown. If the total footage on the right of the rows of seats is 232.5 ft, find the length of each section. (Lesson 7-4) 11. Jake wants to know the height of the oak tree in his front yard. He measured his height as 68 inches and his shadow as 34 inches. At the same time, the tree has a shadow of 5.5 feet. How tall is the tree? (Lesson 7-5) 12. Recreation The kiddie pool and the lap pool at Centerville Park are similar rectangles. The lap pool measures 25 ft wide by 48 ft long. The kiddie pool is 8 ft long. How wide is the kiddie pool to the nearest tenth? (Lesson 7-5) 13. Melissa is enlarging her 4-by-6 photo by 150%. Find the coordinates of the enlarged photo. (Lesson 7-6) Hobbies Use the following information for Exercises 1–4. Jason and Matthew share 210 CDs. The ratio between Matthew’s CDs and Jason’s CDs is 4:3. (Lesson 7-1) 1. Write a proportion that can be used to find the number of CDs each one has. 2. How many CDs does Jason have? 3. The number of Matthew’s CDs is what fraction of the total number of CDs? 4. Jason wants to have the most CDs. What is the least number of CDs he would have to purchase to have more than Matthew? 5. Carpentry Ava’s dollhouse is a scale model of a castle. The great room of
the castle has a width of 40 ft and a length of 50 ft. The width of the great room of the dollhouse is 8 in. What is the length of the great room of the dollhouse? (Lesson 7-1) 6. Travel A map is a scale model of a real city. The scale on the map is 1 in.:30 mi. Two cities are 165 mi apart. How far apart will the cities be on the map? (Lesson 7-2) 7. Recreation The sails on the sailboat below have the given dimensions. Use similar triangles to prove △ABC ∼ △DEF. (Lesson 7-3) � � � � � � �� 8. Graphics A photograph shows a smaller version of the real item. The height of the Washington Monument is approximately 555 ft. The monument in a photo is 5 in. tall. What is the scale factor of the actual monument to the monument in the photo? (Lesson 7-3) S34S34 TEKS TAKS Practice �� Chapter 8 Applications Practice 1. Diving To estimate the height of a diving platform, a spectator stands so that his lines of sight to the top and bottom of the platform form a right angle as shown. The spectator’s eyes are 5 ft above the ground. He is standing 15 ft from the diving platform. How high is the platform? (Lesson 8-1) 6. Safety A lifeguard sees a swimmer struggling in the water at an angle of depression of 15°. The stand is 10 feet tall. What is the horizontal distance from the stand to the swimmer? Round to the nearest foot. (Lesson 8-4) �� 2. Recreation A neighborhood park has a 15-foot-long space available to install a playground slide. If the maximum height of the slide is 6 ft, what are the lengths of the slide x and ladder y that should be installed? Round to the nearest tenth of a foot. (Lesson 8-1) 3. Building The escalator at the mall forms a 35° angle with the floor. The vertical distance from the bottom of the escalator to the top is 25 ft. How long is the escalator? Round to the nearest foot. (Lesson 8-2) 4. Sports A 3-foot-long skateboard ramp forms a 40° angle with the ground. How far
above the ground is the end of the ramp? Round to the nearest foot. (Lesson 8-2) 5. Running A race includes a 0.25-mile hill on which runners travel from 510 ft of elevation to 570 ft of elevation. What angle does the hill form? Round to the nearest degree. (Lesson 8-3) 7. Aviation A helicopter pilot flying at an altitude of 1200 ft sees two landing pads directly in front of him. The angle of depression to the first landing pad is 40°. The angle of depression to the second pad is 28°. What is the distance between the two pads? Round to the nearest foot. (Lesson 8-4) 8. Carpentry Sean is creating a triangular frame from three wooden dowels, which are 18 in., 12 in., and 15 in. long. What are the measures of each angle of the triangle? Round to the nearest degree. (Lesson 8-5) 9. Sports To estimate the width of the sand trap on a golf course, Matthew locates three points and measures the distances shown. What is the width, XZ, of the sand trap to the nearest foot? (Lesson 8-5) 10. Recreation Jill swims due east across a river at 2 mi/h. The river is flowing north at 1.5 mi/h. What are Jill’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. (Lesson 8-6) TEKS TAKS Practice S35 S35 �� Chapter 9 Applications Practice 1. Recreation Kathy is making a kite with diagonals of lengths 30 inches and 20 inches. How many square inches of fabric will she need? (Lesson 9-1) Agriculture Use the following information for Exercises 2 and 3. An acre is 43,560 square feet. (Lesson 9-1) 2. If a one-acre piece of land is a rectangle with a base of 100 ft, what is its height? 3. If a one-acre piece of land is a square, what is the length of each side? Round to the nearest tenth. 4. The garden shown is a regular hexagon with a circular fountain at the center. What is the area of the garden? Round to the nearest square foot. (Lesson 9-2) �� 5. Food A bakery has chees
ecake pans with three diameters: 18 cm, 22 cm, and 26 cm. Find the area of the bottom of each pan. Round to the nearest square centimeter. (Lesson 9-2) 6. Recreation A track for a toy car is a 2 ft by 2 ft square with a semicircle at each end. What is the distance around the track? Round to the nearest foot. (Lesson 9-3) 7. Art Jonas is painting the shape shown on his ceiling. If a quart of paint covers 75 square feet, will one quart be enough to paint the entire shape? Explain. (Lesson 9-3) Transportation Use the following information for Exercises 8 and 9. The graph shows the speed of a car versus time. The base of each square on the graph represents 10 minutes, and the height represents 10 miles per hour. (Lesson 9-4) 8. What is the area of one square on the graph? 9. Estimate the shaded area of the graph. 10. Art Rasha is cutting a mat for a poster with an area of 480 in 2. To find the dimensions of the mat, she multiplies the dimensions of the poster by 1.2. To find the dimensions of the opening, she multiplies the dimensions of the poster by 0.9. What is the area of the remaining part of the mat? (Lesson 9-5) 11. Food A restaurant sells two sizes of pizzas. The smaller pizza has a 12-inch diameter. If the area of the larger pizza is twice the area of the smaller pizza, what is the diameter of the larger pizza? Round to the nearest inch. (Lesson 9-5) 12. Transportation A commuter train stops at a station every 3 minutes and stays at the station for 20 seconds. If you arrive at the station at a random time, what is the probability that you will have to wait more than one minute for a train? Round to the nearest hundredth. (Lesson 9-6) 13. Sports A skydiver is delivering the game ball for a baseball game. Suppose he lands at a random point on the field. What is the probability that he will not land on the pitcher’s mound? Round to the nearest hundredth. (Lesson 9-6) �� S36S36 TEKS TAKS Practice �� Chapter 10 Applications Practice 1. Food Cookie dough is rolled in the
shape of a cylinder. How can the dough be sliced to make circular cookies? (Lesson 10-1) 2. Recreation The tent shown is in the shape of a pentagonal prism. If a wall is used to divide the tent into two rooms, what shapes could the wall be? (Lesson 10-1) 7. Camping The tent structure shown is in the shape of a square pyramid. How many square inches of canvas are required to cover the tent? Round to the nearest square inch. (Lesson 10-5) 3. Business Eli is creating a logo for his business by drawing his name in block capital letters using one-point perspective. Draw Eli’s logo. (Lesson 10-2) 4. Recreation Two hot air balloons were launched from the same location. The first balloon is 5 miles north, 9 miles east, and 0.5 mile above the launching point. The second balloon is 9 miles north, 5 miles east, and 0.8 mile above the launching point. How far apart are the two balloons? Round to the nearest tenth. (Lesson 10-3) 5. Manufacturing The two packages shown hold the same amount of food. Which requires a greater amount of material to produce? (Lesson 10-4) Recreation Use the following information for Exercises 8 and 9. A cylindrical pool has a 10 ft diameter. (Lesson 10-6) 8. How many gallons of water are needed to fill the pool to a depth of 4 feet? Round to the nearest gallon. (Hint: 1 gallon ≈ 0.134 cubic feet.) 9. If the pool is filled to a depth of 4 feet, how much will the water weigh? Round to the nearest pound. (Hint: 1 gallon weighs about 8.34 pounds.) 10. Hobbies The greenhouse shown is in the shape of a cube with a square pyramid on top. What is the volume of the greenhouse? (Lesson 10-7) 6. Hobbies Ashley is using the pattern shown to make cones to protect her plants from freezing. How tall can the plants be to fit in the cone? Round to the nearest tenth. (Lesson 10-5) 11. Food A snow-cone cup has a 3-inch diameter and is 4 inches tall. Another snow-cone cup has a 4-inch diameter and is 3 inches tall. Which cup will hold more? (Lesson 10-7) 12. Sports The circumference of a size
3 soccer ball is 24 in. The circumference of a size 5 soccer ball is 28 in. How many times as great is the volume of a size 5 ball as the volume of a size 3 ball? (Lesson 10-8) TEKS TAKS Practice S37 S37 �� Chapter 11 Applications Practice 1. Measurement There is a water tower near Peter’s house in the shape of a cylinder. He wants to find the diameter of the tank. Peter stands 25 feet from the tower. The distance from Peter to a point of tangency on the tower is 80 feet. What is the diameter of the tank? (Lesson 11-1) 2. Travel Pikes Peak is 14,110 feet above sea level. What is the distance from the summit to the horizon, to the nearest mile? (Hint: Earth’s radius ≈ 4000 mi) (Lesson 11-1) Art Use the diagram to find each value for Exercises 13–15. The diagram represents an engraving on a stained glass window. (Lesson 11-4) 13. x 14. y 15. m ⁀ FE Hobbies Use the circle graph to find each measure for Exercises 3–6 to the nearest degree. Eric collects baseball cards. He has 85 cards from the 1970s, 95 cards from the 1980s, and 125 cards from the 1990s. (Lesson 11-2) 3. ⁀ AB 4. ⁀ AC 5. ∠CDB 6. ∠ADC Data Use the circle graph to find each measure for Exercises 7–10 to the nearest degree. The circle graph shows the color of cars in a parking lot at the mall. (Lesson 11-2) 7. ⁀ HG 8. ⁀ CD 9. ∠AJH 10. ∠FJE 16. Astronomy Two satellites are orbiting Earth. Satellite A is 10,000 km above Earth, and satellite B is 13,000 km above Earth. How many arc degrees of Earth does each satellite see? (Lesson 11-5) �� 17. Entertainment A group of friends ate most of a pepperoni pizza. All that was left was a piece of crust. What was the diameter of the original pizza? (Lesson 11-6) Hobbies Use the following information
to find each area to the nearest tenth for Exercises 11 and 12. A sprinkler system has three types of sprinkler heads: a quarter circle, a semicircle, and a full circle. The sprinkler will spray a distance of 15 feet from the sprinkler head. (Lesson 11-3) 11. What is the area of the sector that will be watered by the quarter circle sprinkler head? 12. What is the area of the sector that will be watered by the semicircle sprinkler head? S38S38 TEKS TAKS Practice 18. Safety Three small towns have agreed to share a new fire station. To make sure each town has equal response time, the station should be the same distance from each town. The three towns are located on a coordinate plane at (0, 0), (6, 0), and (0, 8). At which coordinates should the station be built? (Lesson 11-7) �� Chapter 12 Applications Practice 1. Transportation Two towns are located on the same side of a river. Two roads are being built to meet at the same point P on the river. Draw a diagram that shows where P should be located in order to make the total length of the roads as short as possible. (Lesson 12-1) Agriculture Use the following information for Exercises 6–8. Cattle ranchers brand their cattle to show ownership. Three different brands are shown. (Lesson 12-5) 2. Fashion A piece of fabric used for a scarf has a repeating pattern of trapezoids. To create the pattern, translate the trapezoid with vertices (-1, 3), (3, 3), (4, 1), (-2, 1) along the vector 〈0, -2〉. Repeat to generate a pattern. What are the vertices of the third trapezoid in the pattern? (Lesson 12-2) 3. Computers A screen saver moves an icon around a screen. The icon starts at (20, 0), and then it is rotated about the origin by 50°. Give the icon’s next position. Round each coordinate to the nearest tenth. (Lesson 12-3) 4. Recreation A hole at a miniature golf
course has a barrier between the tee T and the hole H. Copy the figure and draw a diagram that shows how to make a hole in one. (Lesson 12-3) 5. Sports A team’s Web site shows a baseball moving across the screen. The ball is reflected over line ℓ and is then reflected over line m. Describe a single transformation that moves the ball from its starting point to its final position. (Lesson 12-4) 6. Which brands have rotational symmetry? 7. Which brands have line symmetry? 8. Which capital letters could be used to create a brand with rotational symmetry? Interior Design Use the following information for Exercises 9–11. Three kitchen backsplash tile patterns are shown. Identify the symmetry in each pattern. (Lesson 12-6) 9. 10. 11. 12. Hobbies Reid has a baseball card that is 2.5-by-3.5 inches. He wants to enlarge it to poster size using a scale factor of 8. What size poster frame should he buy? (Lesson 12-7) 13. Hobbies A 40 in. by 30 in. piece of art is being made into a 1 in. by 3 __ 4 in. postage stamp. What scale factor should be used to reduce the art? (Lesson 12-7) TEKS TAKS Practice S39 S39 �� Problem-Solving Handbook Draw a Diagram When a problem involves objects, distances, or places, drawing a diagram can make the problem clearer. You can draw a diagram to help understand and solve the problem. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E During a team-building activity, five people stand in a circle. Pieces of ribbon will be used to connect each person to each of the other four people in the circle. How many pieces of ribbon are needed to connect all five people in this way? Understand the Problem List the important information. • There are five people standing in a circle. • Each person should be connected to each of the other four people with a piece of ribbon. The answer is the number of pieces of ribbon needed to connect all five people. Make a Plan Draw a diagram to represent the information in the problem. Solve
Draw a circle. Add five points to the circle to represent the five people in the problem. Then draw segments to connect each point to each of the other four points. Count the number of segments in the final diagram. The total number of segments is the answer to the problem. It takes 10 pieces of ribbon to connect each person to each of the other four people. Look Back Check that the diagram is drawn correctly and that you counted the number of line segments accurately. PRACTICE 1. A delivery truck driver travels 15 miles south to deliver his first package. He then goes 9 miles east and 6 miles north to deliver his next package. From there, the driver travels 12 miles east to make his last delivery. How far is the driver from his starting point? Round to the nearest tenth of a mile. S40 S40 Problem-Solving Handbook 1234 Make a Model When a problem involves manipulating objects, you can use those or similar objects to make a model. This can help you to understand the problem and find the solution. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E During a geometry class, Zach cuts out a parallelogram with base 12 cm and height 6 cm. Catherine cuts out a rectangle with the same base and height. Show that the two shapes have the same area. Understand the Problem List the important information. • There are two geometric shapes, a parallelogram and a rectangle. • The base and the height of the two shapes are the same. To solve the problem, you need to show that the areas of the two shapes are equal. Make a Plan You can make a model of the figures by cutting them out of paper or cardboard. Then compare the areas by placing one on top of the other. Solve If the shaded area of the parallelogram is cut and moved to the opposite side, the figure becomes a rectangle. Place the two shapes on top of each other to compare the area. The shapes have the same base and height, so these shapes have the same area. Look Back Check that your models have the correct dimensions. Use the formulas for the area of a rectangle and a parallelogram to confirm that the shapes have the same area. PRACTICE 1. Find the dimensions of a rectangular prism made up of 16 1-inch cubes. 2. Two
triangles are formed by cutting a rectangle along its diagonal. What possible shapes can be formed by arranging these triangles? Problem-Solving Handbook S41 S41 12��3��4 Guess and Test For complex problems, you can use clues to make guesses and narrow your choices for the solution. Test whether your guess solves the problem, and then continue guessing until you find the solution. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Edgar is designing a party invitation in the shape of a right triangle. If all three side measures are to be whole numbers of inches, what is the smallest possible perimeter for the birthday card? Understand the Problem List the important information. • The invitation is to be a right triangle. • The legs and hypotenuse must be whole numbers. To solve the problem, you need to find the smallest possible perimeter for the right triangle. Make a Plan You can guess and test, starting with the smallest possible whole numbers. Solve Let a and b be the legs of the right triangle, and let c be the hypotenuse. So the relationship a 2 + b 2 = c 2 must hold. Start by using (1, 1) for (a, b) and solve for c 2. Since c must be a whole number, continue to guess and test until c 2 is a perfect square. Guess (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) Test = 10 ✘ 1 2 + 4 2 = 17 ✘ 1 2 + 5 2 = 26 ✘ Guess (2, 2) (2, 3) (2, 4) (2, 5) Test = 13 ✘ 2 2 + 4 2 = 20 ✘ 2 2 + 5 2 = 29 ✘ Guess (3, 3) (3, 4) Test 3 2 + 3 2 = 18 ✘ 3 2 + 4 2 = 25 ✓ Based on the tables, 5 is the smallest possible whole number for c, 3 for a, and 4 for b. So the smallest possible perimeter for the card is 3 in. + 4 in. + 5 in. = 12 in. Look Back Since 3 2 + 4 2 = 5 2, these are reasonable dimensions for the card. The problem asks for the perimeter, which is
12 inches. PRACTICE 1. The sum of Cary’s age and his brother’s age is 34. The difference between their ages is 4. How old are Cary and his brother? 2. Adult tickets for a theater performance cost $8 and children’s tickets cost $3. A group with twice as many adults as children attends the performance and spends $133 on tickets. How many people are in the group? S42 S42 Problem-Solving Handbook 1234 Work Backward Some problems involve a series of events, giving you information about the last event, and then ask you to solve something related to the initial situation. You can work backward to solve these problems. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Sandy is creating a pattern made from isosceles right triangles as shown below. If the hypotenuse of the fifth triangle is 4 in., what are the dimensions of the smallest triangle? Understand the Problem List the important information. • Each triangle is an isosceles right triangle, so each triangle’s legs are congruent. • The hypotenuse of one triangle is equal to the leg length of the next triangle. • The fifth triangle’s hypotenuse is 4 in. You must work backward to find the dimensions of the first triangle. Make a Plan Let h be the hypotenuse and s be the leg length of each triangle. Start with a hypotenuse length of 4 and work backward using the Pythagorean Theorem, which states that. Solve Triangle 5: h = 4, so s = √  8 Triangle 4 The hypotenuse of Triangle 4 equals the leg length of = 2 s 2, so s = 2. Triangle 5. ( √  8 ) 2 Triangle 3: h = 2, so s = √  2. Triangle 2, so s = 1. Triangle 1: h = 1; s = √  0.5 1 2 = 2 s 2, so s = √  0.5. The first triangle should have a leg length of √  0.5 and a hypotenuse of 1. Look Back Recreate the diagram
starting with the dimensions you found for the first triangle, and confirm that the fifth triangle has a hypotenuse of 4 inches. PRACTICE 1. In a trivia game, each question is worth twice as many points as the one before it. Chelsea answers 5 questions and earns 1550 points. How many points was her first question worth? 2. Sheryl has 4 siblings. She is 4 years younger than her sister Meagan. Meagan is twice as old as Tina. Jack is 3 years older than Tina, and Tina is 1 year older than Bryan, who is 9. How old is Sheryl? Problem-Solving Handbook S43 S43 ��1234 Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Find a Pattern In some problems, there is a relationship between different pieces of information. You can find a pattern to help solve these problems. E X A M P L E Frank plants turnips in rows and columns. Each year, he increases the size of his turnip patch by adding one row and one column, as shown in the diagram. How many turnip plants will Frank have after year 5? Understand the Problem List the important information. • In year 1, Frank has 3 turnip plants. • In year 2, he has 8 turnip plants. • In year 3, he has 15 turnip plants. The answer will be the number of turnip plants in year 5. Make a Plan Find the pattern based on the diagram. Solve Make a table of the given information and find a pattern. Number of Turnip Plants Possible Pattern Year 1 Year 2 Year 3 3 8 15 0 + 3 3 + 5 8 + 7 The pattern seems to be the number of turnip plants in the previous year plus the next odd number. So in year 4, Frank will have 15 + 9 = 24 turnip plants, and in year 5, he will have 24 + 11 = 35 turnip plants. Look Back By thinking of the number of plants as the product of the number of rows and columns, you might notice another pattern, n (n + 2), where n is the year number. Use this to confirm your answer. Year 4: 4 (4 + 2) = 24 turnip plants Year 5: 5 (5 + 2) = 35 turnip plants PRACTICE 1. Use the key
GDB = DAY to decode the sentence DQ DSSOH D GDB NHHSV WKH GRFWRU DZDB. 2. Describe the pattern 15, 22, 29, 36, 43,... and find the next two numbers. S44 S44 Problem-Solving Handbook ��1234 Make a Table To solve a problem that involves a relationship between two sets of numbers, you can make a table to organize and analyze the data. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Roy’s Geometry class is playing a game to practice identifying shapes. There are eight shapes in the game: an acute triangle, a right triangle, a square, a rectangle, a rhombus, a parallelogram, a kite, and an isosceles trapezoid. On Roy’s turn, the teacher reads the following clues: The shape is a quadrilateral with four right angles in which all sides are not congruent. Which shape should Roy select? Understand the Problem List the important information. • The possible shapes are an acute triangle, a right triangle, a square, a rectangle, a rhombus, a parallelogram, a kite, and an isosceles trapezoid. • Roy’s shape is a quadrilateral with four right angles. • Roy’s shape does not have four congruent sides. The answer will be the shape that matches Roy’s clues. Make a Plan Make a table and use the given information to identify Roy’s shape. Solve Use the given clues to complete a table and identify Roy’s shape. Shape Quadrilateral? 4 Right Angles? Sides Not Congruent? Acute triangle Right triangle Square Rectangle Rhombus Parallelogram Kite Isosceles trapezoid The rectangle is the only shape that satisfies the given clues. Look Back Make sure that your answer satisfies the given clues. N N N Y N N Y Y PRACTICE 1. Katie gets the following clues: The shape has at least one right angle, has no parallel sides, and is not the kite. Which shape should Katie select? 2. Mary gets the following clues: The shape has no con
gruent sides. How many possible shapes might Mary select? Problem-Solving Handbook S45 S45 1234 Solve a Simpler Problem A problem with many steps or involving very large numbers can be overwhelming. Sometimes it helps to solve a simpler problem first, or to break the complex problem into multiple simpler ones. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Tom plans to repaint his patio, which has the measurements shown below. What is the total area that Tom needs to paint? Understand the Problem List the important information. • AH = FG = 4 ft • DE = 3 ft • CD = 20 ft • GH = EF = 5 ft The answer will be the total area of the patio. Make a Plan To simplify the problem, divide the patio into basic geometric shapes and add their areas together. Solve Find the area of the patio as if it were a complete rectangle, and then subtract the area of the smaller rectangle that is not part of the patio. Step 1: Find the area of each rectangle. Larger rectangle Smaller rectangle Length CD = 20 ft AH + FG + DE = 11 ft GH = 5 ft FG = 4 ft Width Area ℓw = (20) (11) = 220 ft 2 ℓw = (5) (4) = 20 ft 2 Step 2: Subtract the areas to find the area of the patio. Area of patio = 220 - 20 = 200 ft 2 Tom needs to paint 200 square feet. Look Back Divide the patio into a different arrangement of smaller shapes to check your answer. For example, by dividing the patio into three rectangles stacked on top of each other, you find that (4) (20) + (4) (15) + (3) (20) = 200 ft 2, which confirms the first answer. PRACTICE 1. How much paint does Rose need to repaint her patio? 2. Rose plans to add a decorative railing around the outer edges of her patio. The railing will cover every edge except the 30-foot side of the patio that joins her house. About how many feet of railing does Rose need? Round to the nearest foot. S46 S46 Problem-Solving Handbook ��1��234 Use Logical
Reasoning Some problems provide clues and facts that you must use to find the solution. To use logical reasoning, identify these facts and draw conclusions from them. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Dawn, Chloe, and Tyra finish first through third in a cross-country race. The girls wear the numbers 7, 8, and 12. Dawn does not wear an even number. The one who wears number 8 comes in first. Chloe comes in third. Who wears which number, and in what place did each runner finish? Understand the Problem List the important information. • Dawn wears an odd number. • The girl who wears number 8 comes in first place. • Chloe comes in third place. The answer will be a list of who wears which number and each girl’s finishing position. Make a Plan Start with the clues given in the problem. Use logical reasoning to determine each girl’s number and finishing position. Solve Make a table. Read the clues one at a time, and mark the table appropriately. • Dawn wears an odd number, so she must wear number 7. No other girl can wear number 7. • The girl who wears number 8 comes in first place. No other number is the first-place winner. Also, since Dawn wears number 7, she didn’t come in first. • Chloe comes in third. By process of elimination, Dawn must have come in second, and Tyra came in first. So Tyra wears number 8, and thus Chloe wears number 12. 7 Dawn ✓ Chloe Tyra 1st 2nd 3rd ✘ ✘ ✘ ✓ ✘ 8 ✘ ✘ ✓ ✓ ✘ ✘ 12 1st 2nd 3rd ✘ ✘ ✓ ✓ ✘ ✘ ✘ ✓ ✘ ✘ ✓ ✘ ✘ ✘ ✓ Look Back Compare your answer to the facts given in the problem. Make sure none of your conclusions conflict with the given clues. PRACTICE 1. Mike, Jack, and Ann each wear a different type of top in three different colors. The tops are a button-down shirt, a pullover, and a sweater. The colors are blue, yellow, and red. Mike wears a blue shirt, and Jack wears a button-down. The yellow top is
a pullover. Who wears the sweater and who wears the red top? 2. The Warriors, Jaguars, and Cougars each have a different-colored shape on their team shirt. The colors are green, purple, and red, and the shapes are a triangle, a rectangle, and a hexagon. The Warriors’ shape has the most sides, the color of the Jaguars’ shape is green, and the rectangle is purple. Which team has which shape and in which color? Problem-Solving Handbook S47 S47 1234 Use a Venn Diagram A Venn diagram can be useful when you solve a problem involving relationships among sets or groups. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E In a class of 15 students, ten play on at least one of the school sports teams—the basketball team or the baseball team. Five of them are on the basketball team. Three students are on both the basketball team and the baseball team. How many of the students play on the baseball team? Understand the Problem List the important information. • 5 students play on the basketball team. • 3 students play on both teams. The answer is the number of students who play on the baseball team. Make a Plan Organize the information by drawing a Venn diagram. Solve Draw and label the Venn diagram. • 3 people will be in the overlapping area. • Since 5 people play on the basketball team, and 3 of them are also on the baseball team, only 2 people play on only the basketball team. There are 10 student players in all, and five are already accounted for. Therefore, the remaining five play only baseball. Adding the three students who also play basketball, a total of eight students in the class play on the baseball team. Look Back Check your Venn diagram to make sure it is an accurate representation of the information given in the problem. Confirm that the numbers in each of the labeled sections add up to the total number of students in the problem. PRACTICE 1. At Lucy’s Home-Style Restaurant, four of the meals include a side salad, six include only soup as a side, and two meals come with both salad and soup as sides. If all meals come with at least one side, how many different meals are on Lucy’s menu?
2. A cupboard contains 12 cups, and each cup has a lid, a handle, or both. There are seven cups with handles, and three cups with both a lid and a handle. How many cups have only a lid? S48 S48 Problem-Solving Handbook ��1234 Make an Organized List If a problem involves multiple outcomes, it may be useful to make an organized list to record the data and count the different outcomes. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Sally randomly selects two shapes from a bag that contains five different cut-outs: a triangle, a square, a rectangle, a pentagon, and a hexagon. The sum of the number of sides of the two shapes is eight. What combinations of shapes might Sally have selected? Understand the Problem List the important information. • The possible shapes are a triangle, a square, a rectangle, a pentagon, and a hexagon. • The sum of the number of the sides is 9. The answer will be the two shapes Sally selected. Make a Plan Make an organized list of the possible combinations of shapes. Then list the number of sides and the sum of the sides. Solve List the possible combinations of shapes, and find the sum of the shapes’ sides. Triangle (3) Square (4) ✘ ✘ Rectangle (4) Pentagon (5) Hexagon (6) Total number of sides ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ 7 7 8 ✘ 9 8 9 10 9 10 11 There are two combinations of shapes that have a total number of eight sides: the triangle and pentagon and the square and rectangle. Look Back Make sure all possible combinations are shown in the table. Check that the total number of sides for both combinations (triangle and pentagon, and square and rectangle) is 8. PRACTICE 1. How many ways can you make $0.30 by using quarters, dimes, nickels, and pennies? 2. Pete’s Pizza Palace has 5 choices of meat, 4 choices of vegetables, and 2 choices of cheese. You want to order a
pizza with one of each. How many combinations can you order? Problem-Solving Handbook S49 S49 1234 Skills Bank Operations with Real Numbers The four basic operations with real numbers are addition, subtraction, multiplication, and division. E X A M P L E Simplify each expression.5 · 3 3.5 · 3 = 10.5 B 0.5 - 4 0.5 - 4 = -3. PRACTICE Simplify each expression. 1. 3 + 4 2. 9 · 9 5. 0.1 ⋅ 0.1 9. -3 · 7 6. 0.5 + 2 3. 3 ÷ 3 7. 4 - 0.3 4. 10 - 8 8. 8 ÷ 2 10. 6.3 - 8.1 11. -12 ÷ (-3) 12. 17.3 + 12.9 Order of Operations When simplifying expressions, follow the order of operations. 1. Simplify within parentheses. 2. Evaluate exponents and roots. 3. Multiply and divide from left to right. 4. Add and subtract from left to right. E X A M P L E Simplify the expression (3 + 2 2 ) · (3 - 1) 2. (3 + 2 2 ) · (3 - 1) 2 (3 + 4) · (3 - 1) 2 Simplify within parentheses. There is an exponent within the first set of parentheses, so simplify it first. 7 · 2 2 7 · 4 28 Simplify within parentheses. Evaluate the exponent. Multiply. PRACTICE Simplify each expression. 1. 3 · (4 + 1) 2 2. (1 - 3) + 4 · 5 4. 6 - 5 + 2 2 · (9 - 7) 3. S50 S50 Skills Bank Skills Bank Properties Below are the basic properties of addition and multiplication, where a, b, and c are real numbers. Addition Multiplication Closure a + b is a real number. Closure a · b is a real number. Commutative a + b = b + a Commutative a · b = b · a Associative (a + b) + c = a + (b + c) Associative (a · b) · c = a · (b · c) Identity Property of Zero a + 0 = a and 0 + a = a Identity Property of One Multiplication Property of Zero a · 1 = a and and
0 · a = 0 Distributive a · (b + c) = a · b + a · c Transitive If a = b and b = c, then a = c. Other Real Number Properties E X A M P L E 1 Name the property shown. A 2 · (3 - 3) = 0 Multiplication Property of Zero B (9 + 3) + 2 = 9 + (3 + 2) Associative Property of Addition E X A M P L E 2 Give an example of each property, using real numbers. A Closure Property of Multiplication 1.8 · 2.4 is a real number. B Commutative Property of Addition 17 + 84 = 84 + 17 PRACTICE Name the property shown, where a, b, and c are real numbers. 1. If 3 + 8 = x and x = y, then 3 + 8 = y. 2. a + b is a real number. 3. 0 · 9 = 0 5. 10 · (b + c) = 10b + 10c 4. 3c · 2a = 2a · 3c 6. 3a is a real number. 7. (2a + 3b) + 2c = 2a + (3b + 2c) 8. 1 · 2a = 2a Give an example of each property, using real numbers. 9. Identity Property of Addition 10. Distributive Property 11. Commutative Property of Multiplication 12. Closure Property of Multiplication 13. Closure Property of Addition 14. Transitive Property Skills Bank S51 S51 Estimation, Rounding, and Reasonableness Estimation involves rounding numbers. To round a number to a given place value, look at the digit to the right of that place value. If it is greater than or equal to 5, round up. If it is less than 5, round down. E X A M P L E 1 Round each number to the given place value. A 1.2941 to the nearest tenth 9 > 5; round up. 1.2941 B 3.14159 to the nearest hundredth 3.14159 1 < 5; round down. 1.3 100 ___ ̶ 3 33.33 33.333 C to the nearest unit 3 Convert to a decimal. 3 < 5; round down. 3.14 D √  5 to the nearest tenth 2.2360 Convert to a decimal. 2.2360
3 < 5; round down. 33 2.2 E X A M P L E 2 Estimate each sum by rounding. A 12.75 + 15.94 13 + 16 29 Add. Round each number. B 182 + 208 + 319 180 + 210 + 320 Round each number. 710 Add. E X A M P L E 3 Tell whether an estimate is sufficient or an exact answer is needed. A The distance from San Antonio to Austin is about 80 miles. The distance from Austin to Dallas is about 190 miles. If you drive from San Antonio to Austin then to Dallas, about how far did you drive? The problems asks “about how far,” so an estimate is sufficient. B Kim buys two shirts for $12.95 each. Sales tax is 8.25%. How much money does she need? The problem asks for the amount of money, so an exact answer is needed. PRACTICE Round each number to the given place value. 1. 285,618 to the nearest hundred 2. 9.7 to the nearest unit 3. 49.249 to the nearest tenth 4. 873.59 to the nearest ten Estimate each sum by rounding. 5. 73.98 + 180.76 6. 251 + 489 7. 45,792 + 13,819 8. 0.034 + 0.015 9. 27.1 + 43.8 10. 862 + 740 Tell whether an estimate is sufficient or an exact answer is needed. 11. Eric buys dinner for $10.75 and wants to leave about an 18% tip. How much money does he need? 12. Find the height of a triangle with A = 24 in 2 and b =16 in. 13. Ginny is planting grass seed in a 6 ft by 10 ft rectangular patch of lawn. A pound of seeds covers about 1000 ft 2. About how much seed will she use? S52 S52 Skills Bank Classify Real Numbers A set is a group of items. Numbers can be organized into sets. Set Examples Venn Diagram The natural numbers are the counting numbers. The whole numbers are the natural numbers plus 0. The integers are the whole numbers and their opposites. Rational numbers can be written as a ratio of two integers. Irrational numbers cannot be written as a ratio of two integers. The real numbers are the rational numbers plus the irrational numbers. {1, 2, 3, 4, …} {0, 1, 2, 3, …} {…
, -1, 0, 1, 2, …}   1 _ ⎬ ⎨, -3.4, 0 Write all of the names that apply to each number. A -79 real number, rational number, integer B √  13 real number, irrational number PRACTICE Write all of the names that apply to each number. ̶ 2. 0. 1. 11 3 3. π 4. -4.6 5. 0 Exponents Exponents are used to describe repeated multiplication. In the expression b n, b is the base and n is the exponent. A negative exponent is used to represent the reciprocal of the base with the opposite exponent. E X A M P L E Evaluate. A 2 5 2 5 = (2) (2) (2) (2) (2) = 32 PRACTICE Evaluate. 3 6 1. 2. 2 10 B 3 -2 3 -3) (3) = 1 _ 9 3. 5 -3 4. 6 4 5. 7 -2 6. 3 4 Skills Bank S53 S53 �� Properties of Exponents The following properties can be used to simplify expressions with exponents. WORDS NUMBERS ALGEBRA The quotient of two nonzero powers with the same base is the base raised to the difference of the exponents. The product of two powers with the same base equals the base raised to the sum of the exponents If a ≠ 0, then 11 If a ≠ 0, then Any nonzero number raised to the zero power is 1 If a ≠ 0, then a 0 = 1. E X A M P L E 1 Simplify PRACTICE Simplify. 1. 3 b _ 3 2 3. ( b m ) ( b -m ) __ n 0 4. ( xyz 3 ) ( z 2 ) Powers of 10 and Scientific Notation Scientific notation is used to write very large numbers, such as the speed of light (about 300,000,000 m/s) and very small numbers, such as the average diameter of an atom (0.000000030 cm). In scientific notation, the speed of light is 3.0 × 10 8 m/s, and the average diameter of an atom is 3.0 × 10 -8 cm. E X A M P L E 1 Write
each number in standard notation. A 2.99 × 10 4 B 3.04 × 10 -6 2.99 × 10,000 29,900 10 4 = 10,000 Move the decimal 3.04 × 0.000001 0.00000304 10 -6 = 0.000001 Move the decimal point 4 places right. point 6 places left. PRACTICE Write each number in standard notation. 1. 10 3 2. 10 8 4. 9.04 × 10 2 5. 9.0 × 10 -4 3. 10 -4 6. 1.0 × 10 0 S54 S54 Skills Bank Square Roots The square root of a number is one of the two equal factors of that number. Every positive number has two square roots, one positive and one negative. E X A M P L E Find the two square roots of each number. A 81 √  81 = 9 9 is a solution, since 9 · 9 = 81. - √  81 = -9 -9 is a solution, since (-9) · (-9) = 81. B 121 121 = 11 √  - √  121 = -11 11 is a solution, since 11 · 11 = 121. -11 is a solution, since (-11) · (-11) = 121. PRACTICE Find the two square roots of each number. 1. 64 2. 49 3. 225 4. 1 Simplifying Square Roots A square root is in simplest form when the radicand contains no perfect squares and no fractions and there are no square roots in a denominator. The following properties are used to simplify square roots. Multiplication Property: √  Division Property Simplify √ _  8. 18  8 _ √ 18 = √  8 _ √  18 = √  4 · 2 _ √  Division Property Factor. Multiplication Property Simplify. Simplify. PRACTICE Simplify each square root. 1. √  640 2. √  936 3. √  242 Skills Bank S55 S55 The Coordinate Plane Recall that to locate a point on a number line with a given coordinate, you move left or right from
0. The coordinate plane is formed by two perpendicular number lines, the x-axis and the y-axis, that intersect at the origin, (0, 0). The location of a point is described by an ordered pair, (x, y), where x is the distance from the y-axis and y is the distance from the x-axis. The coordinate plane is divided into four quadrants. Quadrant I: x and y are both positive. Quadrant II: x is negative, and y is positive. Quadrant III: x and y are both negative. Quadrant IV: x is positive, and y is negative. E X A M P L E 1 Graph each point. A (3, -1) B (-2, 3) Start at the origin. Move right 3 and then down 1. Start at the origin. Move left 2 and then up 3. E X A M P L E 2 In which quadrant is each ordered pair located? A (1, 1) Quadrant I x and y are both positive. B (3, -3) Quadrant IV x is positive, and y is negative. PRACTICE Graph each point. 1. (5, 1) 2. (3, -1) 3. (-2, 0) 4. (-4, -3) In which quadrant is each ordered pair located? 5. 6. (-2, -6) (2, 6) 7. (-5, 1) 8. (9, -2) S56 S56 Skills Bank �� Connecting Words with Algebra Word phrases or sentences can be written as algebraic expressions. Symbol Word Phrases Algebraic Expressions + - × or · ÷ • a number plus 3 • 3 more than a number • a number minus 2 • the difference of a number and 2 • 3 times a number • the product of 3 and a number • a number divided by 2 • the quotient of a number and 2 n + 3 m - 2 3x or 3 · x k _ or Write an algebraic expression for each word phrase. A a pie cut into 8 equal slices B 3 sheets of paper added to a stack p ÷ 8 s + 3 PRACTICE Write an algebraic expression for each word phrase. 1. 3 lb
more than an apple 2. 10 times as heavy as a horse 3. 3 years less than 9 times Gwen’s age Variables and Expressions A variable is a letter that represents a value that can change or vary. An algebraic expression has one or more variables. To evaluate an algebraic expression, substitute the given value for each variable, and simplify the expression. E X A M P L E Evaluate each expression for the given values of the variables. A 3n for n = 2 B a + b for a = 2 and b = 3 3n 3 (2) 6 Substitute. Simplify. a + b (2) + (3) 5 Substitute. Simplify. PRACTICE Evaluate each expression for the given values of the variables. 1. 2k for k = 6 2. 3 - n for n = 4 3. 3x for x = 0 4. xy for x = 2 and y = 3 5. i + 2 for i = 1.7 6. -1k for k = -1 7. 2x for x = 0.5 8. 4n + 7 for n = 13 9. u 2 v for u = 3 and v = 7 Skills Bank S57 S57 Solving Linear Equations The following properties can be used to solve linear equations in one variable. PROPERTY WORDS NUMBERS ALGEBRA Addition Property of Equality Subtraction Property of Equality You can add the same number to both sides of an equation, and the statement will still be true. You can subtract the same number from both sides of an equation, and the statement will still be true. Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true. Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true3) = 6 (3) 18 = 18 ac = bc (c ≠ 0 Solve. A x + 5 = 11 x + 5 = 11 - 5 - 5 ̶̶̶ ̶̶̶̶ x = 6 C 6y - 4 = 38 6y - 4 = 38 + 4 + 4 ̶̶̶ ̶̶̶̶̶ 6y = 42 6y = 42 _ _ 6 6 y = 7 PRACTICE Solve. 1. x + 7 = 2 Subtract 5 from both sides. B D _ u = 10
2 2 · u _ = 2 · 10 2 u = 20 _ 3 + n 8 = -2 Multiply both sides by 2. Add 4 to both sides. Divide both sides by 6 (-2) 3 + n = -16 - 3 ̶̶̶̶̶ - 3 ̶̶̶ n = -19 Multiply both sides by 8. Subtract 3 from both sides. 2. 2u = 6 3. n _ 3 = 21 4. 13 = x - 16 5. 1.5k = 27 6. 18 + p = 16 8. b - 2.7 = 3.4 9. 2w + 7 = 18 11. 4z - 8 = 18 12. a-2 _ = 1 7 7. 15 = h _ 30 10. d _ 5 + 4 = 17 S58 S58 Skills Bank Solving Equations for a Variable Equations with more than one variable are sometimes called literal equations. Sometimes it is necessary to solve literal equations for one variable in terms of the others. This is also called isolating the variable. E X A M P L E Solve for y. 3y + 2x = y + 2 3y + 2x = -y ̶̶̶̶̶̶ 2y + 2x = - 2x ̶̶̶̶̶̶ y + 2 - y ̶̶̶̶̶ 2 2y - 2x ̶̶̶̶̶̶ = - 2x + 2 2y 2x + 2 _ _ 2 2 y = -x + 1 = - To get the y-terms together, subtract y from both sides. Subtract 2x from both sides. Divide both sides by 2. PRACTICE Solve for y. 1. 3x = 2y - 3 2. y - x _ 2 = 1 3. x + y = 4y + 3 - 2x Writing and Graphing Inequalities An inequality compares two quantities by using one of these symbols: < > ≤ ≥ is less than is greater than is less than or equal to is greater than or equal to E X A M P L E Write each expression as an inequality. Graph the inequality. A q is less than or equal to 3 q ≤ 3 Use ≤ for “is less than or equal to.” Use a solid circle for ≤ or ≥. Shade the side of the line that contains the solutions. B s is greater than 18 s > 18 Use > for “is
greater than.” Use an empty circle for < or >. Shade the side of the line that contains the solutions. PRACTICE Write each expression as an inequality. Graph the inequality. 1. u is less than 0 2. n is greater than or equal to 15 3. x is less than 3 4. b is greater than 5 5. y is less than or equal to 4 6. m is greater than -3 Skills Bank S59 S59 �� Solving Linear Inequalities The following properties are used to solve linear inequalities. PROPERTY WORDS NUMBERS ALGEBRA Addition Property of Inequality You can add the same number to both sides of an inequality, and the statement will still be true Subtraction Property of Inequality You can subtract the same number from both sides of an inequality, and the statement will still be true. Multiplication and Division Properties of Inequality (by a positive number) You can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true. Multiplication and Division Properties of Inequality (by a negative number) If you divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true. 5 < 10 9 < 12 a < b 9 - 5 < 12 - < 12 7 · 3< 12 · 3 21 < 36 4 < 12 < 12 ___ 4 ___ -4 -4 -1 > -3 If a < b and c > 0, then ac < bc. If a < b and c < 0, then a __ c > b __ c. These properties are also true for inequalities that use the symbols, >, ≥, and ≤. E X A M P L E Solve and graph. A 8m < 96 8m < 96 < 96 _ 8m _ 8 8 m < 12 B 3 - 2x ≤ 7 Divide both sides by 8. - 3 ̶̶̶̶̶̶ 3 - 2x ≤ 7 - 3 ̶̶̶ -2x ≤ 4 -2x _ ≥ 4 _ -2 -2 x ≥ -2 PRACTICE Solve and graph. 1. 3x + 2 > 11 4. -n < 3 7. z + 7 > 5 S60 S60 Skills Bank Subtract 3 from both sides. Divide both sides by -2. Reverse the direction of the inequality. 2. -4 < 8 - 3y
5. 2g + 13 ≥ -1 8. k _ 3 + 4 < 12 3. 6x ≥ 5x + 4 6. -4a ≥ 18 - w _ 9. 3 _ 4 4 ≥ 7 �� Absolute Value The absolute value of a number is its distance from zero on the number line. The absolute value of a number a is represented by ⎜a⎟. E X A M P L E Simplify. A ⎜-4⎟ ⎜-4⎟ = 4 B ⎜3 - 9⎟ ⎜3 - 9⎟ = ⎜-6⎟ = 6 PRACTICE Simplify. 1. ⎜7⎟ 2. ⎜-1⎟ 5. ⎜-9 + 2⎟ 6. ⎜-3 + 4⎟ 3. ⎜10 - 15⎟ 7. ⎜-20 + 4⎟ 4. ⎜-4 - 4⎟ 8. ⎜-2 - 4⎟ - ⎜-5⎟ Relations and Functions A relation is a rule that relates two quantities. A function is a relation in which each input value corresponds to exactly one output value. The domain is the set of all possible input values, and the range is the set of all possible output values. E X A M P L E Determine whether each relation is a function.   (1, 2), (2, 4), (3, 6), (4, 8) A S: ⎨ ⎬   yes Each x-value in the set corresponds to exactly one y-value. B ⎜y⎟ = x no C y = x 2 yes ⎜2⎟ = 2 and ⎜-2⎟ = 2, so (2, 2) and (2, -2) satisfy the equation. These two points have the same x-value but different y-values, so the relation is not a function. For each value of x, there is only one value of x 2, so each x-value corresponds to exactly one y-value. PRACTICE Determine whether each relation is a function. 1. y = 3x + 4 4. 10y = -x
2. y 2 = x 5. 9y = 3   (0, 0), (1, 2), (0, 2) 3. S: ⎬ ⎨     (-5, 1), (-4, 1), (-3, 1) 6. S: ⎬ ⎨   Skills Bank S61 S61 �� Inverse Functions A function is a rule that relates two quantities, the input and the output. Each input value corresponds to only one output value. The inverse of a function is a rule that reverses the function. To find the rule for the inverse, switch the x- and y-values. The rule for the inverse is also a function if each input corresponds to only one output. �� E X A M P L E Find the inverse of each function, and state whether the inverse is also a function. ��   (0, 1), (1, 2), (3, 7), (9, 9) A S: ⎬ ⎨   The rule for the inverse is   (1, 0), (2, 1), (7, 3), (9, 9) ⎨. ⎬   It is a function, because each x-value corresponds to only one y-value. PRACTICE B y = x 2 The rule for the inverse is x = y 2. The inverse is not a function, because each x-value corresponds to two y-values, √  x and - √  x. Find the inverse of each function, and state whether the inverse is also a function.   (0, 0), (3, 1), (2, 2), (5, 7) 1. K: ⎬ ⎨   2. y = 5x 3. y = ⎜x⎟ Direct Variation Direct variation is a relationship between two quantities in which one quantity is a constant multiple of the other. The constant is called the constant of variation. The relationship “y varies directly as x, where
k is the constant of variation” is written as y = kx. E X A M P L E Find the constant of variation. A y varies directly as x, and y = 7 B t varies directly as c, and t = 1 when x = 3. y = kx 7 = k (3) 7 _ = k 3 Substitute 7 for y and 3 for x. when c = 0.1. t = kc (1) = k (0.1) Substitute 1 for t and 0.1 for c. Solve for k. 10 = k Solve for k. PRACTICE Find the constant of variation. 1. m varies directly as n, and m = 12 when n = 3. 2. y varies directly as x, and y = 24 when x = 8. 3. s varies directly as t, and s = 9 when t = 36. S62 S62 Skills Bank Functional Relationships in Formulas A formula is an equation that is solved for one variable in terms of the others. By rearranging the terms in a formula, you can see how each variable depends on the others. E X A M P L E Solve the formula P = 2ℓ + 2w for ℓ. If the width stays the same, what happens to the length of a rectangle as its perimeter increases? P = 2ℓ + 2w P - 2w = 2ℓ 1 _ P - w = ℓ 2 Subtract 2w from both sides. Divide both sides by 2. As the perimeter of a rectangle increases, the length also increases. PRACTICE 1. Solve the formula A = 1 __ 2 ( b 1 + b 2 ) h for h. If the base lengths stay the same, what happens to the height h of a trapezoid as its area A increases? 2. Solve the formula I = Prt for t. If the interest rate and amount stay the same, what happens to the time t as the principal amount P decreases? Transformations of Functions The basic transformations of the parent function y = f (x) are given below. Transformation Transformation Function Reflection Across the x-axis y = -ƒ (x) Reflection Across the y-axis y = ƒ (-x) Vertical Translation y = ƒ (x) + k If k > 0, translate k units up. If K < 0, translate k units down. Horizontal
Translation y = ƒ (x - h) If h > 0, translate h units right. If h < 0, translate h units left. E X A M P L E Describe the transformation given by the equation y = (x - 3) 2. Step 1 Identify the parent function. The parent function is y = x 2. Step 2 Identify the transformation. The equation represents a horizontal translation with h = 3. So the transformation is a horizontal translation 3 units to the right. PRACTICE Describe the transformation given by each equation. 1. y = x - 6 2. y = √  -x 3. y = (-x) 2 - 1 4. y - 2 = x 2 Skills Bank S63 S63 Polynomials A monomial is a number or a product of numbers and variables with whole-number exponents. A polynomial is a monomial or the sum or difference of monomials. The degree of a polynomial is the highest power of the variables in all the terms. Polynomial Number of Terms Example Monomial Binomial Trinomial 1 2 3 3 a 2 x 2y - 7 x 2 2n + 3m - Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. A 3x + 2 y 6 + 7 q 2 e trinomial Polynomial with 3 terms B -5z monomial _ 5 c 2 d C 1.5b - 2 binomial D 1 _ x Polynomial with 1 term Polynomial with 2 terms not a polynomial A variable is in the denominator. E X A M P L E 2 Find the degree of each polynomial. A 7 v 3 - 8 w 2 - 9u 7v 3 Degree 3 The degree is 3. B -5z + 1 -5z 1 Degree 1 The degree is 1. -8w 2 Degree 2 -9u 1 Degree 1 +1 0 Degree 0 PRACTICE Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1. e 2 r 3 2. x -2 3. b 2 - 4ac 5. 5 x 2 - 2x + 7 6. 4 7. y - 3 Find the degree of each polynomial. 9. x 5 - x 3 + 3 10. 9 a 3 - 10 a 2 +
a 4 11. 2 - 4x 4. 12 S64 S64 Skills Bank Quadratic Functions A quadratic function is a function that can be written in the form y = a x 2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola, an almost U-shaped graph. Graph of a Quadratic Function y = a x 2 + bx + c • If a > 0, the parabola opens upward. • If a < 0, the parabola opens downward. • The axis of symmetry of the parabola is the vertical line x = - b _. 2a • The vertex of the parabola is (- b _, y). 2a E X A M P L E Tell whether the graph of each quadratic function opens upward or downward. Write an equation for the axis of symmetry, and find the coordinates of the vertex. A y = -2 x 2 + 4x + 2 Step 1 Tell whether the graph opens upward or downward. Since a = -2 < 0, the graph opens downward. Step 2 Find the x-coordinate of the vertex. - b _ 2a = - 4 _ 2 (-2 ) = 1 Substitute -2 for a and 4 for b. The equation for the axis of symmetry is x = 1. Step 3 Find the y-value when x =1. y = -2 (1 ) 2 + 4 (1 ) + 2 = 4 Substitute 1 for x in the original equation. The coordinates of the vertex are (1, 4 Step 1 Tell whether the graph opens upward or downward. Since a = 3 > 0, the graph opens upward. Step 2 Find the x-coordinate of the vertex. - b _ 2a = - 0 _ 2 (3 ) = 0 Substitute 3 for a and 0 for b. The equation for the axis of symmetry is x = 0. Step 3 Find the y-value when x = 0. y = 3 (0 ) 2 - 1 = -1 Substitute 0 for x in the original equation. The coordinates of the vertex are (0, -1). PRACTICE Tell whether the graph of each quadratic function opens upward or downward. Find the coordinates of the vertex, and write an equation for the axis of symmetry. 1. y = -4 x 2 - 8x 2. y = - x 2 - 4x + 2 3. y
= 2x 2 + 4 4. y = 3 x 2 - 6x + 8 5. y = - 5x 2 + 10 6. y = 0.5x 2 + x + 2 Skills Bank S65 S65 �� Factoring to Solve Quadratic Equations One method of solving quadratic equations is to apply the Zero Product Property, which states that if ab = 0, then a = 0 or b = 0. First write the quadratic equation in standard form and then factor it. E X A M P L E Solve the quadratic equation 2 x 2 - 5x + 6 = 9 by factoring. 2 x 2 - 5x -3 = 0 (2x + 1) (x - 3) = 0 Write the equation in standard form. Factor the trinomial. 2x + 1 = 0 or x - 3 = 0 Use the Zero Product Property. x = - 1 _ 2 or x = 3 Solve each equation. PRACTICE Solve each quadratic equation by factoring. 1. x 2 - 3x + 2 = 0 3. 4 x 2 - 8x = -4 5. 4 x 2 + 8x = 32 2. x 2 + 4 = -4x 4. x 2 = 9 6. 3x + 4 = x 2 The Quadratic Formula For a quadratic equation in the form a x 2 + bx + c = 0, you can use the Quadratic Formula, x = -b ± √  b 2 - 4ac __ 2a, to solve for x. E X A M P L E Use the Quadratic Formula to solve each equation. A 2 x 2 + 3 = 7x 2 x 2 - 7x + 3 = 0 Write the equation in standard form. a = 2, b = -7, c = 3 Find a, b, and c. B x 2 - 4x = -6 1x 2 - 4x + 6 = 0 a = 1, b = -4, c = 6 x = - (-7 ) ± √  (-7 ) 2 - 4 (2 ) (3 ) ___ 2 (2 ) Substitute into the Quadratic Formula. x = - (-4 ) ± √ �
�� (-4 ) 2 - 4 (1 ) (6 ) ___ 2 (1 ) x = x = 7 ± √  25 _ 4 7 + 5 _ 4 or x = 7 - 5 _ 4 Simplify. Simplify. x = 4 ± √  -8 _ 2 Since you cannot take the square root of a negative number, there is no solution. x = 3 or x = 1 _ 2 Write the solution. PRACTICE Use the Quadratic Formula to solve each equation. 1. x 2 + 2x = -1 2. 3 x 2 + 2x = 4 4. 4 x 2 + 8x - 12 = 0 5. 2 x 2 + 5x = 3 3. 7 x 2 + 3x = -5 6. 2 x 2 - 7x = 12 S66 S66 Skills Bank Solving Systems of Equations To solve a system of equations, you can use either the substitution method or the elimination method. E X A M P L E ⎧ 3x + 2y = 0 ⎨ Solve the following system of equations. 2x + y = 3 ⎩ Method 1 Use substitution. y = -2x + 3 3x + 2 (-2x + 3 ) = 0 3x - 4x + 6 = 0 -2 (6 ) + 3 = -9 (6, -9 ) Method 2 Use elimination. -2 (2x + y) = -2 (3) -4x - 2y = -6 3x + 2y = 0 -4x - 2y = -6 -x = -6 x = 6 y = -2 (6 ) + 3 = -9 (6, -9 ) PRACTICE Solve each system of equations. 1. ⎧ ⎪ ⎨ ⎪ ⎩ 3x - 3y = 4 x + y = 10 _ 3 5. ⎧ 4x - 6y = 1 ⎨ 3y - x = 2 ⎩ 7. ⎧ 3x - y = 6 ⎨ ⎩ y = 2x + 3 9. ⎧ -3x + 2y = 31 ⎨ x = 0.5y + 6 ⎩ Solve the second equation for y. Substitute -2x + 3 for y into the
first equation. Distribute 2. Simplify. Solve for x. Substitute 6 for x into the second equation and simplify. Simplify. Write the solution as an ordered pair. Multiply each term in the second equation by -2 to get opposite y-coefficients. Simplify. Write the system using the new equation so that like terms are aligned. Add like terms on both sides of the equations. Solve for x. Substitute 6 for x into the second equation and simplify. Simplify. Write the solution as an ordered pair. ⎧ 3x + y = 1 2. ⎨ x + y = -3 ⎩ ⎧ ⎪ 4 2x - 3y = 2 ⎧ x - y = 0 6. ⎨ 2x + 3y = 0 ⎩ ⎧ 2x + 5y = 14 8. ⎨ y = 5 ⎩ ⎧ 3x + y = 4 10. ⎨ x - 2y = 6 ⎩ Skills Bank S67 S67 Solving Systems of Linear Inequalities A system of linear inequalities is a set of two or more inequalities with two or more variables. The solution to a system of inequalities consists of all the ordered pairs that satisfy all the inequalities in the system ⎨ Solve the following system of linear inequalities. + x > -1 ⎪ ⎩ 2 y > -2x - 2 Write the second inequality in slope-intercept form. Graph the solution of each inequality. The solutions of the system are represented by the overlapping shaded regions. PRACTICE Solve each system of linear inequalities. 1. ⎧ y ≤ 2x ⎨ y ≤ x ⎩ ⎧ y ≤ 2x + 1 2. ⎨ y ≤ x - 1 ⎩ Solving Radical Equations A radical equation is an equation that has a variable within a radical, such as a square root. To solve a square-root equation, square both sides and solve the resulting equation. E X A M P L E ( √  x - 9 ) 2 = (1) 2 Square both sides. Solve the equation √  x - 9 = 1. Check your answer. x - 9 = 1 Simplify. x = 10 10 - 9 = √  1 = 1 ✔ Solve
for x. Check √  PRACTICE Solve each equation. Check your answer. 1. √  x + 1 = 4 3. √  1 - x = 3 5. √  7 + x = 0 7. √  3 - 2x = 3 S68 S68 Skills Bank 2. √  2x - 1 = 5 4. √  -6 - 5x = 2 6. √  4x + 4 = 2 8. √  60 - 2x = 8 �� Matrix Operations A matrix is a rectangular array of numbers enclosed in brackets. The entries in a matrix are arranged in rows and columns. Operations such as addition, subtraction, and multiplication by a constant can be performed on matrices. E X A M P L E Simplify. A ⎡ 1 ⎢ 2 -3 ⎣ ⎡ 1 ⎢ 2 -3 ⎣ -2 0 1 -2 0 1 ⎤ 2 ⎥ 3 -4 ⎦ ⎤ 2 ⎥ 3 -3 + 3 ⎥ 11 5 ⎦ Add corresponding entries. B ⎡ 1 ⎢ 2 -3 ⎣ -2 0 1 ⎤ 2 ⎥ 3 -6 ⎣ -6 -5 -5 ⎤ -5 ⎥ -5 -13 ⎦ Subtract corresponding entries3 - 3 -3 ⎣ -2 0 1 ⎤ 2 ⎥ 3 -4 ⎦ 2 (1) 2 (-2) 2 (2) 2 (0) ⎡ ⎢ ⎣ 2 (-3) 2 (1) 2 (-4) ⎦ ⎤ 2 (2) 2 (3) = ⎥ ⎡ ⎢ 2 4 -6 ⎣ -4 0 2 ⎤ 4 6 -8 ⎦ ⎥ Multiply each entry by 2. PRACTICE Simplify. ⎡
1 1. ⎢ 0 ⎣ ⎡ ⎤ 0 -1 ⎦ 4. ⎡ 1 ⎢ 0 ⎣ ⎡ ⎤ -1 -3 ⎦ ⎡ 0 7.5 0 ⎣ ⎦ ⎤ -9 ⎥ -1 ⎦ 2. 2 ⎢ 7 ⎣ ⎤ 2 ⎥ -1 ⎦ 8. ⎡ 8 ⎢ 0 ⎣ ⎡ ⎤ -2 -1 - ⎢ ⎥ 4 -3 ⎣ ⎦ ⎤ 1 ⎥ -7 ⎦ ⎡ 3 3. -. 1 _ ⎢ 2 0 ⎣ ⎤ -8 ⎥ 2 ⎦ Skills Bank S69 S69 Structure of Measurement Systems The metric system of measurement is used worldwide. In the United States, we most commonly use the customary system of measurement but still use the metric system in most science applications. Customary System 12 in. = 1 ft 8 oz = 1 c 16 oz = 1 lb Length Capacity Weight/Mass 3 ft = 1 yd 5280 ft = 1 mi 2 c = 1 pt 2 pt = 1 qt 4 qt = 1 gal 2000 lb = 1 ton Metric System 1000 mm = 1 m 1000 mL = 1 L 1000 mg = 1 g 100 cm = 1 m 1000 L = 1 kL 1000 g = 1 kg 1000 m = 1 km E X A M P L E Complete each conversion. A 1560 mL = L B 2 mi = yd 1560 mL × 1L _ 1000 mL = 1.56 L 2 mi × 5280 ft _ × 1 mi 1 yd _ 3 ft = 3520 yd PRACTICE Complete each conversion. 1. 2.8 m = cm 2. 128 oz = lb 4. 87 ft = yd 5. 2.6 kL = mL 3. 4 1 _ 2 gal = c 6. 108 mg = g Rates and Derived Measurements A rate is the ratio of the change in one measurement to the change in another measurement, usually time. The units of a rate are derived units, or the ratio of two different units, such as miles per hour (mi/h) and kilograms per meter (kg/m car travels 1 km every 5 min. What is the speed of the car in meters per second
? The speed of the car is the ratio of the change in distance to the change in time. 1 km _ 5 min × 1000 m _ 1 km Convert km to m and min to s. × 1 min _ 60 s ≈ 3.33 m/s PRACTICE 1. The mass of a small meteor is decreasing at a rate of 6000 g every 2 min. What is the rate of decrease in kilograms per second? 2. The temperature increases at a rate of 2°F every half hour. What is the rate of increase in degrees Fahrenheit per minute? Round to the nearest thousandth. 3. An athlete runs at a rate of 9.5 m/s. What is the runner’s rate in kilometers per hour? S70 S70 Skills Bank Unit Conversions To convert a measurement in one system to a measurement in another system, multiply by a conversion factor, such as 1 cm ≈ 0.394 in., written as a fraction. If you are converting ______ from inches to centimeters, “cm” goes in the numerator — 1 cm 0.394 in. ______ from centimeters to inches, “in.” goes in the numerator — 0.394 in.. 1 cm. If you are converting Common Conversion Factors Length Capacity Mass/Weight Temperature Metric to Customary 1 cm ≈ 0.394 in. 1 L ≈ 4.227 c 1 g ≈ 0.0353 oz 1 m ≈ 3.281 ft 1 L ≈ 1.057 qt 1 kg ≈ 2.205 lb F = 9 _ 5 C + 32 1 m ≈ 1.094 yd 1 L ≈ 0.264 gal 1 km ≈ 0.621 mi 1 mL ≈ 0.034 oz Customary to Metric 1 in. ≈ 2.540 cm 1 c ≈ 0.237 L 1 oz ≈ 28.350 g 1 ft ≈ 0.305 m 1 qt ≈ 0.946 L 1 lb ≈ 0.454 kg C = 5 _ 9 (F - 32) 1 yd ≈ 0.914 m 1 gal ≈ 3.785 L 1 mi ≈ 1.609 km 1 oz ≈ 29.574 mL E X A M P L E Complete each conversion. If necessary, round to the nearest hundredth. ° F A 20° C = F = 9 _ (20 ) + 32 5 F = 68 20° C is equivalent to
68° F. Simplify. Substitute 20 for C. B 25 lb ≈ g 25 lb × 0.454 kg _ × 1 lb 1000 g _ 1 kg ≈ 11,350 g C 32 ft 2 ≈ m 2 32 ft 2 × 0.305 m _ × 0.305 m _ 1 ft ≈ 2.98 m 2 1 ft Use the conversion factor for pounds to kilograms. Then convert kilograms to grams. The units are squared, so apply the conversion factor for feet to meters twice. PRACTICE Complete each conversion. If necessary, round to the nearest hundredth. 1. 40° C = 3. 2 3 _ 4 5. 18. 12,300 mg ≈ lb 9. 64 gal ≈ kL 11. 98° F ≈ ° C 2. 15 in. ≈ cm 4. 86°F ≈ ° C 6. 18 kg ≈ lb 8. 150 mL ≈ c 10. 98 lb ≈ kg 12. 100 yd ≈ m Skills Bank S71 S71 Accuracy, Precision, and Tolerance The accuracy of a measurement is the numerical measure of how close the measured value is to the actual value of the quantity. A measurement of 133 cm for an actual object with a length of 132.7 cm is accurate to the nearest centimeter. The precision of a measurement is the level of detail determined by the number of decimal places to which the measurement is taken. The measurements 27 cm and 39.48 m, or 3948 cm, are both precise to the nearest centimeter. Tolerance is the range of values within which a measurement lies. In the measurement 16.3 ± 0.1 mm, the ± 0.1 mm is the tolerance of the measurement. So if the measurement is accurate, the actual value of the object being measured is from 16.2 mm to 16.4 mm. E X A M P L E The weight of an object is 3.72 lb. Three measurements of the object were recorded: 3.75 ± 0.02 lb, 3.718 ± 0.002 lb, and 3.73 ± 0.001 lb. Which measurement is the most accurate? Which is the most precise? Which has the smallest tolerance? 3.75 lb ± 0.02 lb 3.718 lb ± 0.002 lb 3.73 lb ± 0.001 lb Ranges from 3.73 lb to 3.77 lb Ranges from 3.716 lb to 3.720 lb Ranges
from 3.729 lb to 3.731 lb Step 1: Find the most accurate measurement. The most accurate measurement is the measurement closest to the actual weight of 3.72 lb. 3.718 ± 0.002 lb Step 2: Find the most precise measurement. The most precise measurement is the measurement (not the tolerance) with the most decimal places. 3.718 ± 0.002 lb Step 3: Find the measurement with the smallest tolerance. The most tolerant measurement is the one with the smallest ± value. 3.73 ± 0.001 lb PRACTICE In each problem, the first value is the actual measure of an object, followed by multiple recorded measurements. Which measurement is the most accurate? Which is the most precise? Which has the smallest tolerance? 1. 1.00 in. {1.0 ± 0.1 in., 1.01 ± 0.01 in., 2 ± 1 in., 1.001 ± 0.1 in.} 2. 2.50 s {2.5 ± 0.1 s, 2.5100 ± 0.0001 s, 2.515 ± 2 s, 2.51 ± 0.01 s} 3. 11.51 m {10.5 ± 0.5 m, 11.51 ± 5 m, 11.500001 ± 0.9 m, 22 ± 12 m} 4. 0.50102041 g {0.5010204 ± 0.5 g, 22.51 ± 0.00000001 g, 51.5120843447 ± 50 g, 0.50102041 ± 5 g} S72 S72 Skills Bank Relative and Absolute Error The absolute error of a measurement is the difference between the measured value and the actual value of the quantity being measured. Absolute error can be misleading when very large or very small numbers are being measured. You can avoid this by using the relative error, which is the absolute error divided by the actual value. Relative error has no units. When expressed as a percent, this is the percent error of a measurement. E X A M P L E Find the absolute, relative, and percent errors. The first number is the actual value, and the second is the measured value. A 5.1 m, 5.0 m B 5.1 ft, 5.5 ft absolute error = 5.0 m - 5.1 m = -0.1 m relative error = 5.0 m - 5.1 m __ 5.1 m = -0
.0196 absolute error = 5.5 ft - 5.1 ft = 0.4 ft relative error = 5.5 ft-5.1 ft __ = 0.0784 5.1 ft percent error = -1.96% percent error = 7.84% PRACTICE Find the absolute, relative, and percent errors. The first value is the actual value, and the second is the measured value. 1. 1.23, 1.00 3. 5.55, 6.00 2. 123, 100 Significant Digits All the digits in a measurement that are known to be exact are called significant digits. Rule Example Number of Significant Digits Significant Digits All nonzero digits Zeros between nonzero digits Zeros after the last nonzero digit that are to the right of the decimal point 14.28 8.002 0.030 4 4 2 14.28 8.002 0.030 Zeros at the end of a whole number are assumed to be nonsignificant. So the measurement 700 has 1 significant digit—the 7. E X A M P L E Determine the number of significant digits in each measurement. A 815 lb B 2 × 10 -2 L C 750 kg D 15.08 m 3 significant digits 1 significant digit 2 significant digits 4 significant digits PRACTICE Determine the number of significant digits in each measurement. 1. 1203 lb 3. 6.003 mi 2. 3.0 cm 4. 5.000 kg 5. 1000 in. 9. 91.0 s 6. 1 × 10 3 L 7. 03.101 g 8. 1.60200 km 10. 2.00 × 10 4 cm 11. 0.10 m 12. 0.00105 lb Skills Bank S73 S73 Choose Appropriate Units When measuring a quantity, it is important to choose the appropriate units so that the measurement will have a reasonable magnitude. E X A M P L E Name the appropriate unit to measure the mass of an elephant (milligram, gram, kilogram, metric tons). The average mass of an elephant is around 5 metric tons, 5000 kg, 5,000,000 g, or 5,000,000,000 mg. Since 5 metric tons has the most reasonable magnitude, the mass of an elephant should be weighed in metric tons. PRACTICE Select the appropriate unit for each measurement. 1. the height of a classroom (millimeter, centimeter, meter, kilometer) 2. the distance from Earth to the
Sun (inches, feet, yards, miles) 3. the length of a decade (seconds, minutes, hours, years) Nonstandard Units There are several nonstandard unit systems. pH, a measure of the concentration of hydrogen ions in a solution, ranges from 0 to 14. Pure water has a pH of 7, which is considered neutral. A pH less than 7 is acidic, and a pH greater than 7 is basic, or alkaline. The Richter scale measures the magnitude of earthquakes. The pH scale and the Richter scale are related by powers of 10. An increase of 1 unit on the scale means an increase by a factor of 10 in the quantity. For example, an earthquake that measures 6.0 on the Richter scale is 10 times as great as one that measures 5.0. E X A M P L E 1 Solutions A and B have the same volume. The pH of solution A is 4, and the pH of solution B is 5. How much more acidic is solution A than solution B? Since 5 - 4 = 1, and 10 1 = 10, solution A is 10 times more acidic than solution B. E X A M P L E 2 Earthquake A had a magnitude of 2.7 on the Richter scale. Earthquake B had a magnitude of 4.7. How much stronger was earthquake B than earthquake A? Since 4 - 2 = 2, and 10 2 = 100, earthquake B was 100 times stronger than earthquake A. PRACTICE 1. Solutions P and Q have the same volume. The pH of solutions P and Q are 1 and 7, respectively. How much more basic is solution Q than solution P? 2. Earthquake R has a magnitude of 5. How much stronger is earthquake R than earthquake S, with magnitude 2? S74 S74 Skills Bank Use Tools for Measurement Length can be measured with tools such as a ruler, a measuring tape, or a micrometer. Weight can be measured with various types of balance scales. Time can be measured with a calendar, a clock, or a stopwatch. Electronic timing devices that measure to a precision of 0.01 s or 0.001 s are used in track events. E X A M P L E What is the width of the bolt? Read the scale on the micrometer. The main scale reads 8.5 mm. The fine scale reads 0.120 mm. The width of the bolt is therefore 8.620 mm. PRACTICE 1. Use a ruler to measure the dimensions
of your Geometry book. 2. Use a stopwatch to measure the time it takes to climb a set of stairs. 3. Use a tape measure to find the height of your classroom doorway. Choose Appropriate Measuring Tools To choose an appropriate measuring tool, consider the following criteria: • How large is the quantity being measured? • How much precision is needed for the measurement? E X A M P L E What instrument would you use to measure the width of a textbook to the nearest sixteenth of an inch: a micrometer, a ruler, or a measuring tape? A measuring tape may be precise only to the nearest quarter of an inch. A micrometer is not large enough to measure a textbook. So a ruler is the most appropriate tool. PRACTICE What instrument would you use to measure each quantity? 1. the duration of a feature film to the nearest minute 2. the length and width of a room to the nearest inch 3. the weight of a feather to the nearest microgram 4. the capacity of a drinking glass to the nearest ounce Skills Bank S75 S75 Measures of Central Tendency A measure of central tendency describes how data clusters around a value in a statistical distribution. The mean is the average value of the data. The median is the middle value when the data is in numerical order or the average of the two middle terms if there is an even number of terms. The mode is the value that occurs most often. There can be one, several, or no modes. E X A M P L E Find the mean, median, and mode of the following data set. {3, 5, 1, 9, 0, 1, 5, 6, 0, 3, 5, 7, 8, 1, 3, 5, 8, 3, 2, 7, 1, 6, 8, 4, 3, 2, 7, 3, 6, 8} Step 1: Find the mean. Add all the data values, and then divide the sum by the number of terms in the set. _ 130 30 mean ≈ 4.33 The sum of the data values is 130, and there are 30 values. = 13 _ 3 Step 2: Find the median. Order the set from the least value to the greatest value. {0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7,
7, 8, 8, 8, 8, 9} To find the median, locate the middle term. Since there are 30 terms, the median will be the average of the 15th and 16th terms. {0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9} 15 terms 4 + 5 = 9 _ _ 2 2 median = 4.5 Step 3: Find the mode. 15 terms The 15th term is 4. The 16th term is 5. Find the average. Data How Often occurs most often. mode = 3 PRACTICE Find the mean, median, and mode of each data set. Round to the nearest hundredth if necessary. 1. {1, 2, 2, 1, 2, 1, 2, 1} 2. {-1, 0, 4, 3, 2, 0, -3, -4, -1, 0} 3. {0, 1, 2, 1, 0, 2, 3, 7, 2} 4. {3, 7, 12, 8, 3, 7, 1, 6, 3, 7, 1, 9, 3, 100, 2} 5. {100, 111, 112, 100, 99, 104, 103, 108} 6. {4, 34, 72, 675, 12, 56, 2, 67, 12, 5, 387, 12, 4, 23, 5, 72, 56, 23, 56, 45, 2, 6} 7. {43, 23, 31, 53, 97, 79, 57, 11, 13, 11, 43, 61, 91, 87, 83, 73, 37, 41, 29} 8. {34, 45, 12, 6, 12, 45, 23, 6, 12, 78, 45, 67, 33} S76 S76 Skills Bank Probability Probability is a measure of how likely an event is to occur. It is calculated by taking the number of outcomes for which the event occurs and dividing by the total number of possible outcomes. Probability can be expressed as a fraction or as a decimal between 0 and 1, or as a percentage between 0% and 100%. The higher the probability, the more likely the event will occur. E X A M P L
E Find the probability of rolling each sum with two number cubes. The two number cubes are independent. In order to roll a 2, both number cubes need to show 1. So there is one possible way to roll a 2. To roll a 3, the first cube can show 1 and the second cube can show 2, or the first cube can show 2 and the second cube can show 1. So there are two possible ways to roll a 3. Repeating this method gives the following table. Sum Number of Outcomes Probability (fraction) Probability (decimal) Probability (percent) 2 3 4 5 6 7 8 9 10 11 12 Total: 36 1 _ 36 = 1 _ 2 _ 18 36 3 _ = 1 _ 12 36 = 1 _ 4 _ 9 36 5 _ 36 6 _ = 1 _ 6 36 5 _ 36 = 1 _ 4 _ 36 9 3 _ = 1 _ 12 36 = 1 _ 2 _ 18 36 1 _ 36 36 _ 36 = 1 0.0278 0.0556 0.0833 0.1111 0.1389 0.1667 0.1389 0.1111 0.0833 0.0556 0.0278 1 2.8% 5.6% 8.3% 11.1% 13.9% 16.7% 13.9% 11.1% 8.3% 5.6% 2.8% 100% PRACTICE Find the probability of each event. Write your answer as a percent rounded to the nearest tenth if necessary. 1. Dave puts 2 red marbles, 3 blue marbles, and 5 green marbles into a bag. Samir then randomly pulls a marble out of the bag and replaces it. What is the probability that Samir chooses a red marble? a blue marble? a green marble? 2. What is the probability of drawing a heart from a standard deck of 52 cards? a jack, queen, or king? an ace? 3. Shireen asks Kate to guess the number she has chosen between 1 and 100. What is the probability that Kate guesses the number correctly? Skills Bank S77 S77 Organizing and Describing Data One way to organize and display data is by using a table. The table can then help you to understand the meaning of the data and identify any relationships within the data. E X A M P L E Organize the given data in a table. Then describe the data. Rick has been tracking his recent test results. He scored 81
% on a Language Arts test, 67% on a History test, 78% on a Science test, 82% on a Spanish test, and 90% on a Geometry test. Test Percent History Science Language Arts Spanish Geometry 67% 78% 81% 82% 90% List each test and its percent score. The table shows that Rick did very well on his Geometry test, but needs to improve on History. All his test scores except for History are above 75%. PRACTICE 1. Organize the given temperatures in a table. Then describe the data. The temperature was 32°F from 6 to 8 A.M., 40°F from 8 to 10 A.M., 56°F from 10 A.M. to 12 P.M., 72°F from 12 to 2 P.M., 65°F from 2 to 4 P.M., 54°F from 4 to 6 P.M., and 40°F from 6 to 8 P.M. Displaying Data Two other ways to display data are to use a bar graph and a histogram. A bar graph is used when the data values are disjoint, and the data represent categories that are not directly related to each other. The bars do not touch. In a histogram, the data categories are usually numerical intervals such as 0–9, 10–19, and so on. The bars in a histogram do touch. E X A M P L E Use the data in the example above to make a vertical bar graph of Rick’s test scores. Choose an appropriate scale and interval. Draw a bar for each test. Title and label the graph. PRACTICE 1. Use the data in Practice Problem 1 above to make a histogram of the temperatures. S78 S78 Skills Bank �� Scatter Plots and Trend Lines A scatter plot is a graph of ordered pairs of data. A scatter plot can help you see any clusters or trends in the data. A trend line is a straight line that accurately expresses the trend of the data on a scatter plot. When drawing a trend line, try to follow clusters of data and the overall pattern of the plot teacher asked 15 students to record the time they spent studying for their geometry test. The results are displayed in the table. Make a scatter plot of the data in the table. Then draw a trend line for the data. Explain the meaning of the trend line.
Time (min) Test Score Graph the ordered pairs (time, score) on a graph using a reasonable scale. Draw a trend line. 0 60 180 160 100 30 15 45 15 0 60 120 90 45 30 50 60 95 100 90 70 60 80 50 60 75 85 80 70 65 The trend line indicates that a student’s scores improved as study time increased. PRACTICE Make a scatter plot of each data set. Then draw a trend line for the data. Explain the meaning of the trend line. 1. A teacher asked nine students to record the number of minutes they spent watching TV the day before a test. The results are displayed in the table. Time Watching TV (min) 0 60 20 45 10 30 60 40 15 Test Score 28 15 20 18 27 29 16 17 24 2. A study was performed to see how age relates to the number of pizza slices someone eats. The results of the study are in the table. Age Pizza Slices 12 4 6 2 18 20 11 12 15 10 10 5 5 3 10 14 17 15 20 6 5 8 6 9 5 4 9 6 14 6 Skills Bank S79 S79 �� Quartiles and Box-and-Whisker Plots A box-and-whisker plot is a graph showing the lower extreme (the least value), the upper extreme (the greatest value), the median, the lower quartile (the median of the lower half of the data), and the upper quartile (the median of the upper half of the data). E X A M P L E 1 Make a box-and-whisker plot of the test scores below. Test Score 9 6 3 8 10 4 7 3 9 7 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 Order the data from least to greatest. Identify the lower and upper extremes. Identify the median. 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 Identify the lower quartile and the upper quartile. Make a box using the median and upper and lower quartiles. Place a bar at the upper and lower extremes. Connect the bars to the box with segments called whiskers. PR
ACTICE 1. Make a box-and-whisker plot of the data set. 12 18 10 17 18 15 17 13 7 14 19 Circle Graphs Circle graphs are used to represent data as percentages of the total. To draw a circle graph, convert the data to percentages, and then make a section of the circle for each category. E X A M P L E 1 Make a circle graph of the data from the study in the table. Team Fans Cougars Panthers Jaguars Tigers Lions 375 375 125 75 50 Percent of fans Degrees of Circle ____ 375 1000 = 37.5% ____ 375 1000 = 37.5% ____ 125 1000 = 12.5% 75 ____ 1000 = 7.5% 50 ____ 1000 = 5% 0.375 (360°) = 135° 0.375 (360°) = 135° 0.125 (360°) = 45° 0.075 (360°) = 27° 0.05 (360°) = 18° Draw a circle, and use the angle degrees to make the sections. PRACTICE 1. Make a circle graph of the data in the table. Favorite Food Number of Students Salad Pizza Smoothie Soup 5 10 4 1 S80 S80 Skills Bank �� Misleading Graphs and Statistics Graphs can be misleading if the related statistics are distorted. E X A M P L E Explain why this graph is misleading. Gary’s score appears to be about three times that of Matt’s. Matt appears to do twice as well as Peter. This is because the scale on the graph begins well above zero. In fact, Gary is only 0.6% above Matt, and Peter is 0.2% below Matt. PRACTICE Explain why each graph is misleading. 1. 2. Venn Diagrams Venn diagrams are used to show relationships between two or more sets of numbers or objects. They show which elements are common between sets. E X A M P L E Draw a Venn diagram to show the relationship between the factors of 24 and factors of 32. Step 1: Find the elements of each set. A: {1, 2, 3, 4, 6, 8, 12, 24} Step 2: Draw a Venn diagram. B: {1, 2, 4, 8, 16, 32} Draw two overlapping ovals, one for each set. Write factors that are in both sets in the
overlapping region. Write factors in one set only in the non-overlapping parts. PRACTICE Draw a Venn diagram to show the relationship between the following sets. 1. factors of 9 and factors of 8 2. factors of 36 and factors of 30 3. factors of 60 and factors of 72 Skills Bank S81 S81 �� Postulates, Theorems, and Corollaries Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. (p. 7) Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. (p. 7) Post. 1-1-3 If two points lie in a plane, then the line containing those points lies in the plane. (p. 7) Post. 1-1-4 If two lines intersect, then they intersect in exactly one point. (p. 8) Post. 1-1-5 If two planes intersect, then they intersect in exactly one line. (p. 8) Post. 1-2-1 Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. (Ruler Post.; p. 13) Post. 1-2-2 Segment Addition Postulate If B is between A and C, then AB + BC = AC. (Seg. Add. Post.; p. 14) Post. 1-3-1 Protractor Postulate Given   AB and a point O on   AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. (Protractor Post.; p. 20) Post. 1-3-2 Angle Addition Postulate If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR. (∠ Add. Post.; p. 22) Thm. 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length
of the hypotenuse. (Pyth. Thm.; p. 45) Chapter 2 Thm. 2-6-1 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. (Lin. Pair Thm.; p. 110) Thm. 2-6-2 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. (≅ Supps. Thm.; p. 111) Thm. 2-6-3 Right Angle Congruence Theorem All right angles are congruent. (Rt. ∠ ≅ Thm.; p. 112) Thm. 2-6-4 Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. (≅ Comps. Thm.; p. 112) S82 S82 Postulates, Theorems, and Corollaries Thm. 2-7-1 Common Segments Theorem Given collinear points A, B, C, and D arranged as shown, if Segs. Thm.; p. 118) ̶̶ BD. (Common ̶̶ CD, then ̶̶ AC ≅ ̶̶ AB ≅ Thm. 2-7-2 Vertical Angles Theorem Vertical angles are congruent. (Vert.  Thm.; p. 120) Thm. 2-7-3 If two congruent angles are supplementary, then each angle is a right angle. (≅  supp. → rt. ; p. 120) Chapter 3 Post. 3-2-1 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. (Corr.  Post.; p. 155) Thm. 3-2-2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. (Alt. Int.  Thm.; p. 156) Thm. 3-2-3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. (Alt. Ext
.  Thm.; p. 156) Thm. 3-2-4 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. (Same-Side Int.  Thm.; p. 156) Post. 3-3-1 Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. (Conv. of Corr.  Post.; p. 162) Post. 3-3-2 Parallel Postulate Through a point P not on line ℓ, there is exactly one line parallel to ℓ. (Parallel Post.; p. 163) Thm. 3-3-3 Converse of the Alternate Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. (Conv. of Alt. Int.  Thm.; p. 163) Thm. 3-3-4 Converse of the Alternate Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. (Conv. of Alt. Ext.  Thm.; p. 163) �� Thm. 3-3-5 Converse of the Same-Side Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel. (Conv. of Same-Side Int.  Thm.; p. 163) Thm. 3-4-1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of ≅  → lines ⊥; p. 173) Thm. 3-4-2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. (⊥ Transv. Thm.; p. 173) Thm. 3-4-3 If two
coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines ⊥ to same line → 2 lines ǁ; p. 173) Thm. 3-5-1 Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. (ǁ Lines Thm.; p. 184) Thm. 3-5-2 Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. (⊥ Lines Thm.; p. 184) Chapter 4 Thm. 4-2-1 Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. (△ Sum Thm.; p. 223) Cor. 4-2-2 The acute angles of a right triangle are complementary. (Acute  of rt. △ are comp.; p. 224) Cor. 4-2-3 The measure of each angle of an equiangular triangle is 60°. (p. 224) Thm. 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. (Ext. ∠ Thm.; p. 225) Thm. 4-2-5 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. (Third  Thm.; p. 226) Post. 4-4-1 Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. (SSS; p. 242) Post. 4-4-2 Side-Angle-Side (SAS) Congruence Postulate 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. (SAS; p. 243) Post. 4-5-1 Angle-Side-Angle (ASA) Congruence Postulate 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. (ASA; p. 252) Thm. 4-5-2 Angle-Angle-Side (AAS) Congruence Theorem 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. (AAS; p. 254) Thm. 4-5-3 Hypotenuse-Leg (HL) Congruence Theorem 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. (HL; p. 255) Thm. 4-8-1 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. (Isosc. △ Thm.; p. 273) Thm. 4-8-2 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. (Conv. of Isosc. △ Thm.; p. 273) Cor. 4-8-3 If a triangle is equilateral, then it is equiangular. (equilateral △ → equiangular; p. 274) Cor. 4-8-4 If a triangle is equiangular, then it is equilateral. (equiangular △ → equilateral; p. 275) Chapter 5 Thm. 5-1-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. (⊥ Bisector Thm.; p. 300) Thm. 5-1-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Conv. of ⊥ Bisector Thm.; p. 300) Thm. 5-1-3 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (∠ Bisector Thm.; p. 301) Postulates, Theorems,
and Corollaries S83 S83 Thm. 5-1-4 Converse of the Angle Bisector Thm. 5-7-2 Pythagorean Inequalities Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. (Conv. of ∠ Bisector Thm.; p. 301) Theorem In △ABC, c is the length of the longest side. If c 2 > a 2 + b 2, then △ABC is an obtuse triangle. If c 2 < a 2 + b 2, then △ABC is an acute triangle. (Pyth. Inequal. Thm.; p. 351) Thm. 5-2-1 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. (Circumcenter Thm.; p. 307) Thm. 5-8-1 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √  2. (45°-45°-90° △ Thm.; p. 356) Thm. 5-2-2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. (Incenter Thm.; p. 309) Thm. 5-3-1 Centroid Theorem The centroid of a triangle is located 2 __ 3 of the distance from each vertex to the midpoint of the opposite side. (Centroid Thm.; p. 314) Thm. 5-4-1 Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side. (△ Midsegment Thm.; p. 323) Thm. 5-5-1 If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. (In △, larger ∠ is opp. longer side; p. 333) Thm. 5-5-2 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (In △, longer side is opp. larger ∠; p.
333) Thm. 5-5-3 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. (△ Inequal. Thm.; p. 334) Thm. 5-6-1 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. (Hinge Thm.; p. 340) Thm. 5-6-2 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. (Conv. of Hinge Thm.; p. 340) Thm. 5-7-1 Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. (Conv. of Pyth. Thm.; p. 350) Thm. 5-8-2 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √  3. (30°-60°-90° △ Thm.; p. 358) Chapter 6 Thm. 6-1-1 Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°. (Polygon ∠ Sum Thm.; p. 383) Thm. 6-1-2 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. (Polygon Ext. ∠ Sum Thm.; p. 384) Thm. 6-2-1 If a quadrilateral is a parallelogram, then its opposite sides are congruent. ( → opp. sides ≅; p. 391) Thm. 6-2-2 If a quadrilateral is a parallelogram, then its opposite angles are congruent
. ( → opp.  ≅; p. 392) Thm. 6-2-3 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( → cons.  supp.; p. 392) Thm. 6-2-4 If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( → diags. bisect each other; p. 392) Thm. 6-3-1 If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (quad. with pair of opp. sides ǁ and ≅ → ; p. 398) Thm. 6-3-2 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp. sides ≅ → ; p. 398) Thm. 6-3-3 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp.  ≅ → ; p. 398) S84 S84 Postulates, Theorems, and Corollaries Thm. 6-3-4 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. (quad. with ∠ supp. to cons.  → ; p. 399) Thm. 6-3-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (quad. with diags. bisecting each other → ; p. 399) Thm. 6-4-1 If a quadrilateral is a rectangle, then it is a parallelogram. (rect. → ; p. 408) Thm. 6-4-2 If a parallelogram is a rectangle, then its diagonals are congruent. (rect. → diags. ≅; p. 408) Thm. 6-4-3 If a quadrilateral is a rhombus, then it is a parallelogram. (rh
ombus → ; p. 409) Thm. 6-4-4 If a parallelogram is a rhombus, then its diagonals are perpendicular. (rhombus → diags. ⊥; p. 409) Thm. 6-4-5 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. (rhombus → each diag. bisects opp. ; p. 409) Thm. 6-5-1 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. ( with one rt. ∠ → rect.; p. 418) Thm. 6-5-2 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. ( with diags. ≅ → rect.; p. 418) Thm. 6-5-3 If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. ( with one pair cons. sides ≅ → rhombus; p. 419) Thm. 6-5-4 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. ( with diags. ⊥ → rhombus; p. 419) Thm. 6-5-5 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. ( with diags. bisecting opp.  → rhombus; p. 419) Thm. 6-6-1 If a quadrilateral is a kite, then its diagonals are perpendicular. (kite → diags. ⊥; p. 427) Thm. 6-6-2 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. (kite → one pair opp.  ≅; p. 427) Thm. 6-6-3 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. (isosc. trap. → base  ≅; p
. 429) Thm. 6-6-4 If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. (trap. with pair base  ≅ → isosc. trap.; p. 429) Thm. 6-6-5 A trapezoid is isosceles if and only if its diagonals are congruent. (isosc. trap ↔ diags. ≅; p. 429) Thm. 6-6-6 Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. (Trap. Midsegment Thm.; p. 431) Chapter 7 Post. 7-3-1 Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (AA ∼ Post.; p. 470) Thm. 7-3-2 Side-Side-Side (SSS) Similarity Theorem If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. (SSS ∼ Thm.; p. 470) Thm. 7-3-3 Side-Angle-Side (SAS) Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. (SAS ∼ Thm.; p. 471) Thm. 7-4-1 Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. (△ Proportionality Thm.; p. 481) Thm. 7-4-2 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. (Conv. of △ Proportionality Thm.; p. 482) Cor. 7-4-3 Two-Transversal Proportionality Corollary If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. (2-Transv. Proportionality Cor.; p. 482) Thm. 7-4-4
Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. (△ ∠ Bisector Thm.; p. 483) Thm. 7-5-1 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a __, then the ratio of their perimeters is b, and the ratio of their areas is a 2 ___ a __ b 2 b or ( a __ ). (p. 490) b 2 Postulates, Theorems, and Corollaries S85 S85 Chapter 8 Thm. 8-1-1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. (p. 518) Cor. 8-1-2 Geometric Means Corollary The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. (p. 519) Cor. 8-1-3 Geometric Means Corollary The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. (p. 519) Thm. 8-5-1 The Law of Sines For any △ABC ____ ____ ____ a = sin B = sin C. c b with side lengths a, b, and c, sin A (p. 552) Thm. 8-5-2 The Law of Cosines For any △ABC with sides a, b, and c, a 2 = b 2 + c 2 - 2bc cos A, b 2 = a 2 + c 2 - 2ac cos B, and c 2 = a 2 + b 2 - 2ab cos C. (p. 553) Chapter 9 Post. 9-1-1 Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. (Area Add. Post.; p. 589) Chapter 11 Thm. 11-1-1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to ⊙ → line ⊥ to radius; p. 748) Thm. 11-1
-2 If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line ⊥ to radius → line tangent to ⊙; p. 748) Thm. 11-1-3 If two segments are tangent to a circle from the same external point, then the segments are congruent. (2 segs. tangent to ⊙ from same ext. pt. → segs. ≅; p. 749) Post. 11-2-1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. (Arc Add. Post.; p. 757) Thm. 11-2-2 In a circle or congruent circles: (1) congruent central angles have congruent chords, (2) congruent chords have congruent arcs, and (3) congruent arcs have congruent central angles. (≅ arcs have ≅ central  have ≅ chords; p. 757) S86 S86 Postulates, Theorems, and Corollaries Thm. 11-2-3 In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. (Diam. ⊥ chord → diam. bisects chord and arc; p. 759) Thm. 11-2-4 In a circle, the perpendicular bisector of a chord is a radius (or diameter). (⊥ bisector of chord is diam.; p. 759) Thm. 11-4-1 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. (Inscribed ∠ Thm.; p. 772) Cor. 11-4-2 If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. (p. 773) Thm. 11-4-3 An inscribed angle subtends a semicircle if and only if the angle is a right angle. (p. 774) Thm. 11-4-4 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. (Quad. inscribed in circle → opp.  supp.; p.
775) Thm. 11-5-1 If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. (p. 782) Thm. 11-5-2 If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. (p. 783) Thm. 11-5-3 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. (p. 784) Thm. 11-6-1 Chord-Chord Product Theorem If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. (p. 792) Thm. 11-6-2 Secant-Secant Product Theorem If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole • outside = whole • outside; p. 793) Thm. 11-6-3 Secant-Tangent Product Theorem If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole • outside = tangent 2 ; p. 794) Thm. 11-7-1 Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k) 2 = r 2. (p. 799) Chapter 12 Thm. 12-4-1 A composition of two isometries is an isometry. (p. 848) Thm. 12-4-2 The composition of two reflections across two parallel lines is equivalent to a translation. The translation vector is perpendicular to the lines. The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. The center of rotation is the intersection of the lines. The angle
of rotation is twice the measure of the angle formed by the lines. (p. 849) Thm. 12-4-3 Any translation or rotation is equivalent to a composition of two reflections. (p. 850) Constructions Angle Bisector............................. p. 23 Parallel Lines............... pp. 163, 170, 171, 179 Center of a Circle......................... p. 774 Parallelogram............................ p. 404 Centroid of a Triangle..................... p. 314 Perpendicular Bisector Circle Circumscribed of a Segment......................... p. 172 About a Triangle................ pp. 313, 778 Perpendicular Lines...................... p. 179 Circle Inscribed in a Triangle............................ p. 313 Circle Through Three Noncollinear Points................... p. 763 Circumcenter of a Triangle...................... pp. 307, 313 Congruent Angles..........................p. 22 Congruent Segmentsp. 14 Congruent Triangles Rectangle................................ p. 424 Reflection.......................
... pp. 824, 829 Regular Decagon......................... p. 381 Regular Dodecagon....................... p. 380 Regular Hexagon......................... p. 380 Regular Octagon. 380 Regular Pentagon. 381 Rhombus.......................... pp. 415, 424 Using ASA............................ p. 253 Right Triangle............................ p. 258 Congruent Triangles Using SAS............................ p. 243 Congruent Triangles Using SSS............................ p. 248 Dilation........................... pp. 872, 878 Equilateral Triangle....................... p. 220 Rotation........................... pp. 839, 844 Segment Bisector...........................p. 16 Square............................. pp. 380, 424 Tangent to a Circle at a Point. 748 Tangent to a Circle from Incenter of a Triangle..................... p. 313 an Exterior Point.
..................... p. 779 Irrational Numbers....................... p. 363 Translation........................ pp. 831, 836 Kite..................................... p. 435 Triangle Circumscribed Midpoint..................................p. 16 Midsegment of a Triangle. 327 Orthocenter of a Triangle.................. p. 320 About a Circle........................ p. 779 Trisecting a Segment...................... p. 487 Constructions S87S87 Selected Answers Chapter 1 43. 14.02 m 45. 12 47. -23 49. -8x + 6 51. ̶̶ BD 53. ̶̶ AD,  CB 1-1 1-3 Check It Out! 1. Possible answer: plane R and plane ABC 2. 3. Possible answer: plane GHF 4. Exercises 3. A, B, C, D, E 5. Possible answer: planes ABC and  7. 9. Possible answer:   AB 11. 13. B, E, A 15. Possible answer: ABC 17. 19. Possible answer: planes  and  21. 23. 25. U 27. U 29. If 2 lines intersect, then they intersect in exactly 1 pt. 31. A 33. A 35. Post. 1-1-3 37. Post. 1-1-2 39. C 41. D n (n - 1) _______ 43. 6 45. 2 twins are 11. 49. no 51. mean: 0.44;
median: 0.442; mode: 0.44 47. Mother is 36; 1-2 Check It Out! 1a. 3 1 __ 2 1b. 4 1 __ 2 3a. 1 2 __ 3 3b. 24 4. 591.25 m 5. RS = 4; ST = 4; RT = 8 ̶̶̶ MY 3. 3.5 ̶̶̶ XM and Exercises 1. 7. 29 9. x = 4; KL = 7; JL = 14 11. 5 11 __ 12 15. 5 17. DE = EF =14; DF = 28 19a. C is the mdpt. of b. 16 21. 7.1 23. 4 25. S 27. Statement A 29. 6.5; -1.5 31. 3.375 33. 9 37. J 39. H ̶̶ AE. 41. S88 S88 Selected Answers Check It Out! 1. ∠RTQ, ∠T, ∠STR, ∠1, ∠2 2a. 40°; acute 2b. 125°; obtuse 2c. 105°; obtuse 3. 62° 4a. 34° 4b. 46° Exercises 1. ∠A, ∠R, ∠O 3. ∠AOB, ∠BOA, or ∠1; ∠BOC, ∠COB, or ∠2; ∠AOC or ∠COA 5. 105°; obtuse 7. 70° 9. 28° 11. ∠1 or ∠JMK; ∠2 or ∠LMK; ∠M or ∠JML 13. 93°; obtuse 15. 66.6° 17. 20° 19. acute 21. acute 27. 67.5°; 22.5° 29. 16 1 __ 3 31. 9 33a. 9 b. 12 c. 0 < x < 15.6 35. m∠COD = 72°; m∠BOC = 90° 37. No; an obtuse ∠ measures greater than 90°, so it cannot be ≅ to an acute ∠ (less than 90°). 41. D 43. C 45. The  are acute. An obtuse ∠ measures between 90° and 180°. Since 1 __ 2 of 180 is 90, the resulting
 must measure less than 90°. 47. 36° or 4° 49. 8100 51. 22.4 53. 55. 57. 6 1-4 Check It Out! 1a. adj.; lin. pair 1b. not adj. 1c. not adj. 2a. (102 - 7x) ° 2b. 63 1 __ 2 ° 3. 68° 4. m∠1 = m∠2 = 62.4°; m∠4 = 27.6° 5. Possible answer: ∠EDG and ∠FDH; m∠EDG ≈ m∠FDH ≈ 45° Exercises 1. (90 - x) °; (180 - x) ° 3. adj.; lin. pair 5. not adj. 7. 98.8° 9. (185 - 6x) ° 11. 69° 13. ∠ABE, ∠CBD; ∠ABC, ∠EBD 15. adj.; lin. pair 17. not adj. 19. 33.6° 21. (94 - 2x) ° 23. m∠2 = 22.3°; m∠3 = m∠4 = 67.7° 25. 1 __ 3 27. 72°; 108° 29. 61°; 29° 31. 10°; 80° 33a. m∠JAH = 64°; m∠KAH = 26° b. m∠JAH = 131.5°; m∠KAH = 48.5° c. m∠JAH = m∠KAH = 7° 35. F 37. T 39. C 41. C 43. 12 45. 30° 47. x = 8 49. y = 3 51. 17 53. 32 55. 52° 1-5 Check It Out! 1. P =14 in.; A = 12.75 in 2 2. 65 i n 2 3. C ≈ 88.0 m; A ≈ 615.8 m 2 Exercises 1. Both terms refer to the dist. around a figure. 3. P = 30 mm; A = 44 m m 2 5. P = (x + 21) m; A = (2x + 6) m 2 7. C ≈ 13.2 m; A ≈ 13.9 m 2 9. C ≈ 50.3 cm; A �
� 201.1 cm 2 11. P = 4x + 12; A = x 2 + 6x 13. 72 in 2 15. C ≈ 39.3 ft; A ≈ 122.7 ft 2 17. 82.81 yd 2 19. 6.1875 in 2 21. 17.1 cm 23. Statement A 25. 9 y 2 π 27. For a square, the length and width are both s, so P = 2l + 2w = 2s + 2s = 4s and A = lw = s (s) = s 2. 29. b = 41 in.; h = 38 in. 31a. ac + ad + bc + bd b. (a + 1) (c + 1) ; ac + a + c + 1 c. (a + 1) 2 ; a 2 + 2a + 1 33. 28 ft 35. 26.46 ft 2 37. 25 2 __ 3 yd 2 or 231 ft 2 39. 10 ft 41. 14 __ π 43. 50 45. Measure any side as the base. Then measure the height of the △ at a rt. ∠ to the base. 47. B 49. A 51. 83.7 in 2 53. 5; 8; 9 55. width = 16 in.; length = 20 in. 57. D: {4, -2, 16}; R: {-2, 8, 0} 59. line or segment 61. 3 1-6 Check It Out! 1. ( 3 __ 2, 0) 2. (4, 3) ̶̶ 3. EF = 5; GH = 5; EF ≅ 4a. 6.7 4b. 8.5 5. 60.5 ft ̶̶̶ GH Exercises 1. hypotenuse 3. (1 1 __ 2, -4) 5. (0, -2) 7. √  29 ; 3 √  5 ; no 9. 15.0 11. 27.2 ft 13. (3 1 __ 2, -4 1 __ 2 ) 15. (8, 4) 17. 2 √  5 ; √  29 ; no 19. 8.9 21. 18 in. 23. 4.47 25. Divide each coord. by 2. 27. 2.5 mi 31. 1 33. Let M be the ̶̶ AC ; AM
= MC = 5.0 ft; mdpt. of MB = MD ≈ 6.4 ft. 35. G 37. J 39. ±2 41. AB = √  x 2 + y 2 43. yes 45. 90°; rt. 47. 135°; obtuse 49. 4 ft 2 �� 1-7 Check It Out! 1a. translation; MNOP → M′N′O′P′ 1b. rotation; △XYZ → △X′Y′Z′ 2. rotation; 90° 3. J′ (-1, -5) ; K′ (1, 5) ; L′ (1, 0) ; M′ (-1, 0) 4. (x, y) → (x - 4, y - 4) Exercises 1. Preimage is △XYZ ; image is △X′Y′Z′ 3. reflection; △ABC → △A′B′C′ 5. reflection across the y-axis 7. (x, y) → (x + 4, y + 4) 9. reflection; WXYZ → W′X′Y′Z′ 11. A′ (-1, -1), B′ (4, -1), C (4, -4), D (-1, -4) 13. reflection 15. reflection 17. acute; ∠XYW: rt.; ∠ZYW: acute; ∠VYW: straight 17. 59° 18. 96° 19. only adj. 20. adj. and a lin. pair 21. not adj. 22. 15.4°; 105.4° 23. (94 - 2x) °; (184 - 2x) ° 24. 73° 25. 14x - 2; 12x 2 - 3x 26. 4x + 16; x 2 + 8x + 16 27. x + 15; 4x - 20 28. 10x + 54; 100x + 140 29. A ≈ 1385.4 m 2 ; C ≈ 131.9 m 30. A ≈ 153.9 ft 2 ; C ≈ 44.0 ft 31. 12 m 32. Y (1, 3) 33. B (- 9, 6) 34. A (0, 2) 35. 8.5 36. 7.3 37. 8.
1 38. 90° rotation; DEFG → D′E′F′G′ 39. translation; PQRS → P′Q′R′S′ 40. X′ (-1, 1) ; Y′ (1, 4) ; Z′ (2, 3) 19. B 21. D 23. R′ (-1, -12) ; S′ (-3, -9) ; T ′ (-7, -7) 25. 29. A 31. A 33a. R″ (1, 0) ; S″ (0, 3) ; T′′ (4, 3) b. (x, y) → ( x + 3, y + 2) 35. 37. (-x, y) 39. x = -6 or x = 3 41. x = 1 or x = 2 43. 13.9° 45. 4.1 47. 6.3 SGR 1. angle bisector 2. complementary angles 3. hypotenuse 4. A, F, E, G or C, G, D, B 5. Possible answer:   GC 6. Possible answer: plane AEG 7. 8. 9. Chapter 2 2-1 Check It Out! 1. 0.0004 2. odd 3. Female whales are longer than male whales. 4a. Possible answer: x = 1 __ 2 4b. Possible answer: 4c. Jupiter or Saturn Exercises 3. 4 __ 6 5. even 7. The number of bacteria doubles every 20 minutes. 9. The 3 pts. are collinear. 11. 5 P.M. 13. ̶̶ 09, 2 __ 11 = 0. 15. n - 1 17. Possible answer: y = -1 19. m∠1 = m∠2 = 90° 21. Possible answer: each term is the previous term multiplied by 1 __ 2 ; 1 __ 16 ; 1 __ 32. 23. 2n + 1 25. F ̶̶ 18, 3 __ 11 27. T 29. 1 __ 11 = 0. ̶̶ = 0. 27,…; the fraction pattern is multiples of 1 __ 11, and the decimal pattern is repeating multiples of 0.09. 31. 34, 55, 89; each term is the sum of the 2 previous terms. 33. odd 37. C 39. D 41. 12 years 43. m∠CAB =
m∠CBA; AC = CB 45. yes 47. no 49. 10x - 6 51. 6πx 53. (3, -2), (4, 0), (8, -1) 2-2 10. 3.5 11. 5 12. 7.6 13. 22 14. 13; 13; 26 15. 18; 18; 36 16. ∠VYX: rt.; ∠VYZ: obtuse; ∠XYZ: Check It Out! 1. Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3. 2. If 2  are comp., then they are acute. 3. F; possible answer: 7 4. Converse: If an animal has 4 paws, then it is a cat; F. Inverse: If an animal is not a cat, then it does not have 4 paws; F. Contrapositive: If an animal does not have 4 paws, then it is not a cat; T. 2 b < a __. 9. T 11. F b Exercises 1. converse 3. Hypothesis: A person is at least 16 years old. Conclusion: A person can drive a car. 5. Hypothesis: a - b < a. Conclusion: b is a positive number. 7. If 0 < a < b, then ( a __ ) 13. Hypothesis: An animal is a tabby. Conclusion: An animal is a cat. 15. Hypothesis: 8 ounces of cereal cost $2.99. Conclusion: 16 ounces of cereal cost $5.98. 17. If the batter makes 3 strikes, then the batter is out. 19. T 21. T 25. T 27. F 29. F 35. If a person is a Texan, then the person is an American. 37a. H: Only you can find it. C: Everything’s got a moral. b. If only you can find it, then everything’s got a moral. 43. If a mineral has a hardness less than 5, then it is not apatite; T. 45. If a mineral is not apatite, then it is calcite; F. 47. If a mineral is calcite, then it has a hardness less than 5; T. 51. H 53. J 55. Some students are adults. Some adults are students.
57. 3 59. y = 2x + 1 61. T 63. T 65. 2 __ 81 2-3 Check It Out! 1. deductive reasoning 2. valid 3. valid 4. Polygon P is not a quad. Exercises 3. deductive reasoning 5. valid 7. invalid 9. deductive reasoning 11. invalid 13. Dakota gets better grades in Social Studies. 15. valid 17. valid 19. yes; no; because the first conditional is false 23. D 25. 196 27a. If you live in San Diego, then you live in the United States. b. If you do not live in California, then you do not live in San Diego. If you do not live in the United States, then you do not live in California. c. If you do not Selected Answers S89 S89 �� live in the United States, then you do not live in San Diego. d. They are contrapositives of each other. 29. 2x + 10 31. -7c + 14 33. (-1.5, 3.5) 2-4 Check It Out! 1a. Conditional: If an ∠ is acute, then its measure is greater than 0° and less than 90°. Converse: If an ∠’s measure is greater than 0° and less than 90°, then the ∠ is acute. 1b. Conditional: If Cho is a member, then he has paid the $5 dues. Converse: If Cho has paid the $5 dues, then he is a member. 2a. Converse: If it is Independence Day, then the date is July 4th. Biconditional: It is July 4th if and only if it is Independence Day. 2b. Converse: If pts. are collinear, then they lie on the same line. Biconditional: Pts. lie on the same line if and only if they are collinear. 3a. T 3b. F; y = 5 4a. A figure is a quad. if and only if it is a 4-sided polygon. 4b. An ∠ is a straight ∠ if and only if its measure is 180°. Exercises 3. Conditional: If your medicine will be ready by 5 P.M., then you dropped your prescription off by 8 A.M. Converse
: If you drop your prescription off by 8 A.M., then your medicine will be ready by 5 P.M. 5. Converse: If 2 segs. are ≅, then they have the same length. Biconditional: 2 segs. have the same length if and only if they are ≅. 7. F 9. An animal is a hummingbird if and only if it is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. 11. Conditional: If a  is a rect., then it has 4 rt. . Converse: If a  has 4 rt. , then it is a rect. 13. Converse: If it is the weekend, then today is Saturday or Sunday. Biconditional: Today is Saturday or Sunday if and only if it is the weekend. 15. Converse: If a △ is a rt. △, then it contains a rt. ∠. Biconditional: A △ contains a rt. ∠ if and only if it is a rt. △. 17. T 19. A player is a catcher if and only if the player is positioned behind home plate and catches throws from the pitcher. 21. yes 23. no 25. A square is a quad. with 4 ≅ sides and 4 rt. . 31. no 33. 5 37a. If I say it, then I mean it. If I mean it, then I say it. 39. G 43a. If an ∠ does not measure 105°, then the ∠ is not obtuse. b. If an ∠ is not obtuse, then it does not measure 105°. c. It is the contrapositive of the original. d. F; the inverse is false, and its converse is true. 47. The graph is reflected across the x-axis and shifted 1 unit down and is narrower than the graph of the parent function. 49. T 51. S 53. F 2-5 Check It Out! 1. 1 __ 2 t = -7 (Given); 2 ( 1 __ 2 t) = 2 (-7) (Mult. Prop. of =); t = -14 (Simplify.) 2. C = 5 __ 9 (F - 32) (Given); C = 5 __ 9 (86 -
32) (Subst.); C = 5 __ 9 (54) (Simplify.); C = 30 (Simplify.) 3. ∠ Add. Post.; Subst.; Simplify.; Subtr. Prop. of =; Mult. Prop. of = 4a. Sym. Prop. of = 4b. Reflex. Prop. of = 4c. Trans. Prop. of = 4d. Sym. Prop. of ≅ Exercises 3. t - 3.2 = -8.3 (Given); t = -5.1 (Add. Prop. of =) 5. x + 3 ____ -2 = 8 (Given); x + 3 = -16 (Mult. Prop. of =); x = -19 (Subtr. Prop. of =) 7. 0 = 2 (r - 3) + 4 (Given); 0 = 2r - 6 + 4 (Distrib. Prop.); 0 = 2r - 2 (Simplify.); 2 = 2r (Add. Prop. of =); 1 = r (Div. Prop. of =) 9. C = $5.75 + $0.89m (Given); $11.98 = $5.75 + $0.89m (Subst.); $6.23 = $0.89m (Subtr. Prop. of =); m = 7 (Div. Prop. of =) 11. Seg. Add. Post.; Subst.; Subtr. Prop. of =; Add. Prop. of =; Div. Prop. of = 13. Trans. Prop. of = 15. Trans. Prop. of ≅ 17. 1.6 = 3.2n (Given); 0.5 = n (Div. Prop. of =) 19. - (h + 3) = 72 (Given); -h - 3 = 72 (Distrib. Prop.); -h = 75 (Add. Prop. of =); h = -75 (Mult. Prop. of =) 21. 1 __ 2 (p - 16) = 13 (Given); 1 __ 2 p - 8 = 13 (Distrib. Prop.); 1 __ 2 p = 21 (Add. Prop. of =); p = 42 (Mult. Prop. of =) 23. ∠ Add. Post.; Subst.; Simplify.; Subtr. Prop. of =; Add. Prop. of =; Div. Prop. of =
S90 S90 Selected Answers 25. Sym. Prop. of ≅ 27. Trans. Prop. of = 29. x = 16; 2 (3.1x - 0.87) = 94.36 (Given); 6.2x - 1.74 = 94.36 (Distrib. Prop.); 6.2x = 96.1 (Add. Prop. of =); x = 15.5 (Div. Prop. of =); possible answer: the exact solution rounds to the estimate. 31. ∠A ≅ ∠T 33. x + 1 ____ 2 x + 1 = 6 (Mult. Prop. of =); x = 5 = 3 (Mdpt. Formula;) 1 + y ____ 2 = 5 (Mdpt. (Subtr. Prop. of =); Formula); 1 + y = 10 (Mult. Prop. of =); y = 9 (Subtr. Prop. of =) 35a. 1733.65 = 92.50 + 79.96 + 983 + 10,820x (Given); 1733.65 = 1155.46 + 10,820x (Simplify.); 578.19 = 10,820x (Subtr. Prop. of =); 0.05 ≈ x (Div. Prop. of =) b. $1.71 37a. x + 15 ≤ 63 (Given); x ≤ 48 (Subtr. Prop. of Inequal.) b. -2x > 36 (Given); x < -18 (Div. Prop. of Inequal.) 39. B 41. D 43. PR = PA + RA (Seg. Add. Post.); PA = QB, QB = RA (Given); PA = RA (Trans. Prop. of =); PR = PA + PA (Subst.); PA = 18 (Given) ; PR = 18 + 18 (Subst.); PR = 36 in. (Simplify.) 45. 7 - 3x > 19 (Given); -3x > 12 (Subtr. Prop. of Inequal.); x < -4 (Div. Prop. of Inequal.) 49. deductive reasoning 2-6 Check It Out! 1. 1. Given 2. Def. of mdpt. 3. Given 4. Trans. Prop. of ≅ 2a. ∠1 and ∠2 are supp., and ∠2 and ∠3 are supp. 2
b. m∠1 + m∠2 = m∠2 + m∠3 2c. Subtr. Prop. of = 2d. ∠1 ≅ ∠3 3. 1. ∠1 and ∠2 are comp. ∠2 and ∠3 are comp. (Given) 2. m∠1 + m∠2 = 90°, m∠2 + m∠3 = 90° (Def. of comp. ) 3. m∠1 + m∠2 = m∠2 + m∠3 (Subst.) 4. m∠2 = m∠2 (Reflex. Prop. of =) 5. m∠1 = m∠3 (Subtr. Prop. of =) 6. ∠1 ≅ ∠3 (Def. of ≅ ) Exercises 1. statements; reasons 3. 1. Given ∠3 ≅ ∠4. By the Trans. Prop. of ≅, ∠2 ≅ ∠4. Similarly, ∠2 ≅ ∠3. the right. The next 2 items are and. 2. Subst. 3. Simplify. 4. Add. Prop. of = 5. Simplify. 6. Def. of supp.  ̶̶ AY. Y is the 5. 1. X is the mdpt. of ̶̶ XB. (Given) ̶̶ ̶̶ ̶̶ YB (Def. of XY ≅ XY, ̶̶ YB (Trans. Prop. of ≅) mdpt. of ̶̶ 2. AX ≅ mdpt.) ̶̶ 3. AX ≅ 7a. m∠1 + m∠2 = 180°, m∠3 + m∠4 = 180° b. Subst. c. m∠1 = m∠4 d. Def. of ≅  ̶̶ ̶̶ AE (Given) CE, ̶̶ DE ≅ 2. BE = CE, DE = AE (Def. of ≅ segs.) 3. AE + BE = AB, CE + DE = CD (Seg. Add. Post.) 4. DE + CE = AB (Subst.) 5. AB = CD (Subst.) ̶̶ 6. CD (Def. of �
� segs.) ̶̶ AB ≅ ̶̶ BE ≅ 9. 1. Exercises 1. flowchart 3. 1. ∠1 ≅ ∠2 (Given) 2. ∠1 and ∠2 are supp. (Lin. Pair Thm.) 3. ∠1 and ∠2 are rt. . (≅  supp. → rt. ) 5. 1. ∠2 ≅ ∠4 (Given) ̶̶ AB ≅ 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 (Vert.  Thm.) 3. ∠1 ≅ ∠4 (Trans. Prop. of ≅) 4. ∠1 ≅ ∠3 (Trans. Prop. of ≅) ̶̶ AC. (Given) 7. 1. B is the mdpt. of ̶̶ 2. BC (Def. of mdpt.) 3. AB = BC (Def. of ≅ segs.) 4. AD + DB = AB, BE + EC = BC (Seg. Add. Post.) 5. AD + DB = BE + EC (Subst.) 6. AD = EC (Given) 7. DB = BE (Subtr. Prop. of =) 11. 132° 13. 59° 17. S 19. N 21. x = 16 25. C 27. D 29. a = 17; 37.5°, 52.5°, and 37.5° 31. 24% 35. Sym. Prop. of ≅ 2-7 Check It Out! 1. 1. RS = UV, ST = TU (Given) 2. RS + ST = TU + UV (Add. Prop. of =) 3. RS + ST = RT, TU + UV = TV (Seg. Add. Post.) 4. RT = TV (Subst.) ̶̶ TV (Def. of ≅ segs.) 5. ̶̶ RT ≅ 2. 3. 1. ∠WXY is a rt. ∠. (Given) 2. m∠WXY = 90° (Def. of rt. ∠) 3. m∠2 + m∠3 = m∠WXY (∠ Add. Post.) 4. m∠2 + m∠3 = 90°
(Subst.) 5. ∠1 ≅ ∠3 (Given) 6. m∠1 = m∠3 (Def. of ≅ ) 7. m∠2 + m∠1 = 90° (Subst.) 8. ∠1 and ∠2 are comp. (Def. of comp. ) 4. It is given that ∠1 ≅ ∠4. By the Vert.  Thm., ∠1 ≅ ∠2 and 9. 1. ∠1 ≅ ∠4 (Given) 2. ∠1 ≅ ∠2 (Vert.  Thm.) 3. ∠4 ≅ ∠2 (Trans. Prop. of ≅) 4. m∠4 = m∠2 (Def. of ≅ ) 5. ∠3 and ∠4 are supp. (Lin. Pair Thm.) 6. m∠3 + m∠4 = 180° (Def. of supp. ) 7. m∠3 + m∠2 = 180° (Subst.) 8. ∠2 and ∠3 are supp. (Def. of supp. ) 11. 13 cm; conv. of the Common Segs. Thm. 13. 37°, Vert.  Thm. 15. y = 11 17. A 21. C 23. D 25. 1. ∠AOC ≅ ∠BOD (Given) 2. m∠AOC = m∠BOD (Def. of ≅ ) 3. m∠AOB + m∠BOC = m∠AOC, m∠BOC + m∠COD = m∠BOD (∠ Add. Post.) 4. m∠AOB + m∠BOC = m∠BOC + m∠COD (Subst.) 5. m∠BOC = m∠BOC (Reflex. Prop. of =) 6. m∠AOB = m∠COD (Subtr. Prop. of =) 7. ∠AOB ≅ ∠COD (Def. of ≅ ) 27. x = 31 and y = 11.5; 86°, 94°, 86°
, and 94° 29. (-4, 5) SGR 1. theorem 2. deductive reasoning 3. counterexample 4. conjecture 5. The rightmost △ is duplicated, rotated 180°, and shifted to and 1. 7. The white section 6. Each item is 1 __ 6 greater than the previous one. The next 2 items are 5 _ 6 is halved. If the white section is a rect. but not a square, it is halved horiz. and the upper portion is colored yellow. If the white section is a square, it is halved vert. and the left portion is colored yellow. The next 2 items are and. 8. odd 9. positive 10. F; 0 11. T 12. T 13. F; during a leap year, there are 29 days in February. 14. Check students’ constructions. Possible answer: The 3 ∠ bisectors of a △ intersect in the int. of the △. 15. If it is Monday, then it is a weekday. 16. If something is a lichen, then it is a fungus. 17. T 18. F; possible answer: √  2 and √  2 19. Converse: If m∠X = 90°, then ∠X is a rt. ∠; T. Inverse: If ∠X is not a rt. ∠, then m∠X ≠ 90°; T. Contrapositive: If m∠X ≠ 90°, then ∠X is not a rt. ∠; T. 20. Converse: If x = 2, then x is a whole number; T. Inverse: If x is not a whole number, then x ≠ 2; T. Contrapositive: If x ≠ 2, then x is not a whole number; F. 21. F 22. T 23. F 24. Sara’s call lasted 7 min. 25. The cost of Paulo’s long-distance call is $2.78. 26. No conclusion; the number and length of calls are unknown. 27. yes 28. no; possible answer: x = 2 29. no; possible answer: a seg. with endpoints (3, 7) and (-5, -1) 30. yes 31. comp. 32. positive 33. greater than 50 mi/h 34. 4s
35. m ___ -5 + 3 = -4.5 (Given); m ___ -5 = -7.5 (Subtr. Prop. of =); m = 37.5 (Mult. Prop. of =) 36. -47 = 3x - 59 (Given); 12 = 3x (Add. Prop. of =); 4 = x (Div. Prop. of =) 37. Reflex. Prop. of = 38. Sym. Prop. of ≅ 39. Trans. Prop. of = 40. figure ABCD 41. m∠5 = ̶̶ ̶̶ m∠2 42. EF 43. I = Prt CD ≅ (Given) ; 4200 = P (0.06) (4) (Subst.); 4200 = P (0.24) (Simplify.); Selected Answers S91 S91 �� 17,500 = P (Div. Prop. of =) 44. 1. Given 2. Def. of comp.  3. Given 4. Def. of ≅  5. Subst. 6. Def. of comp.  45a. Given b. TU = UV c. SU + UV = SV d. Subst. 46. z = 22.5 47. x = 17 48. 49. It is given that ∠ADE and ∠DAE are comp. and ∠ADE and ∠BAC are comp. By the ≅ Comps. Thm., ∠DAE ≅ ∠BAC. By the Reflex. Prop. of ≅, ∠CAE ≅ ∠CAE. By the Common  Thm., ∠DAC ≅ ∠BAE. 50. w = 45; Vert.  Thm. 51. x = 45; ≅  supp. → rt.  Chapter 3 3-1 ̶̶ EJ ̶̶ BF and ̶̶ BF ⊥ ̶̶ BF ǁ ̶̶ DE are skew. Check It Out! 1–2. Possible answers given. 1a. 1b. 1c. BCD 2a. ∠1 and ∠3 2b. ∠2 and ∠7 2c. ∠1 and ∠8
2d. ∠2 and ∠3 3. transv. n; same-side int.  ̶̶ FJ 1d. plane FJH ǁ plane ̶̶ AB and ̶̶ AB and Exercises 1. alternate interior angles 3–9. Possible answers ̶̶̶ given. 3. DH are skew. 5. plane ABC ǁ plane EFG 7. ∠6 and ∠8 9. ∠2 and ∠3 11. transv. m; alt. ext.  13. transv. p, sameside int.  15–21. Possible answers given. 15. 17. plane ABC ǁ plane DEF 19. ∠1 and ∠8 21. ∠2 and ∠5 23. transv. q; alt. int.  25. transv. p; sameside int.  27. corr. 29. sameside int.  31. Possible answer: ̶̶ FG 33a. plane MNR ǁ plane and KLP; plane LMQ ǁ plane KNP; plane PQR ǁ plane KLM ̶̶ CF are skew. ̶̶ CD b. same-side int.  35–39. Possible answers given. 35. ∠5 and ∠8 37. ∠1 and ∠5 39. transv. n; alt. int.  41. The lines are skew. 45. G 47. F 49. transv. m: ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8; transv. n: ∠9 and ∠11, ∠10 and ∠12, ∠13 and ∠15, ∠14 and ∠16; transv. p: ∠1 and ∠9, ∠2 and ∠10, ∠5 and ∠13, ∠6 and ∠14; transv. q: ∠3 and ∠11, ∠4 and ∠12, ∠7 and ∠15, ∠8 and ∠16 51. transv. m: ∠1 and ∠8, ∠4 and ∠5; transv. n
: ∠9 and ∠16, ∠12 and ∠13; transv. p: ∠1 and ∠14, ∠2 and ∠13; transv. q: ∠3 and ∠16, ∠4 and ∠15 53. corr.  55. -3; -7; -3; 9; 29 57. -8; -9; -8; -5; 0 59. C = 11.9 m; A = 11.3 m 2 61. Lin. Pair Thm. 3-2 Check It Out! 1. m∠QRS = 62° 2. m∠ABD = 60° 3. 55° and 60° Exercises 1. m∠JKL = 127° 3. m∠1 = 90° 5. x = 8; y = 9 7. m∠VYX = 100° 9. m∠EFG = 102° 11. m∠STU = 90° 13. 120°; Corr.  Post. 15. 60°; Same-Side Int.  Thm. 17. 60°; Lin. Pair Thm. 19. 120°; Vert.  Thm. 21. x = 4; Same-Side Int.  Thm.; m∠3 = 103°; m∠4 = 77° 23. x = 3; Corr.  Post.; m∠1 = m∠4 = 42° 25a. ∠1 ≅ ∠3 b. Corr.  Post. c. ∠1 ≅ ∠2 d. Trans. Prop. of ≅ 29a. same-side int.  b. By the Same-Side Int.  Thm., m∠QRT + m∠STR = 180°. m∠QRT = 25° + 90° = 115°, so m∠STR = 65°. 31. A 35. J 37. m∠1 = 75° 39. x = 4; y = 12 41. increase 43. m∠1 + m∠2 = 180° 45–47. Possible answers given. 45. ∠3 and ∠6 47. ∠3 and ∠5 3-3 Check It Out! 1a. ∠1
≅ ∠3, so ℓ ǁ m by the Conv. of Corr.  Post. 1b. m∠7 = 77° and m∠5 = 77°, so ∠7 ≅ ∠5. ℓ ǁ m by the Conv. of Corr.  Post. 2a. ∠4 ≅ ∠8, so r ǁ s by the Conv. of Alt. Int.  Thm. 2b. m∠3 = 100° and m∠7 = 100°, so ∠3 ≅ ∠7. r ǁ s by the Conv. of Alt. Int.  Thm. 3. 1. ∠1 ≅ ∠4 (Given) 2. m∠1 = m∠4 (Def. ≅ ) 3. ∠3 and ∠4 are supp. (Given) 4. m∠3 + m∠4 = 180° (Def. supp. ) 5. m∠3 + m∠1 = 180° (Subst.) 6. m∠2 = m∠3 (Vert.  Thm.) 7. m∠2 + m∠1 = 180° (Subst.) 8. ℓ ǁ m (Conv. of Same-Side Int.  Thm.) 4. 4y - 2 = 4 (8) - 2 = 30°; 3y + 6 = 3 (8) + 6 = 30°; The  are ≅, so the oars are ǁ by the Conv. of Corr.  Post. Exercises 1. ∠4 ≅ ∠5, so p ǁ q by the Conv. of Corr.  Post. 3. m∠4 = 47°, and m∠5 = 47°, so ∠4 ≅ ∠5. p ǁ q by the Conv. of Corr.  Post. 5. ∠3 and ∠4 are supp., so r ǁ s by the Conv. of Same-Side Int.  Thm. 7. m∠4 = 61°, and m∠8 = 61°, so ∠4
≅ ∠8. r ǁ s by the Conv. of Alt. Int.  Thm. 9. m∠2 = 132°, and m∠6 = 132°, so ∠2 ≅ ∠6. r ǁ s by the Conv. of Alt. Ext.  Thm. 11. m∠1 = 60°, and m∠2 = 60°, so ∠1 ≅ ∠2. By the Conv. of Alt. Int.  Thm., the landings are ǁ. 13. m∠4 = 54°, and m∠8 = 54°, so ∠4 ≅ ∠8. ℓ ǁ m by the Conv. of Corr.  Post. 15. m∠1 = 55°, and m∠5 = 55°, so ∠1 ≅ ∠5. ℓ ǁ m by the Conv. of Corr.  Post. 17. ∠2 ≅ ∠7, so n ǁ p by the Conv. of Alt. Ext.  Thm. 19. m∠1 = 105°, and m∠8 = 105°, so ∠1 ≅ ∠8. n ǁ p by the Conv. of Alt. Ext.  Thm. 21. m∠3 = 75°, and m∠5 = 105°. 75° + 105° = 180°, so ∠3 and ∠5 are supp. ℓ ǁ m by the Conv. of Same-Side Int.  Thm. 23. If x = 6, then m∠1 = 20° ̶̶ and m∠2 = 20°. So EK by the Conv. of Corr.  Post. 25. Conv. of Alt. Ext.  Thm. 27. Conv. of Corr.  Post. 29. Conv. of Same-Side Int.  Thm. 31. m ǁ n; Conv. of SameSide Int.  Thm. 33. m ǁ n; Conv. of Alt. Ext.  Thm. 35. ℓ ǁ n; Conv. of Same
-Side Int.  Thm. 37a. ∠URT ; m∠URT = m∠URS + m∠SRT by the ∠ Add. Post. It is given that m∠SRT = 25° and m∠URS = 90°, so m∠URT = 25° + 90° = 115°. b. It is given that m∠SUR = 65°. From part a, m∠URT = 115°. 65° + 115° = 180°, so   SU ǁ   RT by the Conv. of Same- ̶̶ DJ ǁ S92 S92 Selected Answers �� Side Int.  Thm. 39. It is given that ∠1 and ∠2 are supp., so m∠1 + m∠2 = 180°. By the Lin. Pair Thm., m∠2 + m∠3 = 180°. By the Trans. Prop. of =, m∠1 + m∠2 = m∠2 + m∠3. By the Subtr. Prop. of =, m∠1 = m∠3. By the Conv. of Corr.  Post., ℓ ǁ m. 41. The Reflex. Prop. is not true for ǁ lines, because a line is not ǁ to itself. The Sym. Prop. is true, because if ℓ ǁ m, then ℓ and m are coplanar and do not intersect. So m ǁ ℓ. The Trans. Prop. is not true for ǁ lines, because if ℓ ǁ m and m ǁ n, then ℓ and n could be the same line. So they would not be ǁ. 43. C 45. 15 47. No lines can be proven ǁ. 49. q ǁ r by the Conv. of Alt. Int.  Thm. 51. s ǁ t by the Conv. of Alt. Ext.  Thm. 53. No lines can be proven
ǁ. 55. By the Vert.  Thm., ∠6 ≅ ∠3, so m∠6 = m∠3. It is given that m∠2 + m∠3 = 180°. By subst., m∠2 + m∠6 = 180°. By the Conv. of Same-Side Int.  Thm., ℓ ǁ m. 57. a = b - c ̶̶ 59. y = - 3 __ 2 x + 3 63. BC ̶̶ 65. AD ̶̶ AB ⊥ ̶̶ AD ǁ 3-4 Check It Out! 1a. 2. 1. ∠EHF ≅ ∠HFG (Given) ̶̶ AB 1b. x < 17 2.   EH ǁ   FG (Conv. of Alt. Int.  Thm.) 3.   FG ⊥   GH (Given) 4.   EH ⊥   GH (⊥ Transv. Thm.) 3. The shoreline and the path of the swimmer should both be ⊥ to the current, so they should be ǁ to each other. ̶̶ Exercises 1. AB and   CD are ̶̶ ̶̶ BC are ≅. 3. x >-5 ⊥. AC and 5. The service lines are coplanar lines that are ⊥ to the same line (the center line), so they must be ǁ to each other. 7. x < 11 9. Both the frets are lines that are ⊥ to the same line (the string), so the frets must be ǁ to each other. 11. x > 8 __ 3 13. x = 6, y = 15 15. x = 60, y = 60 17. no 19. no 21. yes 23a. It is ̶̶ ̶̶ RS, PQ and given that ̶̶ ̶̶ RS by the ⊥ Transv. Thm. QR ⊥ so ̶̶ ̶̶
QR QR. Since It is given that ̶̶ ̶̶ RS by the ⊥ Transv. ⊥ PS ⊥ ̶̶ Thm. b. It is given that QR ̶̶ QR ⊥ ̶̶ PQ ǁ ̶̶ PS ǁ ̶̶ PS ǁ ̶̶ RS, ̶̶ PQ ⊥ ̶̶ PQ. So ̶̶ PS by the ̶̶ QR ⊥ and ⊥Transv. Thm. 25. Possible answer: 1.6 cm 31. C 33. D 35a. n ⊥ p b. AB; AB; the shortest distance from a point to a line is measured along a perpendicular segment. c. The distance between two parallel lines is the length of a segment that is perpendicular to both lines and has one endpoint on each line. 39. 30 games 41. 25° 43. Conv. of Alt. Ext. ∠ Thm. 45. Conv. of Same-Side Int. ∠ Thm. 3-5 Check It Out! 1. m = 2 2. 390 m 3a. ⊥ 3b. neither 3c. ǁ Exercises 1. rise; run 3. m = - 5 __ 9 5. m = 5 __ 2 7. ǁ 9. neither 11. m = 0 13. m = - 7 __ 3 15. ǁ 17. ⊥ 19. m = 1 __ 10 21. m = 1 __ 2 23. m <-1 25a. 66 ft/s b. 45 mi/h 27. F 29.   JK is a vert. line. 33. Possible answer: x = 1, y = -6 35. x-int.: 0.25; y-int.: 1 37. 1. ∠1 is supp. to ∠3. (Given) 2. ∠1 and ∠2 are supp. (Lin. Pair Thm.) 3. ∠2 ≅ ∠3 (≅ Supps. Thm.) 39. T: Corr.  Post. 3-6 Check It Out! 1a. y = 6 1b. y - 2 = 0 2a. 2b. 2c. 3. parallel 4. The lines would be ǁ. Exercises 1. The slope
-intercept form of an equation is solved for y. The x term is first, and the constant term is second. 3. y - 2 = 3 __ 4 (x + 4) 7. 5. 9. intersect 11. ǁ 13. y + 2 = 2x 15. y + 4 = 2 __ 3 (x - 6) 17. 19. intersect 21. coincide 23. $1000 per week 33. no 35. yes 37. ǁ line: y = 3x - 3; ⊥ line: y = - 1 __ 3 x + 11 __ 3 39. ǁ line: y = - 4 __ 3 x + 10 __ 3 ; ⊥ line: y = 3 __ 4 x - 5 41. yes; ∠B 43. no 45. For 4 toppings, both pizzas will cost $14. 47. y = - 1 __ 2 x + 17 __ 2 49. y = 2x + 7 __ 2 51. y = 2x - 1; (2, 3) ; √  5 units 53a–b. b. the time when the car has traveled 300 ft c. Possible answer: 3.5 s 59. J 61. J 63. Possible answer: y = - 8 __ 15 x + 8 65. no 67. 6 69. (1, 0) 71. m = 2 __ 5 73. m = - 4 __ 3 SGR ̶̶ BC are ̶̶ DE ̶̶ DE and ̶̶ AD ⊥ 1. alternate interior angles 2. skew lines 3. transversal 4. point-slope form 5. rise; run 6. Possible answer: ̶̶ ̶̶ skew. 7. DE 8. AB ǁ 9. plane ABC ǁ plane DEF 10. ℓ; alt. int.  11. n; corr.  12. ℓ; sameside int.  13. m; alt. ext.  14. m∠WYZ = 90° 15. m∠KLM = 100° 16. m∠DEF = 79° 17. m∠QRS = 76° 18. ∠4 ≅ ∠6, so c ǁ d by the Conv. of Alt. Int.  Thm. 19. m∠1 = 107° and m∠
5 = 107°, so ∠1 ≅ ∠5. c ǁ d by the Conv. of Corr.  Post. 20. m∠6 = 66°, m∠3 =114°, and 66° + 114° ≠ 180°, so ∠6 and ∠3 are supp. c ǁ d by the Conv. of Same-Side Int.  Thm. 21. m∠1 ≠ 99°, and m∠7 = 99°, so ∠1 ≅ ∠7. c ǁ d by the Conv. of Alt. Ext.  Thm. 22. ̶̶̶ KM 23. x < 13 ̶̶ AD ⊥ ̶̶ BC, ̶̶ BC 24. 1. ̶̶ AB, ̶̶ DC ⊥ ̶̶ BC (⊥ Transv. Thm.); ̶̶ CD (2 lines ⊥ to same ̶̶ AD ǁ (Given); ̶̶ 2. AB ⊥ ̶̶ 3. AB ǁ line → 2 lines ǁ) 25. m = - 1 __ 7 26. m = 5 __ 3 27. neither 28. ǁ 29. ⊥ 30. y = - 4 __ 9 x + 11 __ 3 31. y = 2 __ 3 x - 2 Selected Answers S93 S93 �� 32. y - 0 = 2 (x - 1) 33. ǁ 34. intersect 35. coincide 55. scalene 57.△ACD is equil. 4-3 Chapter 4 4-1 Check It Out! 1. equiangular 2. scalene 3. 17; 17; 17 4a. 4 4b. 3 Exercises 1. An equilateral △ has 3 ≅ sides. 3. rt. 5. obtuse 7. scalene 9. 36; 36; 36 11. 6 13. obtuse 15. equil. 17. scalene 19. 8.6; 8.6 21. 18 ft; 18 ft; 24 ft 23. 25. 27. not possible 29. 35 in. 31. isosc. rt. 33a. 173 ft; 87 ft b. scal
ene 35. S 37. A 41. D 43. D 45. It is an isosc. △ since 2 sides of the △ have length a. It is a rt. △ since 2 sides of the △ lie on the coord. axes and form a rt. ∠. 47. y = -3 49. y = x 2 51. y = x 2 53. T 55. ǁ 57. coincides 4-2 Check It Out! 1. 32° 2a. 26.3° 2b. (90 - x) ° 2c. 41 3 __ 5 ° 3. 141° 4. 32°; 32° Exercises 3. auxiliary lines 5. 36°; 80°; 64° 7. (90 - y) ° 9. 28° 11. 52°; 63° 13. 89°; 89° 15. 84° 17. (90 - 2x) ° 19. 162° 21. 48°; 48° 23. 15°; 60°; 105° 29. 36° 31. 48° 33. 120°; 360° 35. 18° 37. The ext.  at the same vertex of a △ are vert. . Since vert.  are ≅, the 2 ext.  have the same measure. 41. C 43. D 45. y = 7 or y = -7 47. Since an ext. ∠ is = to a sum of 2 remote int. , it must be greater than either ∠. Therefore it cannot be ≅ to a remote int. ∠. 49. 38° 51. 53. 6 in.; Seg. Add. Post. S94 S94 Selected Answers Check It Out! 1. ∠L ≅ ∠E, ∠M ≅ ̶̶ ∠F, ∠N ≅ ∠G, ∠P ≅ ∠H, EF, ̶̶ ̶̶̶ MN ≅ NP ≅ 2a. 4 2b. 37° 3. 1. ∠A ≅ ∠D (Given) ̶̶̶ LM ≅ ̶̶ EH ̶̶ LP ≅ ̶̶̶ GH, ̶̶ FG, ̶̶ DE (Given) ̶̶ AB ≅ ̶̶ AD bisects ̶̶ AD. (Given) ̶̶ ̶̶
AC ≅ EC, 2. ∠BCA ≅ ∠ECD (Vert.  are ≅.) 3. ∠ABC ≅ ∠DEC (Third  Thm.) 4. 5. bisects ̶̶ 6. BC ≅ bisector) 7. △ABC ≅ △DEC (Def. of ≅ ) ̶̶ DC (Def. of ̶̶ BE, and ̶̶ BE 4. 1. ̶̶ JK ǁ ̶̶̶ ML (Given) ̶̶̶ ML (Given) 2. ∠KJN ≅ ∠MLN, ∠JKN ≅ ∠LMN (Alt. Int.  Thm.) 3. ∠JNK ≅ ∠LNM (Vert.  Thm.) 4. 5. bisects ̶̶ 6. JN ≅ bisector) 7. △JKN ≅ △MLN (Def. of ≅ ) ̶̶ JK ≅ ̶̶̶ MK bisects ̶̶̶ MK. (Given) ̶̶̶ ̶̶ MN ≅ LN, ̶̶ KN (Def. of ̶̶ JL, and ̶̶ JL ̶̶ BE ; ̶̶ CE, ̶̶ PN ≅ Exercises 1. You find the  and sides that are in the same, or ̶̶̶ matching, places in the 2 . 3. LM 5. ∠M 7. ∠R 9. KL = 9 11a. Given b. Alt. Int.  Thm. c. Given ̶̶ ̶̶ d. Given e. DE ≅ AE ≅ f. Vert.  Thm. g. Def. of ≅  ̶̶̶ 13. LM 15. ∠N 17. m∠C = 31° 19a. Given b. Given c. ∠NMP ≅ ∠RMP d. ∠NPM ≅ ∠RPM e. Given ̶̶ f. PR g. Given h. Reflex. Prop. of ≅ 21. △GSR ≅ △KPH; △SGR ≅ △PHK; △RGS ≅ �

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