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regular decagon. What is the measure of ∠CBD? � � � � ∠CBD is an exterior angle of a regular decagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angle measures is 360°. m∠CBD = 360° _ = 36° 10 A regular decagon has 10 ≅ ext. , so divide the sum by 10. 5. What if…? Suppose the shutter were formed by 8 blades. What would the measure of each exterior angle be? THINK AND DISCUSS 1. Draw a concave pentagon and a convex pentagon. Explain the difference between the two figures. 2. Explain why you cannot use the expression 360° ____ n to find the measure of an exterior angle of an irregular n-gon. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the formula for finding the indicated value for a regular convex polygon with n sides. 6- 1 Properties and Attributes of Polygons 385 385 �� 6-1 Exercises Exercises KEYWORD: MG7 6-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why an equilateral polygon is not necessarily a regular polygon. 382 Tell whether each outlined shape is a polygon. If it is a polygon, name it by the number of its sides. 2. 2. 3. 3. 4. 5. 5 Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. p. 383 6. 7. 8. Find the measure of each interior angle of pentagon ABCDE. p. 384 10. Find the measure of each interior angle of a regular dodecagon. 11. Find the sum of the interior angle measures of a convex 20-gon 12. Find the value of y in polygon JKLM. �� p. 384 13. Find the measure of each exterior angle of a regular pentagon. �� . 385 Safety Use the photograph of the traffic sign for Exercises 14 and 15. 14. Name the polygon by the number of its sides. 15. In the polygon, ∠P, ∠R, and ∠
T are right angles, and ∠Q ≅ ∠S. What are m∠Q and m∠S? � � � � PRACTICE AND PROBLEM SOLVING Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 16. 17. 18. Independent Practice For See Exercises Example 16–18 19–21 22–24 25–26 27–28 1 2 3 4 5 Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. TEKS TEKS TAKS TAKS 19. Skills Practice p. S14 Application Practice p. S33 20. 21. 386 386 Chapter 6 Polygons and Quadrilaterals �� 22. Find the measure of each interior angle of quadrilateral RSTV. � 23. Find the measure of each interior angle of a regular 18-gon. 24. Find the sum of the interior angle measures of a convex heptagon. 25. Find the measure of each exterior angle of a regular nonagon. �� 26. A pentagon has exterior angle measures of 5a°, 4a°, 10a°, 3a°, and 8a°. Find the value of a. Crafts The folds on the lid of the gift box form a regular hexagon. Find each measure. 27. m∠JKM 28. m∠MKL � � � � Algebra Find the value of x in each figure. 29. 30. 31. Find the number of sides a regular polygon must have to meet each condition. 32. Each interior angle measure equals each exterior angle measure. 33. Each interior angle measure is four times the measure of each exterior angle. 34. Each exterior angle measure is one eighth the measure of each interior angle. Name the convex polygon whose interior angle measures have each given sum. 35. 540° 36. 900° 37. 1800° 38. 2520° Multi-Step An exterior angle measure of a regular polygon is given. Find the number of its sides and the measure of each interior angle. 39. 120° 40. 72° 41. 36° 42. 24° 43. /////ERROR ANALYSIS///// Which conclusion is incorrect? Explain the error. 44. Estimation Graph the polygon formed by the points A (-2, -6
), B (-4, -1), C (-1, 2), D (4, 0), and E (3, -5). Estimate the measure of each interior angle. Make a conjecture about whether the polygon is equiangular. Now measure each interior angle with a protractor. Was your conjecture correct? 45. This problem will prepare you for the Multi-Step TAKS Prep on page 406. In this quartz crystal, m∠A = 95°, m∠B = 125°, m∠E = m∠D = 130°, and ∠C ≅ ∠F ≅ ∠G. a. Name polygon ABCDEFG by the number of sides. b. What is the sum of the interior angle measures � � of ABCDEFG? c. Find m∠F. � � � � � 6- 1 Properties and Attributes of Polygons 387 387 �� 46. The perimeter of a regular polygon is 45 inches. The length of one side is 7.5 inches. Name the polygon by the number of its sides. Draw an example of each figure. 47. a regular quadrilateral 48. an irregular concave heptagon 49. an irregular convex pentagon 50. an equilateral polygon that is not equiangular 51. Write About It Use the terms from the lesson to describe the figure as specifically as possible. 52. Critical Thinking What geometric figure does a regular polygon begin to resemble as the number of sides increases? 53. Which terms describe the figure shown? I. quadrilateral II. concave III. regular I only II only I and II I and III 54. Which statement is NOT true about a regular 16-gon? It is a convex polygon. It has 16 congruent sides. The sum of the interior angle measures is 2880°. The sum of the exterior angles, one at each vertex, is 360°. 55. In polygon ABCD, m∠A = 49°, m∠B = 107°, and m∠C = 2m∠D. What is m∠C? 24° 68° 102° 136° CHALLENGE AND EXTEND 56. The interior angle measures of a convex pentagon are consecutive multiples of 4. Find the measure of
each interior angle. 57. Polygon PQRST is a regular pentagon. Find the values of x, y, and z. 58. Multi-Step Polygon ABCDEFGHJK is a regular decagon. ̶̶ DE are extended so that they meet at point L Sides in the exterior of the polygon. Find m∠BLD. ̶̶ AB and 59. Critical Thinking Does the Polygon Angle Sum Theorem work for concave polygons? Draw a sketch to support your answer. SPIRAL REVIEW Solve by factoring. (Previous course) 60. x 2 + 3x - 10 = 0 61. x 2 - x - 12 = 0 62. x 2 - 12x = -35 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. (Lesson 5-5) 63. 4, 4 64. 6, 12 65. 3, 7 Find each side length for a 30°-60°-90° triangle. (Lesson 5-8) 66. the length of the hypotenuse when the length of the shorter leg is 6 67. the length of the longer leg when the length of the hypotenuse is 10 388 388 Chapter 6 Polygons and Quadrilaterals �� Relations and Functions Algebra Many numeric relationships in geometry can be represented by algebraic relations. These relations may or may not be functions, depending on their domain and range. A relation is a set of ordered pairs. All the first coordinates in the set of ordered pairs are the domain of the relation. All the second coordinates are the range of the relation. A function is a type of relation that pairs each element in the domain with exactly one element in the range. See Skills Bank page S61 Example Give the domain and range of the relation y = 6_ x - 6. Tell whether the relation is a function. Step 1 Make a table of values for the relation. x y -6 -0.5 0 -1 5 -6 6 Undefined 7 6 12 1 Step 2 Plot the points and connect them with smooth curves. Step 3 Identify the domain and range. Since y is undefined at x = 6, the domain of the relation is the set of all real numbers except 6. Since there is no x-value such that y = 0, the range of the relation is the set of all real numbers except 0. Step 4 Determine whether the relation is a function
. From the graph, you can see that only one y-value exists for each x-value, so the relation is a function. Try This TAKS Grades 9–11 Obj. 2 Give the domain and range of each relation. Tell whether the relation is a function. 1. y = (x - 2) 180 2. y = 360 4. y = 360_ x 7. x = -2 5. x = 3y - 10 8. y = x 2 + 4 3. y = (x - 2) 180_ x 6. x 2 + y 2 = 9 9. -x + 8y = 5 On Track for TAKS 389 389 �� 6-2 Use with Lesson 6-2 Activity Explore Properties of Parallelograms In this lab you will investigate the relationships among the angles and sides of a special type of quadrilateral called a parallelogram. You will need to apply the Transitive Property of Congruence. That is, if figure A ≅ figure B and figure B ≅ figure C, then figure A ≅ figure C. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about... polygons... based on explorations and concrete models. Also G.2.B, G.3.B, G.10.A 1 Use opposite sides of an index card to draw a set of parallel lines on a piece of patty paper. Then use opposite sides of a ruler to draw a second set of parallel lines that intersects the first. Label the points of intersection A, B, C, and D, in that order. Quadrilateral ABCD has two pairs of parallel sides. It is a parallelogram. 2 Place a second piece of patty paper over the first and trace ABCD. Label the points that correspond to A, B, C, and D as Q, R, S, and T, in that order. The parallelograms ABCD and QRST are congruent. Name all the pairs of congruent corresponding sides and angles. 3 Lay ABCD over QRST so that ̶̶ AB overlays ̶̶ ST. What do ̶̶ AB and you notice about their lengths? What does this tell ̶̶ you about DA ̶̶ RS. What do you notice about their lengths? overlays ̶̶ DA and What does this tell you about ̶̶ CD? Now move ABCD
so that ̶̶ BC? 4 Lay ABCD over QRST so that ∠A overlays ∠S. What do you notice about their measures? What does this tell you about ∠A and ∠C? Now move ABCD so that ∠B overlays ∠T. What do you notice about their measures? What does this tell you about ∠B and ∠D? 5 Arrange the pieces of patty paper so that ̶̶ QR and ̶̶ RS overlays ̶̶ AB? What does this ̶̶ AD. What do you notice about tell you about ∠A and ∠R? What can you conclude about ∠A and ∠B? 6 Draw diagonals ̶̶ AC and ̶̶ BD. Fold ABCD so that A matches C, making a crease. Unfold the paper and fold it again so that B matches D, making another crease. What do you notice about the creases? What can you conclude about the diagonals? Try This 1. Repeat the above steps with a different parallelogram. Do you get the same results? 2. Make a Conjecture How do you think the sides of a parallelogram are related to each other? the angles? the diagonals? Write your conjectures as conditional statements. 390 390 Chapter 6 Polygons and Quadrilaterals 6-2 Properties of Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B Objectives Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems. Vocabulary parallelogram Who uses this? Race car designers can use a parallelogram-shaped linkage to keep the wheels of the car vertical on uneven surfaces. (See Example 1.) Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names. Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side. A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol . Parallelog
ram ABCD ABCD ̶̶ AB ǁ ̶̶ CD, ̶̶ BC ǁ ̶̶ DA Theorem 6-2-1 Properties of Parallelograms THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is a parallelogram, then its opposite sides are congruent. ( → opp. sides ≅) ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ CD ̶̶ DA PROOF PROOF Theorem 6-2-1 Given: JKLM is a parallelogram. Prove: ̶̶ KL ≅ ̶̶ JK ≅ ̶̶̶ LM, ̶̶ MJ Proof: Statements Reasons 1. JKLM is a parallelogram. ̶̶ KL ǁ ̶̶ JK ǁ ̶̶̶ LM, ̶̶ MJ 2. 3. ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ JL ≅ ̶̶ JL 4. 5. △JKL ≅ △LMJ ̶̶ JK ≅ ̶̶̶ LM, ̶̶ KL ≅ ̶̶ MJ 6. 1. Given 2. Def. of  3. Alt. Int.  Thm. 4. Reflex. Prop. of ≅ 5. ASA Steps 3, 4 6. CPCTC 6- 2 Properties of Parallelograms 391 391 �� Theorems Properties of Parallelograms THEOREM HYPOTHESIS CONCLUSION 6-2-2 6-2-3 6-2-4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. ( → opp.  ≅) If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( → cons.  supp.) If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( → diags. bisect each other) ∠A ≅ ∠C ∠B ≅ ∠D m∠A + m∠B = 180° m∠B + m∠C = 180° m∠C + m∠D =
180° m∠D + m∠A = 180° ̶̶ AZ ≅ ̶̶ BZ ≅ ̶̶ CZ ̶̶ DZ You will prove Theorems 6-2-3 and 6-2-4 in Exercises 45 and 44. E X A M P L E 1 Racing Application Racing The diagram shows the parallelogram-shaped linkage that joins the frame of a race car to one wheel of the car. In PQRS, QR = 48 cm, RT = 30 cm, and m∠QPS = 73°. Find each measure. ̶̶ QR A PS ̶̶ PS ≅ PS = QR PS = 48 cm B m∠PQR  → opp. sides ≅ Def. of ≅ segs. Substitute 48 for QR. Texas Motor Speedway, located in Fort Worth, is home to both NASCAR and Indy car racing events. With seating for 150,061 spectators, it is the second-largest sporting facility in the country. m∠PQR + m∠QPS = 180°  → cons.  supp. m∠PQR + 73 = 180 m∠PQR = 107° Substitute 73 for m∠QPS. Subtract 73 from both sides. ̶̶ RT C PT ̶̶ PT ≅ PT = RT PT = 30 cm  → diags. bisect each other Def. of ≅ segs. Substitute 30 for RT. In KLMN, LM = 28 in., LN = 26 in., and m∠LKN = 74°. Find each measure. 1a. KN 1b. m∠NML 1c. LO � � � � � 392 392 Chapter 6 Polygons and Quadrilaterals ��PQRST E X A M P L E 2 Using Properties of Parallelograms to Find Measures ABCD is a parallelogram. Find each measure. A AD ̶̶ ̶̶ AD ≅ BC AD = BC 7x = 5x + 19 2x = 19 x = 9.5 AD = 7x = 7 (9.5) = 66.5  → opp. sides ≅ Def. of ≅ segs. Substitute the given values. Subtract 5x from both sides. Divide both sides
by 2. B m∠B m∠A + m∠B = 180°  → cons.  supp. (10y - 1) + (6y + 5) = 180 16y + 4 = 180 16y = 176 y = 11 ⎤ ⎦ ⎡ ⎣ m∠B = (6y + 5) ° = 6 (11) + 5 ° = 71° Substitute the given values. Combine like terms. Subtract 4 from both sides. Divide both sides by 16. EFGH is a parallelogram. Find each measure. 2a. JG 2b. FH E X A M P L E 3 Parallelograms in the Coordinate Plane Three vertices of ABCD are A (1, -2), B (-2, 3), and D (5, -1). Find the coordinates of vertex C. Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step 1 Graph the given points. ̶̶ AB by counting Step 2 Find the slope of the units from A to B. The rise from -2 to 3 is 5. The run from 1 to -2 is -3. Step 3 Start at D and count the same number of units. A rise of 5 from -1 is 4. A run of -3 from 5 is 2. Label (2, 4) as vertex C. When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices. Step 4 Use the slope formula to verify that ̶̶ BC ǁ ̶̶ AD. slope of ̶̶ BC = 4 - 3 _ = 1 _ 4 2 - (-2) -1 - (-2) _ 5 - 1 = 1 _ 4 slope of ̶̶ AD = The coordinates of vertex C are (2, 4). 3. Three vertices of PQRS are P (-3, -2), Q (-1, 4), and S (5, 0). Find the coordinates of vertex R. 6- 2 Properties of Parallelograms 393 393 �� E X A M P L E 4 Using Properties of Parallelograms in a Proof Write a two-column proof. A Theorem 6-2-2 Given:
ABCD is a parallelogram. Prove: ∠BAD ≅ ∠DCB, ∠ABC ≅ ∠CDA Proof: Statements Reasons 1. ABCD is a parallelogram. ̶̶̶ DA ≅ ̶̶ BC 2. ̶̶ AB ≅ ̶̶ BD ≅ ̶̶ CD, ̶̶ BD 3. 4. △BAD ≅ △DCB 5. ∠BAD ≅ ∠DCB ̶̶ AC ≅ ̶̶ AC 6. 7. △ABC ≅ △CDA 8. ∠ABC ≅ ∠CDA 1. Given 2.  → opp. sides ≅ 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of ≅ 7. SSS Steps 2, 6 8. CPCTC B Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: ∠G ≅ ∠L Proof: Statements Reasons 1. GHJN and JKLM are parallelograms. 1. Given 2. ∠HJN ≅ ∠G, ∠MJK ≅ ∠L 3. ∠HJN ≅ ∠MJK 4. ∠G ≅ ∠L 2.  → opp.  ≅ 3. Vert.  Thm. 4. Trans. Prop. of ≅ 4. Use the figure in Example 4B to write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: ∠N ≅ ∠K THINK AND DISCUSS 1. The measure of one angle of a parallelogram is 71°. What are the measures of the other angles? 2. In VWXY, VW = 21, and WY = 36. Find as many other measures as you can. Justify your answers. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, draw a figure with markings that represents the given property. 394 394 Chapter 6 Polygons and Quadrilaterals ��
�� 6-2 Exercises Exercises KEYWORD: MG7 6-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why the figure at right is NOT a parallelogram. 2. Draw PQRS. Name the opposite sides and opposite angles. 392 Safety The handrail is made from congruent parallelograms. In ABCD, AB = 17.5, DE = 18, and m∠BCD = 110°. Find each measure. 3. BD 5. BE 7. m∠ADC 4. CD 6. m∠ABC 8. m∠DAB � � � � � JKLM is a parallelogram. Find each measure. p. 393 9. JK 11. m∠L 10. LM 12. m∠ 13. Multi-Step Three vertices of DFGH are D (-9, 4), F (-1, 5), and G (2, 0). p. 393 Find the coordinates of vertex H 14. Write a two-column proof. p. 394 Given: PSTV is a parallelogram. Prove: ∠STV ≅ ∠R ̶̶ PQ ≅ ̶̶ RQ Independent Practice Shipping Cranes can be used to load PRACTICE AND PROBLEM SOLVING � For See Exercises Example 15–20 21–24 25 26 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S14 Application Practice p. S33 cargo onto ships. In JKLM, JL = 165.8, JK = 110, and m∠JML = 50°. Find the measure of each part of the crane. 15. JN 17. LN 16. LM 18. m∠JKL 19. m∠KLM 20. m∠MJK � � � � WXYZ is a parallelogram. Find each measure. 21. WV 23. XZ 22. YW 24. ZV 25. Multi-Step Three vertices of PRTV are P (-4, -4), R (-10, 0), and V (5,
-1). Find the coordinates of vertex T. 26. Write a two-column proof. Given: ABCD and AFGH are parallelograms. Prove: ∠C ≅ ∠G 6- 2 Properties of Parallelograms 395 395 �� Algebra The perimeter of PQRS is 84. Find the length of each side of PQRS under the given conditions. 28. QR = 3 (RS) 27. PQ = QR 31. Cars To repair a large truck, a mechanic might use a parallelogram lift. In the lift, ̶̶ ̶̶ ̶̶ HJ. KJ, and FG ≅ a. Which angles are congruent to ∠1? ̶̶̶ GH ≅ ̶̶ GK ≅ ̶̶ LK ≅ ̶̶ FL ≅ 29. RS = SP - 7 Justify your answer. b. What is the relationship between ∠1 and each of the remaining labeled angles? Justify your answer. � � � � 30. SP = RS 2 � � � � � � � � � � Complete each statement about KMPR. Justify your answer. 32. ∠MPR ≅ ̶̶ PR ≅ 35.? ̶̶̶̶? 33. ∠PRK ≅ ̶̶̶̶? 34. ̶̶̶̶ 36. ̶̶̶ MP ǁ? ̶̶̶̶ ̶̶̶ MT ≅ ̶̶̶ MK ǁ? ̶̶̶̶? ̶̶̶̶ 37. 38. ∠MPK ≅? 39. ∠MTK ≅ ̶̶̶̶? 40. m∠MKR + m∠PRK = ̶̶̶̶? ̶̶̶̶ Find the values of x, y, and z in each parallelogram. 41. 42. 43. 44. Complete the paragraph proof of Theorem 6-2-4 by filling in the blanks. Given: ABCD is a parallelogram. ̶̶ BD bisect each other at E. ̶̶ AC and Prove: Proof: It is given that ABCD is a parallelogram. By the definition of a parallelogram, ̶̶ AB ǁ a. ∠
3 ≅ c. by e. bisect each other at E by the definition of g.?. By the Alternate Interior Angles Theorem, ∠1 ≅ b. ̶̶̶̶?. ̶̶̶̶?, and ̶̶̶̶ ̶̶ CD because d.?. This means that △ABE ≅ △CDE ̶̶̶̶ ̶̶ ̶̶ DE. Therefore CE, and?, ̶̶̶̶ ̶̶ AB ≅?. So by f. ̶̶̶̶ ̶̶ AC and ̶̶ BE ≅ ̶̶ AE ≅ ̶̶ BD?. ̶̶̶̶ 45. Write a two-column proof of Theorem 6-2-3: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Algebra Find the values of x and y in each parallelogram. 46. 47. 48. This problem will prepare you for the Multi-Step TAKS Prep on page 406. In this calcite crystal, the face ABCD is a parallelogram. a. In ABCD, m∠B = (6x + 12) °, and m∠D = (9x - 33) °. Find m∠B. b. Find m∠A and m∠C. Which theorem or theorems did you use to find these angle measures? � � � � 396 396 Chapter 6 Polygons and Quadrilaterals �� 49. Critical Thinking Draw any parallelogram. Draw a second parallelogram whose corresponding sides are congruent to the sides of the first parallelogram but whose corresponding angles are not congruent to the angles of the first. a. Is there an SSSS congruence postulate for parallelograms? Explain. b. Remember the meaning of triangle rigidity. Is a parallelogram rigid? Explain. 50. Write About It Explain why every parallelogram is a quadrilateral but every quadrilateral is not necessarily a parallelogram. 51. What is the value of x in PQRS? 15 20 30 70 52. The diagonals of JKLM intersect at Z. Which statement is true? JL = 1 _ 2 JL = 1 _
2 JL = KM KM JZ JL = 2JZ 53. Gridded Response In ABCD, BC = 8.2, and CD = 5. What is the perimeter of ABCD? CHALLENGE AND EXTEND The coordinates of three vertices of a parallelogram are given. Give the coordinates for all possible locations of the fourth vertex. 54. (0, 5), (4, 0), (8, 5) 55. (-2, 1), (3, -1), (-1, -4) 56. The feathers on an arrow form two congruent parallelograms that share a common side. Each parallelogram is the reflection of the other across the line they share. Show that y = 2x. 57. Prove that the bisectors of two consecutive angles of a parallelogram are perpendicular. SPIRAL REVIEW Describe the correlation shown in each scatter plot as positive, negative, or no correlation. (Previous course) 58. 59. Classify each angle pair. (Lesson 3-1) 60. ∠2 and ∠7 61. ∠5 and ∠4 62. ∠6 and ∠7 63. ∠1 and ∠3 An interior angle measure of a regular polygon is given. Find the number of sides and the measure of each exterior angle. (Lesson 6-1) 64. 120° 65. 135° 66. 156° 6- 2 Properties of Parallelograms 397 397 �� 6-3 Conditions for Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objective Prove that a given quadrilateral is a parallelogram. Who uses this? A bird watcher can use a parallelogram mount to adjust the height of a pair of binoculars without changing the viewing angle. (See Example 4.) You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you can use the definition of a parallelog
ram or the conditions below. Theorems Conditions for Parallelograms THEOREM EXAMPLE In the converse of a theorem, the hypothesis and conclusion are exchanged. 6-3-1 6-3-2 6-3-3 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 ≅ → ) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp. sides ≅ → ) If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp.  ≅ → ) You will prove Theorems 6-3-2 and 6-3-3 in Exercises 26 and 29. PROOF PROOF Theorem 6-3-1 ̶̶ KL ǁ ̶̶ MJ, ̶̶ KL ≅ ̶̶ MJ Given: Prove: JKLM is a parallelogram. Proof: ̶̶ KL ≅ ̶̶ MJ. Since ̶̶ KL ǁ ̶̶ MJ, ∠1 ≅ ∠2 by the It is given that Alternate Interior Angles Theorem. By the Reflexive Property ̶̶ ̶̶ JL ≅ JL. So △JKL ≅ △LMJ by SAS. By CPCTC, of Congruence, ̶̶̶ ̶̶ ∠3 ≅ ∠4, and LM by the Converse of the Alternate JK ǁ Interior Angles Theorem. Since the opposite sides of JKLM are parallel, JKLM is a parallelogram by definition. 398 398 Chapter 6 Polygons and Quadrilaterals �� The two theorems below can also be used to show that a given quadrilateral is a parallelogram. Theorems Conditions for Parallelograms THEOREM EXAMPLE 6-3-4 6-3-5 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. �
�� → ) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (quad. with diags. bisecting each other → ) You will prove Theorems 6-3-4 and 6-3-5 in Exercises 27 and 30. E X A M P L E 1 Verifying Figures are Parallelograms A Show that ABCD is a parallelogram for x = 7 and y = 4. Step 1 Find BC and DA. BC = x + 14 BC = 7 + 14 = 21 Step 2 Find AB and CD. AB = 5y - 4 AB = 5 (4) - 4 = 16 Given Substitute and simplify. DA = 3x DA = 3x = 3 (7) = 21 Given Substitute and simplify. CD = 2y + 8 CD = 2 (4) + 8 = 16 Since BC = DA and AB = CD, ABCD is a parallelogram by Theorem 6-3-2. B Show that EFGH is a parallelogram for z = 11 and w = 4.5. m∠F = (9z + 19) ° ⎤ ⎦ ⎡ ⎣ 9 (11) + 19 ° = 118° m∠F = m∠H = (11z - 3) ° ⎤ ⎦ ⎡ ⎣ 11 (11) - 3 ° = 118° m∠H = m∠G = (14w - 1) ° ⎤ ⎦ ⎡ ⎣ 14 (4.5) - 1 ° = 62° m∠G = Given Substitute 11 for z and simplify. Given Substitute 11 for z and simplify. Given Substitute 4.5 for w and simplify. Since 118° + 62° = 180°, ∠G is supplementary to both ∠F and ∠H. EFGH is a parallelogram by Theorem 6-3-4. 1. Show that PQRS is a parallelogram for a = 2.4 and b = 9. 6- 3 Conditions for Parallelograms 399 399 �� E X A M P L E 2 Applying Conditions for Parallelograms Deter
mine if each quadrilateral must be a parallelogram. Justify your answer. A B No. One pair of opposite sides are parallel. A different pair of opposite sides are congruent. The conditions for a parallelogram are not met. Yes. The diagonals bisect each other. By Theorem 6-3-5, the quadrilateral is a parallelogram. Determine if each quadrilateral must be a parallelogram. Justify your answer. 2a. 2b. E X A M P L E 3 Proving Parallelograms in the Coordinate Plane Show that quadrilateral ABCD is a parallelogram by using the given definition or theorem. A A (-3, 2), B (-2, 7), C (2, 4), D (1, -1) ; definition of parallelogram Find the slopes of both pairs of opposite sides. To say that a quadrilateral is a parallelogram by definition, you must show that both pairs of opposite sides are parallel. slope of ̶̶ AB = 7 - 2 _ -2 - (-3) slope of slope of ̶̶ CD = -1 - 4 _ 1 - 2 ̶̶ BC = 5 _ = 5 -1 _ = - 3 = -4 2 - (-2) 2 - (-1) _ -3 - 1 slope of ̶̶ DA = Since both pairs of opposite sides are parallel, ABCD is a parallelogram by definition. B F (-4, -2), G (-2, 2), H (4, 3), J (2, -1) ; Theorem 6-3-1 Find the slopes and lengths of one pair of opposite sides. = 1 _ 6 ̶̶ GH = 3 - 2 _ slope of slope of ̶̶ JF = 4 - (-2) -2 - (-1) _ -3 - 2) 2 = √  37 4 - (-2) = -1 _ -6 = 1 _ 6 GH = √  ⎤ ⎦ ⎡ ⎣ 2 = √  37 (-4 - 2) 2 + -2 - (-1) JF = √ �
�� ̶̶̶ GH and Since GH = JF, FGHJ is a parallelogram. ̶̶ JF have the same slope, so ̶̶ JF. So by Theorem 6-3-1, ̶̶̶ GH ≅ ̶̶̶ GH ǁ ̶̶ JF. 400 400 Chapter 6 Polygons and Quadrilaterals �� 3. Use the definition of a parallelogram to show that the quadrilateral with vertices K (-3, 0), L (-5, 7), M (3, 5), and N (5, -2) is a parallelogram. You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply. Conditions for Parallelograms Both pairs of opposite sides are parallel. (definition) One pair of opposite sides are parallel and congruent. (Theorem 6-3-1) Both pairs of opposite sides are congruent. (Theorem 6-3-2) Both pairs of opposite angles are congruent. (Theorem 6-3-3) One angle is supplementary to both of its consecutive angles. (Theorem 6-3-4) The diagonals bisect each other. (Theorem 6-3-5) To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions. E X A M P L E 4 Bird-Watching Application Bird-Watching The westernmost bald eagle nest in Texas is 9 miles north of Llano, where a family of bald eagles can be seen from the side of the highway during their winter nesting season. In the parallelogram mount, there are bolts at P, Q, R, and S such that PQ = RS and QR = SP. The frame PQRS moves when you raise or lower the binoculars. Why is PQRS always a parallelogram? When you move the binoculars, the angle measures change, but PQ, QR, RS, and SP stay the same. So it is always true that PQ = RS and QR = SP. Since both pairs of opposite sides of the quadrilateral are congruent, PQRS is always a
parallelogram. � � � � � � 4. The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same? THINK AND DISCUSS 1. What do all the theorems in this lesson have in common? 2. How are the theorems in this lesson different from the theorems in Lesson 6-2? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write one of the six conditions for a parallelogram. Then sketch a parallelogram and label it to show how it meets the condition. 6- 3 Conditions for Parallelograms 401 401 �� 6-3 Exercises Exercises KEYWORD: MG7 6-3 KEYWORD: MG7 Parent GUIDED PRACTICE. Show that EFGH is a parallelogram 2. Show that KLPQ is a parallelogram p. 399 for s = 5 and t = 6. for m = 14 and n = 12.5 Determine if each quadrilateral must be a parallelogram. Justify your answer. p. 400 3. 4. 5. 400 Show that the quadrilateral with the given vertices is a parallelogram. 6. W (-5, -2), X (-3, 3), Y (3, 5), Z (1, 0) 7. R (-1, -5), S (-2, -1), T (4, -1), U (5, -5. Navigation A parallel rule can be used p. 401 Independent Practice For See Exercises Example 9–10 11–13 14–15 16 1 2 3 4 TEKS TEKS TAKS TAKS ̶̶ BC. You place the edge of one ruler on to plot a course on a navigation chart. The tool is made of two rulers connected ̶̶ AD at hinges to two congruent crossbars and your desired course and then move the second ruler over the compass rose on the chart to read the bearing for your course. ̶̶ AB always parallel to If ̶̶ BC, why is ̶̶ AD ǁ ̶̶ CD? � � � � PRACTICE AND PROBLEM SOLVING 9. Show that BC
GH is a parallelogram 10. Show that TUVW is a parallelogram for for x = 3.2 and y = 7. for a = 19.5 and b = 22. Determine if each quadrilateral must be a parallelogram. Justify your answer. Skills Practice p. S14 Application Practice p. S33 11. 12. 13. Show that the quadrilateral with the given vertices is a parallelogram. 14. J (-1, 0), K (-3, 7), L (2, 6), M (4, -1) 15. P (-8, -4), Q (-5, 1), R (1, -5), S (-2, -10) 402 402 Chapter 6 Polygons and Quadrilaterals �� 16. Design The toolbox has cantilever trays � that pull away from the box so that you can reach the items beneath them. Two congruent brackets connect each tray to the box. Given that AD = BC, ̶̶ CD keep how do the brackets the tray horizontal? ̶̶ AB and � � � Determine if each quadrilateral must be a parallelogram. Justify your answer. 17. 18. 19. Algebra Find the values of a and b that would make the quadrilateral a parallelogram. 20. 22. 21. 23. 24. Critical Thinking Draw a quadrilateral that has congruent diagonals but is not a parallelogram. What can you conclude about using congruent diagonals as a condition for a parallelogram? 25. Social Studies The angles at the corners of the flag of the Republic of the Congo are right angles. The red and green triangles are congruent isosceles right triangles. Why is the shape of the yellow stripe a parallelogram? 26. Complete the two-column proof of Theorem 6-3-2 by filling in the blanks. Given: ̶̶ CD, ̶̶ DA Prove: ABCD is a parallelogram. ̶̶ AB ≅ ̶̶ BC ≅ Proof: Statements Reasons ̶̶ AB ≅ ̶̶ BD ≅ ̶̶ CD, ̶̶ BD 1. 2. ̶̶ BC ≅ ̶
̶̶ DA 3. △DAB ≅ b.? ̶̶̶̶̶ 4. ∠1 ≅ d. ̶̶ CD, ̶̶ AB ǁ?, ∠4 ≅ e. ̶̶̶̶̶ ̶̶ ̶̶̶ DA BC ǁ 5.? ̶̶̶̶̶ 6. ABCD is a parallelogram. 1. Given 2. a. 3. c.? ̶̶̶̶̶? ̶̶̶̶̶ 4. CPCTC 5. f. 6. g.? ̶̶̶̶̶? ̶̶̶̶̶ 6- 3 Conditions for Parallelograms 403 403 �� Measurement Ancient balance scales had one beam that moved on a single hinge. The stress on the hinge often made the scale imprecise. 27. Complete the paragraph proof of Theorem 6-3-4 by filling in the blanks. Given: ∠P is supplementary to ∠Q. ∠P is supplementary to ∠S. Prove: PQRS is a parallelogram. Proof: It is given that ∠P is supplementary to a.?. ̶̶̶̶ By the Converse of the Same-Side Interior Angles Theorem,? and b. ̶̶̶̶ ̶̶ QR ǁ c.? and ̶̶̶̶ by the definition of e. ̶̶ PQ ǁ d.?. ̶̶̶̶?. So PQRS is a parallelogram ̶̶̶̶ 28. Measurement In the eighteenth century, Gilles Personne de Roberval designed a scale with two beams and two hinges. In ABCD, ̶̶ AB, and F is the midpoint E is the midpoint of ̶̶ CD. Write a paragraph proof that AEFD and of EBCF are parallelograms. Prove each theorem. 29. Theorem 6-3-3 Given: ∠E ≅ ∠G, ∠F ≅ ∠H Prove: EFGH is a parallelogram. Plan: Show that the sum of the interior angles of EFGH is 360°. Then apply properties of equality to show that m
∠E + m∠F = 180° and m∠E + m∠H = 180°. ̶̶ FG ǁ Then you can conclude that ̶̶̶ GH and ̶̶ EF ǁ ̶̶ HE. � � � � � � 30. Theorem 6-3-5 ̶̶ JL and Given: Prove: JKLM is a parallelogram. ̶̶̶ KM bisect each other. Plan: Show that △JNK ≅ △LNM and △KNL ≅ △MNJ. Then use the fact that the corresponding angles are congruent to show ̶̶ JK ǁ ̶̶̶ LM and ̶̶ KL ǁ ̶̶ MJ. 31. Prove that the figure formed by two midsegments of a triangle and their corresponding bases is a parallelogram. 32. Write About It Use the theorems from Lessons 6-2 and 6-3 to write three biconditional statements about parallelograms. 33. Construction Explain how you can construct a parallelogram based on the conditions of Theorem 6-3-1. Use your method to construct a parallelogram. 34. This problem will prepare you for the Multi-Step TAKS Prep on page 406. A geologist made the following observations while examining this amethyst crystal. Tell whether each set of observations allows the geologist to conclude that PQRS is a parallelogram. If so, explain why. ̶̶ PQ ≅ ̶̶ SR, and ̶̶ PS ǁ ̶̶ QR. a. b. ∠S and ∠R are supplementary, and ̶̶ PQ ǁ c. ∠S ≅ ∠Q, and ̶̶ SR. ̶̶ PS ≅ ̶̶ QR. � � � � 404 404 Chapter 6 Polygons and Quadrilaterals �� 35. What additional information would allow you to conclude that WXYZ is a parallelogram? ̶̶ XY ≅ ̶̶̶ WX ≅ ̶̶̶ ZW ̶̶ YZ ̶̶̶ WY ≅ ̶̶̶ WZ ∠XWY ≅ ∠ZYW 36. Which could be the coordinates of the fourth vertex of ABCD with A (-1, -1
), B (1, 3), and C (6, 1)? D (8, 5) D (4, -3) D (13, 3) D (3, 7) 37. Short Response The vertices of quadrilateral RSTV are R (-5, 0), S (-1, 3), T (5, 1), and V (2, -2). Is RSTV a parallelogram? Justify your answer. CHALLENGE AND EXTEND 38. Write About It As the upper platform of the movable staircase is raised and lowered, the height of each step changes. How does the upper platform remain parallel to the ground? 39. Multi-Step The diagonals of a parallelogram intersect at (-2, 1.5). Two vertices are located at (-7, 2) and (2, 6.5). Find the coordinates of the other two vertices. 40. Given: D is the midpoint of ̶̶ AC, and E is the midpoint of ̶̶ BC. Prove: ̶̶ DE ǁ ̶̶ AB, DE = 1 _ 2 AB (Hint: Extend ̶̶ ̶̶ EF ≅ DE to form Then show that DFBA is a parallelogram.) ̶̶ DF so that ̶̶ DE. SPIRAL REVIEW  . -5, -2, 0, 0.5 ⎬ ⎨ Complete a table of values for each function. Use the domain   (Previous course) 41. f (x) = 7x - 3 42. f (x) = x + 2 _ 2 43. f (x) = 3x 2 + 2 Use SAS to explain why each pair of triangles are congruent. (Lesson 4-4) 44. △ABD ≅ △CDB 45. △TUW ≅ △VUW For JKLM, find each measure. (Lesson 6-2) 46. NM 48. JL 47. LM 49. JK 6- 3 Conditions for Parallelograms 405 405 �� SECTION 6A Polygons and Parallelograms Crystal Clear A crystal is a mineral formation that has polygonal faces. Geologists classify crystals based on the types
of polygons that the faces form. 1. What type of polygon is ABCDE in the fluorite crystal? Given that m∠B = 120°, m∠E = 65°, and ∠C ≅ ∠D, find m∠A. ̶̶ AE ǁ ̶̶ CD, � � � � � � � � � 2. The pink crystals are called rhodochrosite. The face FGHJ is a parallelogram. Given that m∠F = (9x - 13) ° and m∠J = (7x + 1) °, find m∠G. Explain how you found this angle measure. 3. While studying the amazonite crystal, a geologist ̶̶̶ MN ≅ ̶̶ QP and ∠NQP ≅ ∠QNM. Can the found that geologist conclude that MNPQ is a parallelogram? Why or why not? Justify your answer. � � � � 406 406 Chapter 6 Polygons and Quadrilaterals SECTION 6A Quiz for Lessons 6-1 Through 6-3 6-1 Properties and Attributes of Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. 4. 5. Find the sum of the interior angle measures of a convex 16-gon. 6. The surface of a trampoline is in the shape of a regular hexagon. Find the measure of each interior angle of the trampoline. 7. A park in the shape of quadrilateral PQRS is bordered by four sidewalks. Find the measure of each exterior angle of the park. 8. Find the measure of each exterior angle of a regular decagon. 6-2 Properties of Parallelograms A pantograph is used to copy drawings. Its legs form a parallelogram. In JKLM, LM = 17 cm, KN = 13.5 cm, and m∠KJM = 102°. Find each measure. 9. KM 10. KJ 11. MN 13. m∠JML 12. m∠JKL 15. Three vertices of ABCD are A (-3, 1), B (5, 7), and C (6, 2). Find the coordinates of vertex D. 14. m∠
KLM WXYZ is a parallelogram. Find each measure. 16. WX 18. m∠X 17. YZ 19. m∠W � � � � � 6-3 Conditions for Parallelograms 20. Show that RSTV is a parallelogram 21. Show that GHJK is a parallelogram for x = 6 and y = 4.5. for m = 12 and n = 9.5. Determine if each quadrilateral must be a parallelogram. Justify your answer. 22. 23. 24. 25. Show that a quadrilateral with vertices C (-9, 4), D (-4, 8), E (2, 6), and F (-3, 2) is a parallelogram. Ready to Go On? 407 407 �� 6-4 Properties of Special Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures.... Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objectives Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems. Vocabulary rectangle rhombus square Who uses this? Artists who work with stained glass can use properties of rectangles to cut materials to the correct sizes. A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles. Theorems Properties of Rectangles THEOREM HYPOTHESIS CONCLUSION 6-4-1 If a quadrilateral is a rectangle, then it is a parallelogram. (rect. → ) 6-4-2 If a parallelogram is a rectangle, then its diagonals are congruent. (rect. → diags. ≅) ABCD is a parallelogram. ̶̶ AC ≅ ̶̶ BD You will prove Theorems 6-4-1 and 6-4-2 in Exercises 38 and 35. Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of paralle
lograms that you learned in Lesson 6-2. E X A M P L E 1 Craft Application An artist connects stained glass pieces with lead strips. In this rectangular window, the strips are cut so that FG = 14 in. and FH = 20 in. Find JG. ̶̶ ̶̶ EG ≅ FH EG = FH = 20 JG = 1 _ 2 JG = 1 _ (20) = 10 in. 2 EG Rect. → diags. ≅ Def. of ≅ segs.  → diags. bisect each other Substitute and simplify. � � � � � Carpentry The rectangular gate has diagonal braces. Find each length. 1a. HJ 1b. HK 408 408 Chapter 6 Polygons and Quadrilaterals ��HGLJK48 in.30.8 in.ge07se_c06l04003aAB A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides. Theorems Properties of Rhombuses THEOREM HYPOTHESIS CONCLUSION 6-4-3 If a quadrilateral is a rhombus, then it is a parallelogram. (rhombus → ) 6-4-4 If a parallelogram is a rhombus, then its diagonals are perpendicular. (rhombus → diags. ⊥) 6-4-5 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. (rhombus → each diag. bisects opp. ) ABCD is a parallelogram. ̶̶ AC ⊥ ̶̶ BD ∠1 ≅ ∠2 ∠3 ≅ ∠4 ∠5 ≅ ∠6 ∠7 ≅ ∠8 You will prove Theorems 6-4-3 and 6-4-4 in Exercises 34 and 37. PROOF PROOF Theorem 6-4-5 Given: JKLM is a rhombus. Prove: ̶̶ JL bisects ∠KJM and ∠KLM. ̶̶̶ KM bisects ∠JKL and ∠JML. Proof: Since JKLM is a rhombus,
̶̶ JK ≅ ̶̶ JM, and ̶̶ KL ≅ ̶̶̶ ML by the definition of a rhombus. By the Reflexive Property of Congruence, Thus △JKL ≅ △JML by SSS. Then ∠1 ≅ ∠2, and ∠3 ≅ ∠4 by CPCTC. ̶̶ JL bisects ∠KJM and ∠KLM by the definition of an angle bisector. So ̶̶̶ KM bisects ∠JKL and ∠JML. By similar reasoning, ̶̶ JL ≅ ̶̶ JL. Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses. E X A M P L E 2 Using Properties of Rhombuses to Find Measures RSTV is a rhombus. Find each measure. A VT ST = SR 4x + 7 = 9x - 11 18 = 5x Def. of rhombus Substitute the given values. Subtract 4x from both sides and add 11 to both sides. Divide both sides by 5. Def. of rhombus 3.6 = x VT = ST VT = 4x + 7 VT = 4 (3.6) + 7 = 21.4 Substitute 4x + 7 for ST. Substitute 3.6 for x and simplify. 6- 4 Properties of Special Parallelograms 409 409 �� RSTV is a rhombus. Find each measure. B m∠WSR m∠SWT = 90° 2y + 10 = 90 y = 40 Rhombus → diags. ⊥ Substitute 2y + 10 for m∠SWT. Subtract 10 from both sides and divide both sides by 2. m∠WSR = m∠TSW m∠WSR = (y + 2) ° m∠WSR = (40 + 2) ° = 42° Rhombus → each diag. bisects opp.  Substitute y + 2 for m∠TSW. Substitute 40 for y and simplify. CDFG is a rhombus. Find each measure. 2a. CD 2b. m∠GCH if m∠G
CD = (b + 3) ° and m∠CDF = (6b - 40) ° Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three. E X A M P L E 3 Verifying Properties of Squares Show that the diagonals of square ABCD are congruent perpendicular bisectors of each other. Step 1 Show that ̶̶ AC and ̶̶ BD are congruent. AC = √  ⎤ ⎦ ⎡ ⎣ 2 + (7 - 0) 2 = √  58 2 - (-1) BD = √  ⎤ ⎦ ⎡ ⎣ 2 + (2 - 5) 2 = √  58 4 - (-3) ̶̶ AC ≅ ̶̶ AC and ̶̶ BD. ̶̶ BD are perpendicular. Since AC = BD, Step 2 Show that slope of 2 - (-1) _ ̶̶ 7 AC = 3 _ 7 7 ̶̶ BD = 2 - 5 _ slope of _ _ ) = -1, ) (- 3 Since ( 7 7 3 4 - (-3) ̶̶ AC ⊥ ̶̶ BD. Step 3 Show that ̶̶ AC and ̶̶ BD bisect each other. mdpt. of mdpt. of ̶̶ AC : ( BD : ( ̶̶ 0 + 7 - Since ̶̶ AC and ̶̶ BD have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other. 3. The vertices of square STVW are S (-5, -4), T (0, 2), V (6, -3), and W (1, -9). Show that the diagonals of square STVW are congruent perpendicular bis
ectors of each other. 410 410 Chapter 6 Polygons and Quadrilaterals �� Special Parallelograms To remember the properties of rectangles, rhombuses, and squares, I start with a square, which has all the properties of the others. To get a rectangle that is not a square, I stretch the square in one direction. Its diagonals are still congruent, but they are no longer perpendicular. Taylor Gallinghouse Central High School To get a rhombus that is not a square, I go back to the square and slide the top in one direction. Its diagonals are still perpendicular and bisect the opposite angles, but they aren’t congruent. E X A M P L E 4 Using Properties of Special Parallelograms in Proofs Given: EFGH is a rectangle. J is the midpoint of Prove: △FJG is isosceles. Proof: ̶̶ EH. Statements Reasons 1. EFGH is a rectangle. J is the midpoint of ̶̶ EH. 2. ∠E and ∠H are right angles. 3. ∠E ≅ ∠H 4. EFGH is a parallelogram. ̶̶ EF ≅ ̶̶ EJ ≅ ̶̶̶ HG ̶̶ HJ 5. 6. 7. △FJE ≅ △GJH ̶̶ GJ ̶̶ FJ ≅ 8. 9. △FJG is isosceles. 1. Given 2. Def. of rect. 3. Rt. ∠ ≅ Thm. 4. Rect. →  5.  → opp. sides ≅ 6. Def. of mdpt. 7. SAS Steps 3, 5, 6 8. CPCTC 9. Def. of isosc. △ 4. Given: PQTS is a rhombus with diagonal ̶̶ PR. Prove: ̶̶ RQ ≅ ̶̶ RS THINK AND DISCUSS 1. Which theorem means “The diagonals of a rectangle are congruent”? Why do you think the theorem is written as a conditional? 2. What properties of a rhombus are the same as the properties of all parallelograms?
What special properties does a rhombus have? 3. GET ORGANIZED Copy and complete the graphic organizer. Write the missing terms in the three unlabeled sections. Then write a definition of each term. 6- 4 Properties of Special Parallelograms 411 411 �� 6-4 Exercises Exercises KEYWORD: MG7 6-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What is another name for an equilateral quadrilateral? an equiangular quadrilateral? a regular quadrilateral. 408 Engineering The braces of the bridge support lie along the diagonals of rectangle PQRS. RS = 160 ft, and QS = 380 ft. Find each length. 2. TQ 4. ST 3. PQ 5. PR ABCD is a rhombus. Find each measure. p. 409 6. AB 7. m∠ABC. Multi-Step The vertices of square JKLM p. 410 are J (-3, -5), K (-4, 1), L (2, 2), and M (3, -4). Show that the diagonals of square JKLM are congruent perpendicular bisectors of each other. 411 Independent Practice For See Exercises Example 10–13 14–15 16 17 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 9. Given: RECT is a rectangle. Prove: △REY ≅ △TCX ̶̶ RX ≅ ̶̶ TY PRACTICE AND PROBLEM SOLVING Carpentry A carpenter measures the diagonals of a piece of wood. In rectangle JKLM, JM = 25 in., and JP = 14 1 __ in. Find each length. 2 10. JL 11. KL 12. KM 13. MP VWXY is a rhombus. Find each measure. 14. VW 15. m∠VWX and m∠WYX if m∠WVY = (4b + 10) ° and m∠XZW = (10b - 5) ° 16. Multi-Step The vertices of square PQRS are P (-4, 0), Q (4, 3), R (7, -5), and S (-1,
-8). Show that the diagonals of square PQRS are congruent perpendicular bisectors of each other. 17. Given: RHMB is a rhombus with diagonal ̶̶ HB. Prove: ∠HMX ≅ ∠HRX Find the measures of the numbered angles in each rectangle. 18. 19. 20. 412 412 Chapter 6 Polygons and Quadrilaterals �� Find the measures of the numbered angles in each rhombus. 21. 22. 23. Tell whether each statement is sometimes, always, or never true. (Hint: Refer to your graphic organizer for this lesson.) 24. A rectangle is a parallelogram. 25. A rhombus is a square. 26. A parallelogram is a rhombus. 27. A rhombus is a rectangle. 28. A square is a rhombus. 29. A rectangle is a quadrilateral. 30. A square is a rectangle. 31. A rectangle is a square. 32. Critical Thinking A triangle is equilateral if and only if the triangle is equiangular. Can you make a similar statement about a quadrilateral? Explain your answer. 33. History There are five shapes of clay tiles in this tile mosaic from the ruins of Pompeii. a. Make a sketch of each shape of tile and tell whether the shape is a polygon. b. Name each polygon by its number of sides. Does each shape appear to be regular or irregular? c. Do any of the shapes appear to be special parallelograms? If so, identify them by name. d. Find the measure of each interior angle of the center polygon. 34. /////ERROR ANALYSIS///// Find and correct the error in this proof of Theorem 6-4-3. Given: JKLM is a rhombus. Prove: JKLM is a parallelogram. Proof: History Pompeii was located in what is today southern Italy. In C.E. 79, Mount Vesuvius erupted and buried Pompeii in volcanic ash. The ruins have been excavated and provide a glimpse into life in ancient Rome. It is given that JKLM is a rhombus. So by the definition of a rhombus, ̶̶ ̶̶ JK ≅ MJ. Theorem 6-2-1
states that if a quadrilateral is a parallelogram, then its opposite sides are congruent. So JKLM is a ̶̶̶ LM, and ̶̶ KL ≅ parallelogram by Theorem 6-2-1. 35. Complete the two-column proof of Theorem 6-4-2 by filling in the blanks. Given: EFGH is a rectangle. ̶̶ GE Prove: ̶̶ FH ≅ Proof: Statements Reasons 1. EFGH is a rectangle. 1. Given 2. EFGH is a parallelogram. ̶̶ EF ≅ b. ̶̶ ̶̶ EH EH ≅ 3. 4.? ̶̶̶̶̶ 5. ∠FEH and ∠GHE are right angles. 2. a.? ̶̶̶̶̶ 3.  → opp. sides ≅ 4. c. 5. d.? ̶̶̶̶̶? ̶̶̶̶̶ 6. ∠FEH ≅ e.? ̶̶̶̶̶ 6. Rt. ∠ ≅ Thm. 7. △FEH ≅ △GHE ̶̶ GE ̶̶ FH ≅ 8. 7. f. 8. g.? ̶̶̶̶̶? ̶̶̶̶̶ 6- 4 Properties of Special Parallelograms 413 413 �� 36. This problem will prepare you for the Multi-Step TAKS Prep on page 436. The organizers of a fair plan to fence off a plot of land given by the coordinates A (2, 4), B (4, 2), C (-1, -3), and D (-3, -1). a. Find the slope of each side of quadrilateral ABCD. b. What type of quadrilateral is formed by the fences? Justify your answer. c. The organizers plan to build a straight path connecting A and C and another path connecting B and D. Explain why these two paths will have the same length. 37. Use this plan to write a proof of Theorem 6-4-4. Given: VWXY is a rhombus. ̶̶̶ WY ̶̶ VX ⊥ Prove: Plan: Use the definition of a rhombus and the properties of parallelograms to show that △W
ZX ≅ △YZX. Then use CPCTC to show that ∠WZX and ∠YZX are right angles. 38. Write a paragraph proof of Theorem 6-4-1. Given: ABCD is a rectangle. Prove: ABCD is a parallelogram. 39. Write a two-column proof. Given: ABCD is a rhombus. E, F, G, and H are the midpoints of the sides. Prove: EFGH is a parallelogram. Multi-Step Find the perimeter and area of each figure. Round to the nearest hundredth, if necessary. 40. 41. 42. 43. Write About It Explain why each of these conditional statements is true. a. If a quadrilateral is a square, then it is a parallelogram. b. If a quadrilateral is a square, then it is a rectangle. c. If a quadrilateral is a square, then it is a rhombus. 44. Write About It List the properties that a square “inherits” because it is (1) a parallelogram, (2) a rectangle, and (3) a rhombus. 45. Which expression represents the measure of ∠J in rhombus JKLM? x° 2x° (180 - x) ° (180 - 2x) ° 46. Short Response The diagonals of rectangle QRST intersect at point P. If QR = 1.8 cm, QP = 1.5 cm, and QT = 2.4 cm, find the perimeter of △RST. Explain how you found your answer. 414 414 Chapter 6 Polygons and Quadrilaterals �� 47. Which statement is NOT true of a rectangle? Both pairs of opposite sides are congruent and parallel. Both pairs of opposite angles are congruent and supplementary. All pairs of consecutive sides are congruent and perpendicular. All pairs of consecutive angles are congruent and supplementary. CHALLENGE AND EXTEND 48. Algebra Find the value of x in the rhombus. 49. Prove that the segment joining the midpoints of two consecutive sides of a rhombus is perpendicular to one diagonal and parallel to the other. 50. Extend the definition of a triangle midsegment to write a definition for the midsegment of a rectangle. Prove that a midse
gment of a rectangle divides the rectangle into two congruent rectangles. 51. The figure is formed by joining eleven congruent squares. How many rectangles are in the figure? SPIRAL REVIEW 52. The cost c of a taxi ride is given by c = 2 + 1.8 (m - 1), where m is the length of the trip in miles. Mr. Hatch takes a 6-mile taxi ride. How much change should he get if he pays with a $20 bill and leaves a 10% tip? (Previous course) Determine if each conditional is true. If false, give a counterexample. (Lesson 2-2) 53. If a number is divisible by -3, then it is divisible by 3. 54. If the diameter of a circle is doubled, then the area of the circle will double. Determine if each quadrilateral must be a parallelogram. Justify your answer. (Lesson 6-3) 55. 56. Construction Rhombus     ̶̶ PS. Set the compass ̶̶ PS. Place Draw to the length of the compass point at P and ̶̶ draw an arc above PS. Label a point Q on the arc. Place the compass point at Q and draw an arc to the right of Q. Place the compass point at S and draw an arc that intersects the arc drawn from Q. Label the point of intersection R. Draw ̶̶ PQ, ̶̶ QR, and ̶̶ RS. 6- 4 Properties of Special Parallelograms 415 415 �� 6-5 Predict Conditions for Special Parallelograms In this lab, you will use geometry software to predict the conditions that are sufficient to prove that a parallelogram is a rectangle, rhombus, or square. Use with Lesson 6-5 Activity 1 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B, G.9.B KEYWORD: MG7 Lab6 1 Construct ̶̶ AB and ̶̶ AD with a common endpoint A. Construct a line through D parallel to Construct a line through B parallel to ̶̶ AB. ̶̶ AD. 2 Construct point C at the intersection of the
̶̶ BC two lines. Hide the lines and construct ̶̶ CD to complete the parallelogram. and 3 Measure the four sides and angles of the parallelogram. 4 Move A so that m∠ABC = 90°. What type of special parallelogram results? 5 Move A so that m∠ABC ≠ 90°. 6 Construct ̶̶ AC and ̶̶ BD and measure their lengths. Move A so that AC = BD. What type of special parallelogram results? Try This 1. How does the method of constructing ABCD in Steps 1 and 2 guarantee that the quadrilateral is a parallelogram? 2. Make a Conjecture What are two conditions for a rectangle? Write your conjectures as conditional statements. 416 416 Chapter 6 Polygons and Quadrilaterals Activity 2 1 Use the parallelogram you constructed in Activity 1. Move A so that AB = BC. What type of special parallelogram results? 2 Move A so that AB ≠ BC. 3 Label the intersection of the diagonals as E. Measure ∠AEB. 4 Move A so that m∠AEB = 90°. What type of special parallelogram results? 5 Move A so that m∠AEB ≠ 90°. 6 Measure ∠ABD and ∠CBD. Move A so that m∠ABD = m∠CBD. What type of special parallelogram results? Try This 3. Make a Conjecture What are three conditions for a rhombus? Write your conjectures as conditional statements. 4. Make a Conjecture A square is both a rectangle and a rhombus. What conditions do you think must hold for a parallelogram to be a square? 6- 5 Technology Lab 417 417 6-5 Conditions for Special Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objective Prove that a given quadrilateral is a rectangle, rhombus, or square. Who uses this? Building contractors and carpenters can use the conditions for rectangles to make sure the frame for a house has the correct shape. When you are given a parallelogram with certain properties, you can use the theorems below
to determine whether the parallelogram is a rectangle. Theorems Conditions for Rectangles THEOREM EXAMPLE 6-5-1 6-5-2 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. ( with one rt. ∠ → rect.) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. ( with diags. ≅ → rect.) ̶̶ AC ≅ ̶̶ BD You will prove Theorems 6-5-1 and 6-5-2 in Exercises 31 and 28. E X A M P L E 1 Carpentry Application ̶̶ WZ and A contractor built a wood frame for the side of a house so that ̶̶ ̶̶ XY ≅ XW ≅ tape measure, the contractor found that XZ = WY. Why must the frame be a rectangle? ̶̶ YZ. Using a Both pairs of opposite sides of WXYZ are congruent, so WXYZ is a parallelogram. Since XZ = WY, the diagonals of WXYZ are congruent. Therefore the frame is a rectangle by Theorem 6-5-2. 418 418 Chapter 6 Polygons and Quadrilaterals �� 1. A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle? Below are some conditions you can use to determine whether a parallelogram is a rhombus. Theorems Conditions for Rhombuses THEOREM EXAMPLE In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. 6-5-3 6-5-4 6-5-5 If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. ( with one pair cons. sides ≅ → rhombus) If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. ( with diags. ⊥ → rhombus) If one diagonal of a parallelog
ram bisects a pair of opposite angles, then the parallelogram is a rhombus. ( with diag. bisecting opp.  → rhombus) You will prove Theorems 6-5-3 and 6-5-4 in Exercises 32 and 30. PROOF PROOF Theorem 6-5-5 Given: JKLM is a parallelogram. ̶̶ JL bisects ∠KJM and ∠KLM. Prove: JKLM is a rhombus. Proof: Statements Reasons 1. JKLM is a parallelogram. ̶̶ JL bisects ∠KJM and ∠KLM. 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ JL ≅ ̶̶ JL 3. 4. △JKL ≅ △JML ̶̶ JK ≅ ̶̶ JM 5. 6. JKLM is a rhombus. 2. Def. of ∠ bisector 3. Reflex. Prop. of ≅ 4. ASA Steps 2, 3 5. CPCTC 6.  with one pair cons. sides ≅ → rhombus To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43. 6- 5 Conditions for Special Parallelograms 419 419 �� E X A M P L E 2 Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. A Given: ̶̶ AB ≅ ̶̶ AD ⊥ ̶̶ BC ≅ ̶̶ AC ⊥ ̶̶ CD, ̶̶ DC, Conclusion: ABCD is a square. Step 1 Determine if ABCD is a parallelogram. ̶̶ AD, ̶̶ BD ̶̶ AB ≅ ̶̶ CD, ̶̶ BC ≅ ̶̶ AD Given ABCD is a parallelogram. Quad. with opp. sides ≅ →  Step 2 Determine if ABCD is a rectangle. ̶̶ AD ⊥ ̶̶ DC, so ∠ADC is a
right angle. Def. of ⊥ ABCD is a rectangle.  with one rt. ∠ → rect. Step 3 Determine if ABCD is a rhombus. ̶̶ AC ⊥ ̶̶ BD ABCD is a rhombus. Given  with diags. ⊥ → rhombus Step 4 Determine if ABCD is a square. Since ABCD is a rectangle and a rhombus, it has four right angles and four congruent sides. So ABCD is a square by definition. The conclusion is valid. ̶̶ BC ̶̶ AB ≅ B Given: You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals. Conclusion: ABCD is a rhombus. The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. To apply this theorem, you must first know that ABCD is a parallelogram. 2. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ∠ABC is a right angle. Conclusion: ABCD is a rectangle. E X A M P L E 3 Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. A A (0, 2), B (3, 6), C (8, 6), D (5, 2) Step 1 Graph ABCD. Step 2 Determine if ABCD is a rectangle. √  (8 - 0) 2 + (6 - 2) 2 AC = = √  80 = 4 √  5 BD = √  (5 - 3) 2 + (2 - 6) 2 = √  20 = 2 √  5 Since 4 √  5 ≠ 2 √ �
� 5, ABCD is not a rectangle. Thus ABCD is not a square. 420 420 Chapter 6 Polygons and Quadrilaterals �� Step 3 Determine if ABCD is a rhombus. slope of ̶̶ AC = 6 - 2 _ ) (-2) = -1, B E (-4, -1), F (-3, 2), G (3, 0), H (2, -3) Since ( 1 _ = 1 _ 2 ̶̶ AC ⊥ 8 - 0 2 slope of ̶̶ BD = 2 - 6 _ = -2 5 - 3 ̶̶ BD. ABCD is a rhombus. Step 1 Graph EFGH. Step 2 Determine if EFGH is a rectangle. EG = √  ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ 2 2 + 0 - (-1) 3 - (-4) = √  50 = 5 √  2 FH = √  ⎤ ⎦ ⎡ ⎣ 2 + (-3 - 2) 2 2 - (-3) = √  50 = 5 √  2 Since 5 √  2 = 5 √  2, the diagonals are congruent. EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus. slope of ̶̶ EG = 0 - (-1) _ 3 - (-4) slope of Since ( 1_ ̶̶ FH = -3 - 2 _ 7)(-1) ≠ -1, 2 - (-3) ̶̶ EG ⊥/ = 1 _ 7 = -5 _ 5 ̶̶ FH. = -1 So EFGH is a not a rhombus and cannot be a square. Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
3a. K (-5, -1), L (-2, 4), M (3, 1), N (0, -4) 3b. P (-4, 6), Q (2, 5), R (3, -1), S (-3, 0) THINK AND DISCUSS 1. What special parallelogram is formed when the diagonals of a parallelogram are congruent? when the diagonals are perpendicular? when the diagonals are both congruent and perpendicular? 2. Draw a figure that shows why this statement is not necessarily true: If one angle of a quadrilateral is a right angle, then the quadrilateral is a rectangle. 3. A rectangle can also be defined as a parallelogram with a right angle. Explain why this definition is accurate. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, write at least three conditions for the given parallelogram. 6- 5 Conditions for Special Parallelograms 421 421 �� 6-5 Exercises Exercises GUIDED PRACTICE. Gardening A city garden club is planting a KEYWORD: MG7 6-5 KEYWORD: MG7 Parent p. 418. 420 square garden. They drive pegs into the ground at each corner and tie strings between each pair. ̶̶̶ ZW. The pegs are spaced so that How can the garden club use the diagonal strings to verify that the garden is a square? ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. 2. Given: ̶̶ AC ≅ ̶̶ BD Conclusion: ABCD is a rectangle. 3. Given: ̶̶ AB ǁ ̶̶ AB ⊥ Conclusion: ABCD is a rectangle. ̶̶ AB ≅ ̶̶ CD, ̶̶ CD, ̶̶ BC. 420 Multi-Step Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 4. P (-5, 2), Q (4, 5), R (
6, -1), S (-3, -4) 5. W (-6, 0), X (1, 4), Y (2, -4), Z (-5, -8) Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 PRACTICE AND PROBLEM SOLVING 6. Crafts A framer uses a clamp to hold together the pieces of a picture frame. ̶̶ RS and The pieces are cut so that ̶̶ ̶̶ QR ≅ SP. The clamp is adjusted so that PZ, QZ, RZ, and SZ are all equal. Why must the frame be a rectangle? ̶̶ PQ ≅ � � � � � Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ FH ̶̶ FH bisect each other. ̶̶ EG and 7. Given: ̶̶ EG ⊥ Conclusion: EFGH is a rhombus. 8. Given: ̶̶ FH bisects ∠EFG and ∠EHG. Conclusion: EFGH is a rhombus. Multi-Step Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 9. A (-10, 4), B (-2, 10), C (4, 2), D (-4, -4) 10. J (-9, -7), K (-4, -2), L (3, -3), M (-2, -8) Tell whether each quadrilateral is a parallelogram, rectangle, rhombus, or square. Give all the names that apply. 11. 12. 13. 422 422 Chapter 6 Polygons and Quadrilaterals ge07sec06l05004aABeckmannXWYZV�� Tell whether each quadrilateral is a parallelogram, rectangle, rhombus, or square. Give all the names that apply. 14. 15. 16. 17. /////ERROR ANALYSIS///// In ABCD, ̶̶ AC ≅ ̶̶ BD. Which conclusion is incorrect? Explain the error.
Give one characteristic of the diagonals of each figure that would make the conclusion valid. 18. Conclusion: JKLM is a rhombus. 19. Conclusion: PQRS is a square. The coordinates of three vertices of ABCD are given. Find the coordinates of D so that the given type of figure is formed. 20. A (4, -2), B (-5, -2), C (4, 4) ; rectangle 21. A (-5, 5), B (0, 0), C (7, 1) ; rhombus 22. A (0, 2), B (4, -2), C (0, -6) ; square 23. A (2, 1), B (-1, 5), C (-5, 2) ; square Find the value of x that makes each parallelogram the given type. 24. rectangle 25. rhombus 26. square 27. Critical Thinking The diagonals of a quadrilateral are perpendicular bisectors of each other. What is the best name for this quadrilateral? Explain your answer. 28. Complete the two-column proof of Theorem 6-5-2 by filling in the blanks. Given: EFGH is a parallelogram. ̶̶ EG ≅ ̶̶ HF Prove: EFGH is a rectangle. Proof: Statements Reasons 1. EFGH is a parallelogram. ̶̶ EG ≅ ̶̶ HF ̶̶ EF ≅ ̶̶̶ HG 2. 3. b.? ̶̶̶̶̶ 4. △EFH ≅ △HGE 5. ∠FEH ≅ d.? ̶̶̶̶̶ 6. ∠FEH and ∠GHE are supplementary. 7. g.? ̶̶̶̶̶ 8. EFGH is a rectangle. 1. Given 2. a.? ̶̶̶̶̶ 3. Reflex. Prop. of ≅ 4. c. 5. e. 6. f.? ̶̶̶̶̶? ̶̶̶̶̶? ̶̶̶̶̶ 7. ≅  supp. → rt.  8. h.? ̶̶̶̶̶ 6- 5 Conditions for Special Parallelograms 423 423 ��
�� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 436. A state fair takes place on a plot of land given by the coordinates A (-2, 3), B (1, 2), C (2, -1), and D (-1, 0). a. Show that the opposite sides of quadrilateral ABCD are parallel. b. A straight path connects A and C, and another path connects B and D. Use slopes to prove that these two paths are perpendicular. c. What can you conclude about ABCD? Explain your answer. 30. Complete the paragraph proof of Theorem 6-5-4 by filling in the blanks. Given: PQRS is a parallelogram. Prove: PQRS is a rhombus. ̶̶ PR ⊥ ̶̶ QS Proof: that ̶̶ PR ⊥ It is given that PQRS is a parallelogram. The diagonals of a?. By the ̶̶̶̶?. It is given ̶̶̶̶ parallelogram bisect each other, so Reflexive Property of Congruence, ̶̶ PT ≅ a. ̶̶ QT ≅ b. ̶̶ QS, so ∠QTP and ∠QTR are right angles by the?. Then ∠QTP ≅ ∠QTR by the d. ̶̶̶̶ definition of c. So △QTP ≅ △QTR by e. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a?, and ̶̶̶̶ ̶̶ QP ≅ f.?. ̶̶̶̶?, by CPCTC. ̶̶̶̶ g.?. Therefore PQRS is rhombus. ̶̶̶̶ 31. Write a two-column proof of Theorem 6-5-1. Given: ABCD is a parallelogram. ∠A is a right angle. Prove: ABCD is a rectangle. 32. Write a paragraph proof of Theorem 6-5-3. Given: JKLM is a parallelogram. Prove: JKLM is a rhombus. ̶̶ JK ≅ ̶̶ KL 33. Algebra Four
lines are represented by the equations below. m: y = -x + 7 ℓ: y = -x + 1 a. Graph the four lines in the coordinate plane. b. Classify the quadrilateral formed by the lines. c. What if…? Suppose the slopes of lines n and p change to 1. n: y = 2x + 1 p: y = 2x + 7 Reclassify the quadrilateral. 34. Write a two-column proof. Given: FHJN and GLMF are parallelograms. ̶̶ FG ≅ ̶̶ FN Prove: FGKN is a rhombus. 35. Write About It Use Theorems 6-4-2 and 6-5-2 to write a biconditional statement about rectangles. Use Theorems 6-4-4 and 6-5-4 to write a biconditional statement about rhombuses. Can you combine Theorems 6-4-5 and 6-5-5 to write a biconditional statement? Explain your answer. Construction Use the diagonals to construct each figure. Then use the theorems from this lesson to explain why your method works. 36. rectangle 37. rhombus 38. square 424 424 Chapter 6 Polygons and Quadrilaterals �� 39. In PQRS, ̶̶ PR and ̶̶ QS intersect at T. What additional information is needed to conclude that PQRS is a rectangle? ̶̶ PT ≅ ̶̶ PT ≅ ̶̶ QT ̶̶ RT ̶̶ ̶̶ PT ⊥ QT ̶̶ PT bisects ∠QPS. 40. Which of the following is the best name for figure WXYZ with vertices W (-3, 1), X (1, 5), Y (8, -2), and Z (4, -6)? Parallelogram Rectangle Rhombus Square 41. Extended Response a. Write and solve an equation to find the value of x. b. Is JKLM a parallelogram? Explain. c. Is JKLM a rectangle? Explain. d. Is JKLM a rhombus? Explain. CHALLENGE AND EXTEND ̶̶ ̶̶ 42. Given: DF, BC, ̶̶ EF, ̶̶ AB ≅
̶̶ BC ǁ ̶̶ AC ≅ ̶̶ BE ⊥ ̶̶ DE, ̶̶ EF ̶̶ AB ⊥ ̶̶ DE ⊥ ̶̶ EF, Prove: EBCF is a rectangle. 43. Critical Thinking Consider the following statement: If a quadrilateral is a rectangle and a rhombus, then it is a square. a. Explain why the statement is true. b. If a quadrilateral is a rectangle, is it necessary to show that all four sides are congruent in order to conclude that it is a square? Explain. c. If a quadrilateral is a rhombus, is it necessary to show that all four angles are right angles in order to conclude that it is a square? Explain. 44. Cars As you turn the crank of a car jack, the platform that supports the car raises. Use the diagonals of the parallelogram to explain whether the jack forms a rectangle, rhombus, or square. SPIRAL REVIEW Sketch the graph of each function. State whether the function is linear or nonlinear. (Previous course) 45. y = -3x + 1 46. y = x 2 - 4 47. y = 3 Find the perimeter of each figure. Round to the nearest tenth. (Lesson 5-7) 48. 49. Find the value of each variable that would make the quadrilateral a parallelogram. (Lesson 6-3) 50. x 51. y 52. z 6- 5 Conditions for Special Parallelograms 425 425 �� 6-6 Explore Isosceles Trapezoids In this lab you will investigate the properties and conditions of an isosceles trapezoid. A trapezoid is a quadrilateral with one pair of parallel sides, called bases. The sides that are not parallel are called legs. In an isosceles trapezoid, the legs are congruent. Use with Lesson 6-6 Activity 1 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B, G.9.B KEYWORD: MG7 Lab6 1 Draw ̶̶ AB and a point C not
on ̶̶ AB. Construct a parallel line ℓ through C. 2 Draw point D on line ℓ. Construct ̶̶ AC and ̶̶ BD. 3 Measure AC, BD, ∠CAB, ∠ABD, ∠ACD, and ∠CDB. 4 Move D until AC = BD. What do you notice about m∠CAB and m∠ABD? What do you notice about m∠ACD and m∠CDB? 5 Move D so that AC ≠ BD. Now move D so that m∠CAB = m∠ABD. What do you notice about AC and BD? Try This 1. Make a Conjecture What is true about the base angles of an isosceles trapezoid? Write your conjecture as a conditional statement. 2. Make a Conjecture How can the base angles of a trapezoid be used to determine if the trapezoid is isosceles? Write your conjecture as a conditional statement. Activity 2 1 Construct ̶̶ AD and ̶̶ CB. 2 Measure AD and CB. 3 Move D until AC = BD. What do you notice about AD and CB? 4 Move D so that AC ≠ BD. Now move D so that AD = BC. What do you notice about AC and BD? Try This 3. Make a Conjecture What is true about the diagonals of an isosceles trapezoid? Write your conjecture as a conditional statement. 4. Make a Conjecture How can the diagonals of a trapezoid be used to determine if the trapezoid is isosceles? Write your conjecture as a conditional statement. 426 426 Chapter 6 Polygons and Quadrilaterals 6-6 Properties of Kites and Trapezoids TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Why learn this? The design of a simple kite flown at the beach shares the properties of the geometric figure called a kite. A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Objectives Use properties of kites to solve problems. Use properties of trapezoids to solve problems. Vocabulary kite trapez
oid base of a trapezoid leg of a trapezoid base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid Theorems Properties of Kites THEOREM HYPOTHESIS CONCLUSION 6-6-1 6-6-2 If a quadrilateral is a kite, then its diagonals are perpendicular. (kite → diags. ⊥) If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. (kite → one pair opp.  ≅) ̶̶ AC ⊥ ̶̶ BD ∠B ≅ ∠D ∠A ≇ ∠C You will prove Theorem 6-6-1 in Exercise 39. PROOF PROOF Theorem 6-6-2 ̶̶ JK ≅ Given: JKLM is a kite with Prove: ∠K ≅ ∠M, ∠KJM ≇ ∠KLM ̶̶ JM and ̶̶ KL ≅ ̶̶̶ ML. Proof: Step 1 Prove ∠K ≅ ∠M. ̶̶ JK ≅ ̶̶ JL ≅ It is given that of Congruence, So ∠K ≅ ∠M by CPCTC. ̶̶ KL ≅ ̶̶ ̶̶̶ ML. By the Reflexive Property JM and ̶̶ JL. This means that △JKL ≅ △JML by SSS. Step 2 Prove ∠KJM ≇ ∠KLM. If ∠KJM ≅ ∠KLM, then both pairs of opposite angles of JKLM are congruent. This would mean that JKLM is a parallelogram. But this contradicts the given fact that JKLM is a kite. Therefore ∠KJM ≇ ∠KLM. 6- 6 Properties of Kites and Trapezoids 427 427 �� E X A M P L E 1 Problem-Solving Application Alicia is using a pattern to make a kite. She has made the frame of the kite by placing wooden sticks along the diagonals. She also has cut four triangular pieces of fabric and has attached them to the frame. To finish the kite, Alicia must cover the outer edges with
a cloth binding. There are 2 yards of binding in one package. What is the total amount of binding needed to cover the edges of the kite? How many packages of binding must Alicia buy? � � � �� Understand the Problem The answer has two parts. • the total length of binding Alicia needs • the number of packages of binding Alicia must buy Make a Plan The diagonals of a kite are perpendicular, so the four triangles are right triangles. Use the Pythagorean Theorem and the properties of kites to find the unknown side lengths. Add these lengths to find the perimeter of the kite. Solve PQ = = √  √  16 2 + 13 2 425 = 5 √  17 in. RQ = PQ = 5 √  17 in. √  16 2 + 22 2 740 = 2 √  PS = = √  185 in. Pyth. Thm. ̶̶ PQ ≅ ̶̶ RQ Pyth. Thm. RS = PS = 2 √  185 in. ̶̶ RS ≅ ̶̶ PS perimeter of PQRS = 5 √  17 + 5 √  17 + 2 √  185 + 2 √  185 ≈ 95.6 in. Kites The Zilker Kite Festival, held in Austin, Texas, has been an annual event since 1929. It is the longest continuously running kite festival in the country. Participants compete in categories such as highest kite, steadiest kite, strongest kite, largest kite, and most unusual kite. Alicia needs approximately 95.6 inches of binding. One package of binding contains 2 yards, or 72 inches. 95.6 _ 72 ≈ 1.3 packages of binding In order to have enough, Alicia must buy 2 packages of binding. Look Back To estimate the perimeter, change the side lengths into decimals and round. 5 √  17 ≈ 21, and 2 √  2 (21) + 2 (27) = 96. So 95.6 is a reasonable answer.
185 ≈ 27. The perimeter of the kite is approximately 1. What if...? Daryl is going to make a kite by doubling all the measures in the kite above. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy? 428 428 Chapter 6 Polygons and Quadrilaterals 1234 E X A M P L E 2 Using Properties of Kites In kite EFGH, m∠FEJ = 25°, and m∠FGJ = 57°. Find each measure. A m∠GFJ m∠FJG = 90° Kite → diags. ⊥ m∠GFJ + m∠FGJ = 90 m∠GFJ + 57 = 90 m∠GFJ = 33° Acute  of rt. △ are comp. Substitute 57 for m∠FGJ. Subtract 57 from both sides. B m∠JFE △FJE is also a right triangle, so m∠JFE + m∠FEJ = 90°. By substituting 25° for m∠FEJ, you find that m∠JFE = 65°. C m∠GHE ∠GHE ≅ ∠GFE m∠GHE = m∠GFE m∠GFE = m∠GFJ + m∠JFE m∠GHE = 33° + 65° = 98° Kite → one pair opp.  ≅ Def. of ≅  ∠ Add. Post. Substitute. In kite PQRS, m∠PQR = 78°, and m∠TRS = 59°. Find each measure. 2a. m∠QRT 2b. m∠QPS 2c. m∠PSR A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base. If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid. Theorems Isosceles Tra
pezoids THEOREM DIAGRAM EXAMPLE 6-6-3 6-6-4 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. (isosc. trap. → base  ≅) If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. (trap. with pair base  ≅ → isosc. trap.) 6-6-5 A trapezoid is isosceles if and only if its diagonals are congruent. (isosc. trap. ↔ diags. ≅) Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.” ∠A ≅ ∠D ∠B ≅ ∠C ABCD is isosceles. ̶̶ AC ≅ ̶̶ DB ↔ ABCD is isosceles. 6- 6 Properties of Kites and Trapezoids 429 429 �� E X A M P L E 3 Using Properties of Isosceles Trapezoids A Find m∠Y. m∠W + m∠X = 180° 117 + m∠X = 180 m∠X = 63° ∠Y ≅ ∠X m∠Y = m∠X m∠Y = 63° Same-Side Int.  Thm. Substitute 117 for m∠W. Subtract 117 from both sides. Isosc. trap. → base  ≅ Def. of ≅  Substitute 63 for m∠X. B RT = 24.1, and QP = 9.6. Find PS. ̶̶ RT ̶̶ QS ≅ QS = RT QS = 24.1 QP + PS = QS 9.6 + PS = 24.1 PS = 14.5 Isosc. trap. → diags. ≅ Def. of ≅ segs. Substitute 24.1 for RT. Seg. Add. Post. Substitute 9.6 for QP and 24.1 for QS. Subtract 9.6 from both sides. 3a. Find m∠F
. 3b. JN = 10.6, and NL = 14.8. Find KM. E X A M P L E 4 Applying Conditions for Isosceles Trapezoids A Find the value of y so that EFGH is isosceles. ∠E ≅ ∠H Trap. with pair base  ≅ m∠E = m∠H 2y 2 - 25 = y 2 + 24 → isosc. trap. Def. of ≅  Substitute 2 y 2 - 25 for m∠E and y 2 + 24 for m∠H. y 2 = 49 Subtract y 2 from both sides and add 25 to both sides. y = 7 or y = -7 Find the square root of both sides. B JL = 5z + 3, and KM = 9z - 12. Find the value of z so that JKLM is isosceles. ̶̶̶ KM ̶̶ JL ≅ JL = KM 5z + 3 = 9z - 12 Diags. ≅ → isosc. trap. Def. of ≅ segs. Substitute 5z + 3 for JL and 9z - 12 for KM. 15 = 4z Subtract 5z from both sides and add 12 to both sides. 3.75 = z Divide both sides by 4. 4. Find the value of x so that PQST is isosceles. 430 430 Chapter 6 Polygons and Quadrilaterals �� The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it. Theorem 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. ̶̶ XY ǁ XY = 1 _ 2 ̶̶ BC, ̶̶ XY ǁ ̶̶ AD (BC + AD ) You will prove the Trapezoid Midsegment Theorem in Exercise 46. E X A M P L E 5 Finding Lengths Using Midsegments Find ST. MN = 1 _ (ST +
RU) 2 31 = 1 _ (ST + 38 ) 2 62 = ST + 38 24 = ST Trap. Midsegment Thm. Substitute the given values. Multiply both sides by 2. Subtract 38 from both sides. 5. Find EH. THINK AND DISCUSS 1. Is it possible for the legs of a trapezoid to be parallel? Explain. 2. How is the midsegment of a trapezoid similar to a midsegment of a triangle? How is it different? 3. GET ORGANIZED Copy and complete the graphic organizer. Write the missing terms in the unlabeled sections. Then write a definition of each term. (Hint: This completes the Venn diagram you started in Lesson 6-4.) 6- 6 Properties of Kites and Trapezoids 431 431 �� 6-6 Exercises Exercises KEYWORD: MG7 6-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 428 1. In trapezoid PRSV, name the bases, the legs, and the midsegment. 2. Both a parallelogram and a kite have two pairs of congruent sides. How are the congruent sides of a kite different from the congruent sides of a parallelogram? 3. Crafts The edges of the kite-shaped glass in the sun catcher are sealed with lead strips. JH, KH, and LH are 2.75 inches, and MH is 5.5 inches. How much lead is needed to seal the edges of the sun catcher? If the craftsperson has two 3-foot lengths of lead, how many sun catchers can be sealed. 429 In kite WXYZ, m∠WXY = 104°, and m∠VYZ = 49°. Find each measure. 4. m∠VZY 5. m∠VXW 6. m∠XWZ. Find m∠A. p. 430 8. RW = 17.7, and SV = 23.3. Find TW. 430 9. Find the value of z so that EFGH is isosceles. 10. MQ = 7y - 6, and LP = 4y + 11
. Find the value of y so that LMPQ is isosceles 11. Find QR. 12. Find AZ. p. 431 432 432 Chapter 6 Polygons and Quadrilaterals �� PRACTICE AND PROBLEM SOLVING 13. Design Each square section in the iron railing contains four small kites. The figure shows the dimensions of one kite. What length of iron is needed to outline one small kite? How much iron is needed to outline one complete section, including the square? In kite ABCD, m∠DAX = 32°, and m∠XDC = 64°. Find each measure. 14. m∠XDA 15. m∠ABC 16. m∠BCD Independent Practice For See Exercises Example 13 14–16 17–18 19–20 21–22 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 17. Find m∠Q. 18. SZ = 62.6, and KZ = 34. Find RJ. 19. Algebra Find the value of a so that XYZW is isosceles. Give your answer as a simplified radical. 20. Algebra GJ = 4x - 1, and FH = 9x - 15. Find the value of x so that FGHJ is isosceles. 21. Find PQ. 22. Find KR. Tell whether each statement is sometimes, always, or never true. 23. The opposite angles of a trapezoid are supplementary. 24. The opposite angles of a kite are supplementary. 25. A pair of consecutive angles in a kite are supplementary. 26. Estimation Hal is building a trapezoid-shaped frame for a flower bed. The lumber costs $1.29 per foot. Based on Hal’s sketch, estimate the cost of the lumber. (Hint: Find the angle measures in the triangle formed by the dashed line.) Find the measure of each numbered angle. 27. 30. 28. 31. 29. 32. 6- 6 Properties of Kites and Trapezoids 433 433 ��6 ft60°20 ft6 ftge07se_c06
l06005aAB�� 33. This problem will prepare you for the Multi-Step TAKS Prep on page 436. The boundary of a fairground is a quadrilateral with vertices at E (-1, 3), F (3, 4), G (2, 0), and H (-3, -2). a. Use the Distance Formula to show that EFGH is a kite. b. The organizers need to know the angle measure at each vertex. Given that m∠H = 46° and m∠F = 62°, find m∠E and m∠G. Algebra Find the length of the midsegment of each trapezoid. 34. 35. 36. Mechanics The Peaucellier cell, invented in 1864, converts circular motion into linear motion. This type of linkage was supposedly used in the fans that ventilated the Houses of Parliament in London prior to the invention of electric fans. 37. Mechanics A Peaucellier cell is made of seven rods ̶̶ OA ≅ connected by joints at the labeled points. AQBP is a ̶̶ OB. As P moves along a circular rhombus, and path, Q moves along a linear path. In the position shown, m∠AQB = 72°, and m∠AOB = 28°. What are m∠PAQ, m∠OAQ, and m∠OBP? 38. Prove that one diagonal of a kite bisects a pair of opposite angles and the other diagonal. 39. Prove Theorem 6-6-1: If a quadrilateral is a kite, then its diagonals are perpendicular. Multi-Step Give the best name for a quadrilateral with the given vertices. 40. (-4, -1), (-4, 6), (2, 6), (2, -4) 41. (-5, 2), (-5, 6), (-1, 6), (2, -1) 42. (-2, -2), (1, 7), (4, 4), (1, -5) 43. (-4, -3), (0, 3), (4, 3), (8, -3) 44. Carpentry The window frame is a regular octagon. It
is made from eight pieces of wood shaped like congruent isosceles trapezoids. What are m∠A, m∠B, m∠C, and m∠D? 45. Write About It Compare an isosceles trapezoid to a trapezoid that is not isosceles. What properties do the figures have in common? What properties does one have that the other does not? � � � � 46. Use coordinates to verify the Trapezoid Midsegment Theorem. a. M is the midpoint of b. N is the midpoint of ̶̶ c. Find the slopes of QR, you conclude? ̶̶ QP. What are its coordinates? ̶̶ RS. What are its coordinates? ̶̶ PS, and ̶̶̶ MN. What can d. Find QR, PS, and MN. Show that MN = 1 __ 2 (PS + QR). 47. In trapezoid PQRS, what could be the lengths of ̶̶ QR and ̶̶ PS? 6 and 10 6 and 26 8 and 32 10 and 24 434 434 Chapter 6 Polygons and Quadrilaterals �� 48. Which statement is never true for a kite? The diagonals are perpendicular. One pair of opposite angles are congruent. One pair of opposite sides are parallel. Two pairs of consecutive sides are congruent. 49. Gridded Response What is the length of the midsegment of trapezoid ADEB in inches? CHALLENGE AND EXTEND 50. Write a two-column proof. (Hint: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Use this fact to draw ̶̶̶ WZ.) auxiliary lines ̶̶ VY so that ̶̶ VY ⊥ ̶̶ UX and ̶̶ UX ⊥ ̶̶ XZ ≅ Given: WXYZ is a trapezoid with Prove: WXYZ is an isosceles trapezoid. ̶̶̶ WZ and ̶̶̶ YW. 51. The perimeter of isosceles trapezoid ABCD is 27.4 inches. If BC = 2 (AB), find AD,
AB, BC, and CD. SPIRAL REVIEW 52. An empty pool is being filled with water. After 10 hours, 20% of the pool is full. If the pool is filled at a constant rate, what fraction of the pool will be full after 25 hours? (Previous course) Write and solve an inequality for x. (Lesson 3-4) 53. 54. Tell whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. (Lesson 6-5) 55. (-3, 1), (-1, 3), (1, 1), and (-1, -1) 55. 56. (1, 1), (4, 5), (4, 0), and (1, -4) Construction Kite     Draw a segment ̶̶ AC. Construct line ℓ as the perpendicular bisector ̶̶ of AC. Label the intersection as X. Draw a point B on ℓ ̶̶ AB above ̶̶ CB. and ̶̶ AC. Draw ̶̶ AC so that Draw a point D on ℓ below DX ≠ BX. Draw and ̶̶ AD ̶̶ CD. 1. Critical Thinking How would you modify the construction above so that ABCD is a concave kite? 6- 6 Properties of Kites and Trapezoids 435 435 �� SECTION 6B Other Special Quadrilaterals A Fair Arrangement The organizers of a county fair are using a coordinate plane to plan the layout of the fairground. The fence that surrounds the fairground will have vertices at A (-1, 4), B (7, 8), C (3, 0), and D (-5, -4). 1. The organizers consider creating two straight paths through the fairground: one from point A to point C and another from point B to point D. Use a theorem from Lesson 6-4 to prove that these paths would be perpendicular. 2. The organizers instead decide to put an entry gate at the midpoint of each side of the fence, as shown. They plan to create straight paths that connect the gates. Show that the paths ̶̶ PQ, ̶̶ SP form a parallelogram. ̶̶ QR,
̶̶ RS, and ̶̶ PR and 3. Use the paths ̶̶ SQ to tell whether PQRS is a rhombus, rectangle, or square. 4. One section of the fair will contain all the rides and games. The organizers will fence off this area within the fairground by using the existing fences along ̶̶ CE, where E has coordinates (-1, 0). What type of along quadrilateral will be formed by these four fences? ̶̶ BC and adding fences ̶̶ AB and ̶̶ AE and 5. To construct the fences, the organizers need to know the angle measures at each vertex. Given that m∠B = 37°, find the measures of the other angles in quadrilateral ABCE. 436 436 Chapter 6 Polygons and Quadrilaterals �� SECTION 6B Quiz for Lessons 6-4 Through 6-6 6-4 Properties of Special Parallelograms The flag of Jamaica is a rectangle with stripes along the diagonals. In rectangle QRST, QS = 80.5, and RS = 36. Find each length. 1. SP 2. QT 3. TR 4. TP GHJK is a rhombus. Find each measure. 5. HJ 6. m∠HJG and m∠GHJ if m∠JLH = (4b - 6) ° and m∠JKH = (2b + 11) ° 7. Given: QSTV is a rhombus. Prove: ̶̶ PQ ≅ ̶̶ RQ ̶̶ PT ≅ ̶̶ RT 6-5 Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ AC ⊥ 8. Given: ̶̶ BD Conclusion: ABCD is a rhombus. 9. Given: ̶̶ AB ≅ ̶̶ AB ǁ Conclusion: ABCD is a rectangle. ̶̶ AC ≅ ̶̶ CD, ̶̶ BD, ̶̶ CD Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 10. W (-2, 2), X (1, 5), Y (7, -1), Z (4
, -4) ̶̶ ZX are midsegments of △TWY. 11. M (-4, 5), N (1, 7), P (3, 2), Q (-2, 0) ̶̶ VX and 12. Given: ̶̶̶ TW ≅ ̶̶ TY Prove: TVXZ is a rhombus. 6-6 Properties of Kites and Trapezoids In kite EFGH, m∠FHG = 68°, and m∠FEH = 62°. Find each measure. 13. m∠FEJ 15. m∠FGJ 14. m∠EHJ 16. m∠EHG 17. Find m∠R. 18. YZ = 34.2, and VX = 53.4. Find WZ. 19. A dulcimer is a trapezoid-shaped stringed instrument. The bases are 43 in. and 23 in. long. If a string is attached at the midpoint of each leg of the trapezoid, how long is the string? Ready to Go On? 437 437 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary base of a trapezoid.......... 429 kite........................ 427 rhombus................... 409 base angle of a trapezoid.... 429 leg of a trapezoid........... 429 side of a polygon............ 382 concave.................... 383 midsegment of a trapezoid.. 431 square..................... 410 convex..................... 383 parallelogram.............. 391 trapezoid...
................ 429 diagonal................... 382 rectangle................... 408 vertex of a polygon.......... 382 isosceles trapezoid.......... 429 regular polygon............. 382 Complete the sentences below with vocabulary words from the list above. 1. The common endpoint of two sides of a polygon is a(n)?. ̶̶̶̶ 2. A polygon is? if no diagonal contains points in the exterior. ̶̶̶̶ 3. A(n)? is a quadrilateral with four congruent sides. ̶̶̶̶ 4. Each of the parallel sides of a trapezoid is called a(n)?. ̶̶̶̶ 6-1 Properties and Attributes of Polygons (pp. 382–388) E X A M P L E S EXERCISES TEKS G.2.B, G.3.B, G.4.A, G.5.A, G.5.B, G.7.A ■ Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. The figure is a closed plane figure made of segments that intersect only at their endpoints, so it is a polygon. It has six sides, so it is a hexagon. ■ Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. The polygon is equilateral, but it is not equiangular. So it is not regular. No diagonal contains points in the exterior, so it is convex. Find each measure. ■ the sum of the interior angle measures of a convex 11-gon (n - 2) 180° (11 - 2) 180° = 1620° Polygon ∠ Sum Thm. Substitute 11 for n. ■ the measure of each exterior angle of a regular pentagon sum of ext.  = 360° Polygon Ext. ∠ SumThm. measure of one ext. ∠ = 360° _ = 72° 5 Tell whether each figure is a polygon
. If it is a polygon, name it by the number of its sides. 5. 6. 7. Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 8. 10. 9. Find each measure. 11. the sum of the interior angle measures of a convex dodecagon 12. the measure of each interior angle of a regular 20-gon 13. the measure of each exterior angle of a regular quadrilateral 14. the measure of each interior angle of hexagon ABCDEF 438 438 Chapter 6 Polygons and Quadrilaterals �� 6-2 Properties of Parallelograms (pp. 391–397) TEKS G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B E X A M P L E S ■ In PQRS, m∠RSP = 99°, PQ = 19.8, and RT = 12.3. Find PT. ̶̶ RT ̶̶ PT ≅ PT = RT PT = 12.3  → diags. bisect each other Def. of ≅ segs. Substitute 12.3 for RT. JKLM is a parallelogram. Find each measure. ■ LK ̶̶ ̶̶ LK JM ≅ JM = LK 2y - 9 = y + 7 y = 16  → opp. sides ≅ Def. of ≅ segs. Substitute the given values. Solve for y. LK = 16 + 7 = 23 ■ m∠M m∠J + m∠M = 180°  → cons.  supp. Substitute the given (x + 4) + 3x = 180 x = 44 m∠M = 3 (44) = 132° values. Solve for x. EXERCISES In ABCD, m∠ABC = 79°, BC = 62.4, and BD = 75. Find each measure. 15. BE 16. AD 17. ED 18. m∠CDA 19. m∠BCD 20. m∠DAB WXYZ is a parallelogram. Find each measure. 21. WX 22. YZ 23. m∠W 25
. m∠Y 24. m∠X 26. m∠Z 27. Three vertices of RSTV are R (-8, 1), S (2, 3), and V (-4, -7). Find the coordinates of vertex T. 28. Write a two-column proof. Given: GHLM is a parallelogram. ∠L ≅ ∠JMG Prove: △GJM is isosceles. 6-3 Conditions for Parallelograms (pp. 398–405) E X A M P L E S ■ Show that MNPQ is a parallelogram for a = 6 and b = 1.6. MN = 2a + 5 MN = 2 (6) + 5 = 17 QP = 4 (6) - 7 = 17 MQ = 7b MQ = 7 (1.6) = 11.2 NP = 2b + 8 NP = 2 (1.6) + 8 = 11.2 QP = 4a - 7 Since its opposite sides are congruent, MNPQ is a parallelogram. ■ Determine if the quadrilateral must be a parallelogram. Justify your answer. No. One pair of opposite angles are congruent, and one pair of consecutive sides are congruent. None of the conditions for a parallelogram are met. TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C EXERCISES Show that the quadrilateral is a parallelogram for the given values of the variables. 29. m = 13, n = 27 30. x = 25, y = 7 Determine if the quadrilateral must be a parallelogram. Justify your answer. 31. 32. 33. Show that the quadrilateral with vertices B (-4, 3), D (6, 5), F (7, -1), and H (-3, -3) is a parallelogram. Study Guide: Review 439 439 �� 6-4 Properties of Special Parallelograms (pp. 408–415) E X
A M P L E S In rectangle JKLM, KM = 52.8, and JM = 45.6. Find each length. ■ KL JKLM is a . KL = JM = 45.6 Rect. →   → opp. sides ≅ ■ NL JL = KM = 52.8 NL = 1_ 2 JL = 26.4 Rect. → diags. ≅  → diags. bisect each other ■ PQRS is a rhombus. Find m∠QPR, given that m∠QTR = (6y + 6) ° and m∠SPR = 3y°. m∠QTR = 90° 6y + 6 = 90 y = 14 m∠QPR = m∠SPR m∠QPR = 3 (14) ° = 42° Rhombus → diags. ⊥ Substitute the given value. Solve for y. Rhombus → each diag. bisects opp.  ■ The vertices of square ABCD are A (5, 0), B (2, 4), C (-2, 1), and D (1, -3). Show that the diagonals of square ABCD are congruent perpendicular bisectors of each other. AC = BD = 5 √  2 ̶̶ AC = - 1_ slope of 7 ̶̶ BD = 7 ̶̶ AC Product of slopes is -1, so diags. are ⊥. Diags. are ≅. slope of mdpt. of = mdpt. of ̶̶ BD = ( 3 _ ), 1 _ 2 2 Diags. bisect each other. TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C EXERCISES In rectangle ABCD, CD = 18, and CE = 19.8. Find each length. 34. AB 35. AC 36. BD 37. BE In rhombus WXYZ, WX = 7a + 1, WZ = 9a - 6, and VZ = 3a. Find each measure. 38. WZ 39. XV 40. XY 41. XZ In rhombus RSTV, m�
�TZV = (8n + 18) °, and m∠SRV = (9n + 1) °. Find each measure. 42. m∠TRS 43. m∠RSV 44. m∠STV 45. m∠TVR Find the measures of the numbered angles in each figure. 46. rectangle MNPQ 47. rhombus CDGH Show that the diagonals of the square with the given vertices are congruent perpendicular bisectors of each other. 48. R (-5, 0), S (-1, -2), T (-3, -6), and U (-7, -4) 49. E (2, 1), F (5, 1), G (5, -2), and H (2, -2) 6-5 Conditions for Special Parallelograms (pp. 418–425) E X A M P L E S EXERCISES TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C ■ Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ LP ⊥ ̶̶ KN Given: Conclusion: KLNP is a rhombus. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. 50. Given: ̶̶ FS, Conclusion: EFRS is a square. ̶̶ ER ⊥ ̶̶ ER ≅ ̶̶ FS The conclusion is not valid. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply this theorem, you must first know that KLNP is a parallelogram. 440 440 Chapter 6 Polygons and Quadrilaterals 51. Given: ̶̶ FS bisect each other. ̶̶ ER and ̶̶ ̶̶ FS ER ≅ Conclusion: EFRS is a rectangle. ̶̶ ES ̶̶ EF ≅ ̶̶ FR ǁ ̶̶ EF ǁ ̶̶ RS, ̶̶ ES, 52. Given: Conclusion: EFRS is a rhombus. �� ■ Use the
diagonals to tell whether a parallelogram with vertices P (-5, 3), Q (0, 1), R (2, -4), and S (-3, -2) is a rectangle, rhombus, or square. Give all the names that apply. PR = √  98 = 7 √  2 QS = √  18 = 3 √  2 Distance Formula Distance Formula Use the diagonals to tell whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 53. B (-3, 0), F (-2, 7), J (5, 8), N (4, 1) 54. D (-4, -3), H (5, 6), L (8, 3), P (-1, -6) 55. Q (-8, -2), T (-6, 8), W (4, 6), Z (2, -4) Since PR ≠ QS, PQRS is not a rectangle and not a square. slope of = -1 ̶̶ PR = 7 _ -7 ̶̶ QS = 3 _ 3 = 1 slope of Slope Formula Slope Formula Since the product of the slopes is -1, the diagonals are perpendicular. PQRS is a rhombus. 6-6 Properties of Kites and Trapezoids (pp. 427–435) E X A M P L E S EXERCISES TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C ■ In kite PQRS, m∠SRT = 24°, and m∠TSP = 53°. Find m∠SPT. △PTS is a right triangle. Kite → diags. ⊥ Acute  of rt. △ are comp. m∠SPT + m∠TSP = 90° m∠SPT + 53 = 90 Substitute 53 for m∠TSP. m∠SPT = 37° Subtract 53 from both sides. ■ Find m∠D. m∠C + m∠D = 180° Same-
Side Int.  Thm. Substitute 51 for m∠C. Subtract. 51 + m∠D = 180 m∠D = 129° ■ In trapezoid HJLN, In kite WXYZ, m∠VXY = 58°, and m∠ZWX = 50°. Find each measure. 56. m∠XYZ 57. m∠ZWV 58. m∠VZW 59. m∠WZY Find each measure. 60. m∠R and m∠S 61. BZ if ZH = 70 and EK = 121.6 62. MN 63. EQ JP = 32.5, and HL = 50. Find PN. ̶̶ ̶̶ JN ≅ HL JN = HL = 50 JP + PN = JN 32.5 + PN = 50 PN = 17.5 ■ Find WZ. Isosc. trap. → diags. ≅ Def. of ≅ segs. Seg. Add. Post. Substitute. Subtract 32.5 from both sides. AB = 1 _ (XY + WZ) Trap. Midsegment Thm. 2 73.5 = 1 _ (42 + WZ) 2 147 = 42 + WZ 105 = WZ Multiply both sides by 2. Solve for WZ. Substitute. 64. Find the value of n so that PQXY is isosceles. Give the best name for a quadrilateral whose vertices have the given coordinates. 65. (-4, 5), (-1, 8), (5, 5), (-1, 2) 66. (1, 4), (5, 4), (5, -4), (1, -1) 67. (-6, -1), (-4, 2), (0, 2), (2, -1) Study Guide: Review 441 441 �� Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. The base of a fountain is in the shape of a quadrilateral, as shown. Find the measure of each interior angle of the fountain. 4. Find the sum
of the interior angle measures of a convex nonagon. 5. Find the measure of each exterior angle of a regular 15-gon. � �� 6. In EFGH, EH = 28, HZ = 9, and m∠EHG = 145°. Find FH and m∠FEH. 7. JKLM is a parallelogram. Find KL and m∠L. 8. Three vertices of PQRS are P (-2, -3), R (7, 5), and S (6, 1). Find the coordinates of Q. 9. Show that WXYZ is a parallelogram for a = 4 and b = 3. 10. Determine if CDGH must be a parallelogram. Justify your answer. 11. Show that a quadrilateral with vertices K (-7, -3), L (2, 0), S (5, -4), and T (-4, -7) is a parallelogram. 12. In rectangle PLCM, LC = 19, and LM = 23. Find PT and PM. 13. In rhombus EHKN, m∠NQK = (7z + 6) °, and m∠ENQ = (5z + 1) °. Find m∠HEQ and m∠EHK. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ NP ≅ 14. Given: 15. Given: ̶̶̶ MQ, Conclusion: MNPQ is a rectangle. ̶̶̶ NM ≅ ̶̶ NP ≅ ̶̶ PQ, ̶̶̶ NQ ≅ ̶̶ ̶̶̶ MN PQ ≅ Conclusion: MNPQ is a square. ̶̶̶ QM ≅ ̶̶̶ MP Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 16. A (-5, 7), C (3, 6), E (7, -1), G (-1, 0) 17. P (4, 1), Q (3, 4), R (-3, 2), S (-2, -1) 18. m
∠JFR = 43°, and m∠JNB = 68°. Find m∠FBN. 20. Find HR. � � � 442 442 Chapter 6 Polygons and Quadrilaterals �� 19. PV = 61.1, and YS = 24.7. Find MY. � � � �� FOCUS ON SAT The scores for each SAT section range from 200 to 800. Your score is calculated by subtracting a fraction for each incorrect multiple-choice answer from the total number of correct answers. No points are deducted for incorrect grid-in answers or items you left blank. If you have time, go back through each section of the test and check as many of your answers as possible. Try to use a different method of solving the problem than you used the first time. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. 1. Given the quadrilateral below, what value of x 3. Which of the following terms best describes the would allow you to conclude that the figure is a parallelogram? (A) -2 (B) 0 (C) 1 (D) 2 (E) 3 2. In the figure below, if ABCD is a rectangle, what type of triangle must △ABE be? (A) Equilateral (B) Right (C) Equiangular (D) Isosceles (E) Scalene figure below? (A) Rhombus (B) Trapezoid (C) Quadrilateral (D) Square (E) Parallelogram 4. Three vertices of MNPQ are M (3, 1), N (0, 6), and P (4, 7). Which of the following could be the coordinates of vertex Q? (A) (7, 0) (B) (–1, 1) (C) (7, 2) (D) (11, 3) (E) (9, 4) 5. If ABCDE is a regular pentagon, what is the measure of ∠C? (A) 45° (B) 60° (C) 90° (D) 108° (E) 120° College Entrance Exam Practice 443 443 �� Multiple Choice: Eliminate Answer
Choices For some multiple-choice test items, you can eliminate one or more of the answer choices without having to do many calculations. Use estimation or logic to help you decide which answer choices can be eliminated. What is the value of x in the figure? 3° 63° 83° 153° The sum of the exterior angle measures of a convex polygon is 360°. By rounding, you can estimate the sum of the given angle measures. 100° + 30° + 140° + 30° = 300° If x = 153°, the sum of the angle measures would be far greater than 360°. So eliminate D. If x = 3°, the sum would be far less than 360°. So eliminate A. From your estimate, it seems likely that the correct choice is B, 63°. Confirm that this is correct by doing the actual calculation. 98° + 32° + 63° + 135° + 32° = 360° The correct answer is B, 63°. What is m∠B in the isosceles trapezoid? 216° 108° 72° 58° Base angles of an isosceles trapezoid are congruent. Since ∠D and ∠B are not a pair of base angles, their measures are not equal. Eliminate G, 108°. ∠D and ∠C are base angles, so m∠C = 108°. ∠B and ∠C are same-side interior angles formed by parallel lines. So they are supplementary angles. Therefore the measure of angle B cannot be greater than 180°. You can eliminate F. m∠B = 180° - 108° = 72° The correct answer is H, 72°. 444 444 Chapter 6 Polygons and Quadrilaterals �� Try to eliminate unreasonable answer choices. Some choices may be too large or too small or may have incorrect units. Item C In isosecles trapezoid ABCD, AC = 18.2, and DG = 6.3. What is GB? Read each test item and answer the questions that follow. Item A The diagonals of rectangle MNPQ intersect at S. If MN = 4.1 meters, MS = 2.35 meters, and MQ = 2.3 meters, what is the area of △MPQ to the nearest tenth? 24.5 11.9 6.3 2.9 6. Will the measure of
̶̶ than, or equal to the measure of AC? What answer choices can you eliminate and why? ̶̶ GB be more than, less 4.7 square meters 5.4 meters 9.4 square meters 12.8 meters 1. Are there any answer choices you can eliminate immediately? If so, which choices and why? 2. Describe how to use estimation to eliminate at least one more answer choice. Item B What is the sum of the interior angles of a convex hexagon? 7. Explain how to use estimation to answer this problem. Item D In trapezoid LMNP, XY = 25 feet. What are two ̶̶ LM and possible lengths for ̶̶ PN? 18 feet and 32 feet 49 feet and 2 feet 10 feet and 15 feet 7 inches and 43 inches 180° 500° 720° 1080° 3. Can any of the answer choices be eliminated immediately? If so, which choices and why? 4. How can you use the fact that 500 is not a multiple of 180 to eliminate choice G? 5. A student answered this problem with J. Explain the mistake the student made. 8. Which answer choice can you eliminate immediately? Why? 9. A student used logic to eliminate choice H. Do you agree with the student’s decision? Explain. 10. A student used estimation and answered this problem with G. Explain the mistake the student made. TAKS Tackler 445 445 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–6 Multiple Choice Use the figure below for Items 6 and 7. 1. The exterior angles of a triangle have measures of (x + 10) °, (2x + 20) °, and 3x°. What is the measure of the smallest interior angle of the triangle? 15° 35° 55° 65° 2. If a plant is a monocot, then its leaves have parallel veins. If a plant is an orchid, then it is a monocot. A Mexican vanilla plant is an orchid. Based on this information, which conclusion is NOT valid? The leaves of a Mexican vanilla plant have parallel veins. A Mexican vanilla plant is a monocot. All orchids have leaves with parallel veins. All monocots are orchids. 3. If △ABC ≅ △PQR and △RPQ ≅ △XYZ, which of the following angles
is congruent to ∠CAB? ∠QRP ∠XZY ∠YXZ ∠XYZ 4. Which line coincides with the line 2y + 3x = 4? x + 2 3y + 2x = 4 y = 2 _ 3 a line through (-1, 1) and (2, 3) a line through (0, 2) and (4, -4) 5. What is the value of x in polygon ABCDEF? 12 18 24 36 446 446 Chapter 6 Polygons and Quadrilaterals 6. If ̶̶ JK ǁ ̶̶̶ ML, what additional information do you need to prove that quadrilateral JKLM is a parallelogram? ̶̶ JM ≅ ̶̶̶ MN ≅ ̶̶ KL ̶̶ LN ∠MLK and ∠LKJ are right angles. ∠JML and ∠KLM are supplementary. 7. Given that JKLM is a parallelogram and that m∠KLN = 25°, m∠JMN = 65°, and m∠JML = 130°, which term best describes quadrilateral JKLM? Rectangle Rhombus Square Trapezoid 8. For two lines and a transversal, ∠1 and ∠2 are same-side interior angles, ∠2 and ∠3 are vertical angles, and ∠3 and ∠4 are alternate exterior angles. Which classification best describes the angle pair ∠2 and ∠4? Adjacent angles Alternate interior angles Corresponding angles Vertical angles 9. For △ABC and △DEF, ∠A ≅ ∠F, and ̶̶ AC ≅ ̶̶ EF. Which of the following would allow you to conclude that these triangles are congruent by AAS? ∠ABC ≅ ∠EDF ∠ACB ≅ ∠EDF ∠BAC ≅ ∠FDE ∠CBA ≅ ∠FED �� 10. The vertices of ABCD are A (1, 4), B (4, y), C (3, -2), and D (0, -3). What is the value of y? 3 4 5 6 STANDARDIZED TEST PREP
Short Response 17. In △ABC, AE = 9x - 11.25, and AF = x + 4. 11. Quadrilateral RSTU is a kite. What is the length ̶̶ RV? of 4 inches 5 inches 6 inches 13 inches 12. What is the measure of each interior angle in a regular dodecagon? 30° 144° 150° 162° 13. The coordinates of the vertices of quadrilateral RSTU are R (1, 3), S (2, 7), T (10, 5), and U (9, 1). Which term best describes quadrilateral RSTU? Parallelogram Rectangle Rhombus Trapezoid �� Mixed numbers cannot be entered into the grid for gridded-response questions. For example, if you get an answer of 7 1 __ 7.25 or 29 __. 4, you must grid either 4 Gridded Response 14. If quadrilateral MNPQ is a parallelogram, what is the value of x? 15. What is the greatest number of line segments determined by six coplanar points when no three are collinear? 16. Quadrilateral RSTU is a rectangle with diagonals SU = 6a - 25, what is the value of a? ̶̶ SU. If RT = 4a + 2 and ̶̶ RT and a. Find the value of x. Show your work and explain how you found your answer. b. If ̶̶ DF ≅ ̶̶ EF, show that △AFD ≅ △CFE. State any theorems or postulates used. 18. Consider quadrilateral ABCD. a. Show that ABCD is a trapezoid. Justify your answer. b. What are the coordinates for the endpoints of the midsegment of trapezoid ABCD? 19. Suppose that ∠M is complementary to ∠N and ∠N is complementary to ∠P. Explain why the measurements of these three angles cannot be the angle measurements of a triangle. Extended Response 20. Given △ABC and △XYZ, suppose that ̶̶ AB ≅ ̶̶ XY and ̶̶ BC ≅ ̶̶ YZ. a. If AB = 5, BC = 6, AC = 8, and m∠B < m∠Y, explain why △XYZ is
obtuse. Justify your reasoning and state any theorems or postulates used. b. If AB = 3, BC = 5, AC = 5, and m∠B > m∠Y, ̶̶ XZ so that △XYZ is a right find the length of triangle. Justify your reasoning and state any theorems or postulates used. c. If AB = 8 and BC = 4, find the range of possible ̶̶ AC. Justify your answer. values for the length of Cumulative Assessment, Chapters 1–6 447 447 �� T E X A S TAKS Grades 9–11 Obj. 10 Arlington Georgetown Southwestern University Southwestern University, a nationally recognized liberal arts university in Georgetown, Texas, is Texas’s oldest university. Southwestern was officially chartered in 1875 ge07ts_c0 6psl001aa but was formed from the resources of four existing schools—Rutersville College 2nd pass (chartered in 1840), Wesleyan College (chartered in 1844), McKenzie College (chartered in 1848), and Soule University (chartered in 1856). 6/20/5 cmurphy Choose one or more strategies to solve each problem. 1. The trusses that line the ceiling of the McCombs Campus Center are made of triangular shapes. The center shape ̶̶ AC of the resembles an equilateral triangle. If the side triangle is 42 inches long, about how tall is the center truss ̶̶ CD? Round to the nearest inch. 2. The floor of the Rockwell Rotunda in the McCombs Campus Center is in the shape of a regular octagon. What is the measure of each interior angle of the rotunda floor? 3. Each section of the stained-glass window is made of five polygonal shapes. Name each polygon by the number of its sides. Tell whether each polygon appears to be regular or irregular, and concave or convex. Identify which of the five polygons appear to be special quadrilaterals. 4. Square ABEG at the center of the marble fireplace is composed of three shapes— a smaller square DCFG and two congruent quadrilaterals ABCD and EBCF. Find the angle measures in ABCD, and explain why ABCD must be a trapezoid. 448 448 Chapter 6 Polygons
and Quadrilaterals �� 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 Titan When it opened in April 2001 at Six Flags Over Texas in Arlington, Titan became the tallest roller coaster in Texas and the third tallest in the world. Titan trains travel up to 85 mi/h and cover over 5312 feet of track. The first hill features a 255-foot drop at a 65° angle into a dark, 120-foot-long tunnel! Choose one or more strategies to solve each problem. 1. If a Titan train takes 3 minutes and 30 seconds to travel the entire track, what is the roller coaster’s average speed in miles per hour? 2. If a Titan train travels through the tunnel at its maximum speed, about how long does it take the train to pass through the tunnel? Round to the nearest hundredth of a second. 3. Titan has three trains, each of which holds 30 passengers. If the roller coaster can accommodate 1600 passengers per hour, about how many trains run each hour? The figure below shows the structure of the first hill of Titan. For 4–6, use the figure. 4. Titan reaches its maximum height at the top of the first hill. The ascent covers a horizontal distance AE of about 350 feet. What is the ̶̶ AB to the nearest foot? length of the ascent 5. The length of the descent ̶̶ CD is about 270 feet. What is FD to the nearest foot? 6. Event organizers plan to hang a banner across the first hill of Titan from X to Y, ̶̶ AB and Y is where X is the midpoint of ̶̶ CD. What is the width the midpoint of of the banner to the nearest foot? 449449 �� Similarity 7A Similarity Relationships 7-1 Ratio and Proportion Lab Explore the Golden Ratio 7-2 Ratios in Similar Polygons Lab Predict Triangle Similarity Relationships 7-3 Triangle Similarity: AA, SSS, and SAS 7B Applying Similarity Lab Investigate Angle Bisectors of a Triangle 7-4 Applying Properties of Similar Triangles 7-5 Using Proportional Relationships 7-6 Dilations and Similarity in the Coordinate Plane KEYWORD: MG7 ChProj The Lighthouse Rock is located in Palo
Duro Canyon or “the Grand Canyon of Texas.” 450 450 Chapter 7 Vocabulary Match each term on the left with a definition on the right. 1. side of a polygon A. two nonadjacent angles formed by two intersecting lines 2. denominator 3. numerator 4. vertex of a polygon 5. vertical angles B. the top number of a fraction, which tells how many parts of a whole are being considered C. a point that corresponds to one and only one number D. the intersection of two sides of a polygon E. one of the segments that form a polygon F. the bottom number of a fraction, which tells how many equal parts are in the whole Simplify Fractions Write each fraction in simplest form. 6. 16_ 20 7. 14_ 21 8. 33_ 121 9. 56_ 80 Ratios Use the table to write each ratio in simplest form. 10. jazz CDs to country CDs 11. hip-hop CDs to jazz CDs 12. rock CDs to total CDs 13. total CDs to country CDs Identify Polygons Ryan’s CD Collection Rock Jazz Hip-hop Country 36 18 34 24 Determine whether each figure is a polygon. If so, name it by the number of sides. 14. 15. 16. 17. Find Perimeter Find the perimeter of each figure. 18. rectangle PQRS 20. rhombus JKLM 19. regular hexagon ABCDEF 21. regular pentagon UVWXY Similarity 451 451 �� Key Vocabulary/Vocabulario dilation dilatación proportion proporción ratio scale razón escala scale drawing dibujo a escala Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. When an eye doctor dilates your eyes, the pupils become enlarged. What might it mean for one geometric figure to be a dilation of another figure? scale factor factor de escala 2. A blueprint is a scale drawing of a building. similar semejante similar polygons polígonos semejantes similarity ratio razón de semejanza What do you think is the definition of a scale drawing? 3. What does the word similar mean in everyday language? What do you think the term similar polygons means? 4
. Bike riders often talk about gear ratios. Give examples of situations where the word ratio is used. What do these examples have in common? Geometry TEKS G.1.B Geometric structure* recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.3.B Geometric structure* construct and justify statements about geometric figures and their properties G.5.B Geometric patterns* use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures... 7-2 Tech. Lab Les. 7-1 7-3 Tech. Lab Les. 7-2 7-4 Tech. Lab Les. 7-3 Les. 7-4 Les. 7-5 Les. 7-.9.B Congruence and the geometry of size* formulate and test ★ ★ ★ ★ ★ conjectures about the properties and attributes of polygons... based on explorations and concrete models G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. ★ ★ ★ ★ ★ ★ G.11.B Similarity and the geometry of shape* use ratios to solve ★ ★ ★ ★ ★ problems involving similar figures G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area... when one or more dimensions of a figure are changed and apply this idea in solving problems ★ * Knowledge and skills are written out completely on pages TX28–TX35. 452 452 Chapter 7 Reading Strategy: Read and Understand the Problem Many of the concepts you are learning are used in real-world situations. Throughout the text, there are examples and exercises that are real-world word problems. Listed below are strategies for solving word problems. Problem Solving Strategies • Read slowly and carefully. Determine what information is given and what you are asked to find. • If a diagram is provided, read the labels and make sure that you understand the information. If you do not, resketch and relabel the diagram so it makes sense to you. If a diagram is not provided, make a quick sketch and label it. • Use the given information to set up and solve the problem. • Decide whether your answer makes sense. From Lesson 6-1: Look at how the Polygon Exterior Angle The
orem is used in photography. Photography Application The aperture of the camera shown is formed by ten blades. The blades overlap to form a regular decagon. What is the measure of ∠CBD? � � � � Step Understand the Problem Procedure Result • List the important information. • The answer will be the measure of ∠CBD. ∠CBD is one of the exterior angles of the regular decagon formed by the apeture. Make a Plan • A diagram is provided, and it is labeled accurately. Solve • You can use the Polygon Exterior Angle Theorem. Then divide to find the measure of one of the exterior angles. m∠CBD = 360° _ = 36° 10 Look Back • The answer is reasonable since a decagon has 10 exterior angles. 10 (36°) = 360° Try This Use the problem-solving strategies for the following problem. 1. A painter’s scaffold is constructed so that the braces lie along the diagonals of rectangle PQRS. Given RS = 28 and QS = 85, find QT. Similarity 453 453 �� 7-1 Ratio and Proportion TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.5.B, G.7.B, G.7.C Objectives Write and simplify ratios. Use proportions to solve problems. Who uses this? Filmmakers use ratios and proportions when creating special effects. (See Example 5.) Vocabulary ratio proportion extremes means cross products The Lord of the Rings movies transport viewers to the fantasy world of Middle Earth. Many scenes feature vast fortresses, sprawling cities, and bottomless mines. To film these images, the moviemakers used ratios to help them build highly detailed miniature models. A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b, a : b, or a__, where b ≠ 0. b For example, the ratios 1 to 2, 1 : 2, and 1 __ 2 all represent the same comparison. E X A M P L E 1 Writing Ratios Write a ratio expressing the slope of ℓ. In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined. Slope = rise = run = 3 - (-1) _ 4 - (-2) = 2 _ = 4 _ 3 6 Substitute the given
values. Simplify. 1. Given that two points on m are C (-2, 3) and D (6, 5), write a ratio expressing the slope of m. A ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3 : 7 : 3 : 7. E X A M P L E 2 Using Ratios The ratio of the side lengths of a quadrilateral is 2 : 3 : 5 : 7, and its perimeter is 85 ft. What is the length of the longest side? Let the side lengths be 2x, 3x, 5x, and 7x. Then 2x + 3x + 5x + 7x = 85. After like terms are combined, 17x = 85. So x = 5. The length of the longest side is 7x = 7 (5) = 35 ft. 2. The ratio of the angle measures in a triangle is 1 : 6 : 13. What is the measure of each angle? 454 454 Chapter 7 Similarity �� A proportion is an equation stating that two ratios are equal. In the proportion a __ = c __, the values a and d are the extremes. The values b and c are d b the means. When the proportion is written as a : b = c : d, the extremes are in the first and last positions. The means are in the two middle positions. In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products. Cross Products Property = c _ In a proportion, if a _ d b and b and d ≠ 0, then ad = bc. The Cross Products Property can also be stated as, “In a proportion, the product of the extremes is equal to the product of the means.” E X A M P L E 3 Solving Proportions Cross Products Prop. Divide both sides by 45. 5 A Simplify. _ y = Solve each proportion. _ 45 63 5 (63) = y (45) 315 = 45y y = 7 = 24 x + 2) 2 = 6 (24) (x + 2) 2 = 144 x + 2 = ±12 x + 2 = 12 or x + 2 = -12 B Cross Products Prop. Simplify. Find the square root of both sides. Rewrite as two eqns. x = 10 or x = -14
Subtract 2 from both sides. Solve each proportion. = x _ 3a. 3 _ 8 56 3c. d _ = 6 _ 2 3 3b. 3d. 2y = 8 _ _ 4y The following table shows equivalent forms of the Cross Products Property. Properties of Proportions ALGEBRA _ _ c = d a The proportion b the following: is equivalent to NUMBERS = 2 _ 3 6 The proportion 1 _ the following: is equivalent to b a ad = bc 6) = 3 (2- 1 Ratio and Proportion 455 455 �� E X A M P L E 4 Using Properties of Proportions Given that 4x = 10y, find the ratio of x to y in simplest form. Since x comes before y in the sentence, x will be in the numerator of the fraction. 4x = 10y x _ y = 10 _ 4 x _ y = 5 _ 2 Divide both sides by 4y. Simplify. 4. Given that 16s = 20t, find the ratio t : s in simplest form. E X A M P L E 5 Problem-Solving Application During the filming of The Lord of the Rings, the special-effects team built a model of Sauron’s tower with a height of 8 m and a width of 6 m. If the width of the full-size tower is 996 m, what is its height? Understand the Problem The answer will be the height of the tower. Make a Plan Let x be the height of the tower. Write a proportion that compares the ratios of the height to the width. height of model tower ___ = width of model tower = x _ 8 _ 996 6 height of full-size tower ___ width of full-size tower Solve = x _ 8 _ 6 996 6x = 8 (996) 6x = 7968 x = 1328 Cross Products Prop. Simplify. Divide both sides by 6. The height of the full-size tower is 1328 m. Look Back Check the answer in the original problem. The ratio of the height to the width of the model is 8 : 6, or 4 : 3. The ratio of the height to the width of the tower is 1328 : 996. In simplest form, this ratio is also 4 : 3. So the ratios are equal, and the answer is correct. 5. What if...? Suppose the special-effects team made a different model with a height
of 9.2 m and a width of 6 m. What is the height of the actual tower? 456 456 Chapter 7 Similarity 1234 THINK AND DISCUSS 1. Is the ratio 6 : 7 the same ratio as 7 : 6? Why or why not? 2. Susan wants to know if the fractions 3 __ 7 and 12 __ 28 are equivalent. Explain how she can use the properties of proportions to find out. 3. GET ORGANIZED Copy and complete the graphic organizer. In the boxes, write the definition of a proportion, the properties of proportions, and examples and nonexamples of a proportion. 7-1 Exercises Exercises KEYWORD: MG7 7-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. = 2 _ 1. Name the means and extremes in the proportion. Write the cross products for the proportion Write a ratio expressing the slope of each line. p. 454 3. ℓ 4. m 5. The ratio of the side lengths of a quadrilateral p. 454 is 2 : 4 : 5 : 7, and its perimeter is 36 m. What is the length of the shortest side? 7. The ratio of the angle measures in a triangle is 5 : 12 : 19. What is the measure of the largest angle. 455 Solve each proportion. = 40 _ 8. x _ 2 16 y = 27 _ y _ 11. 3 9. 7 _ y = 21 _ 27 = x - 1 _ 12. 16 _ 4 x - 1 10. 6 _ 58 13. x 2 _ 18 = t _ 29 = 14. Given that 2a = 8b, find the ratio of a p. 456 to b in simplest form. 15. Given that 6x = 27y, find the ratio y : x in simplest form 16. Architecture The Arkansas State p. 456 Capitol Building is a smaller version of the U.S. Capitol Building. The U.S. Capitol is 752 ft long and 288 ft tall. The Arkansas State Capitol is 564 ft long. What is the height of the Arkansas State Capitol? 7- 1 Ratio and Proportion 457 457 �� PRACTICE AND PROBLEM SOLVING Independent Practice Write a ratio expressing the slope of each line. For See Exercises Example 17. ℓ
18. m 19. n 17–19 20–21 22–27 28–29 30 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 20. The ratio of the side lengths of an isosceles triangle is 4 : 4 : 7, and its perimeter is 52.5 cm. What is the length of the base of the triangle? 21. The ratio of the angle measures in a parallelogram is 2 : 3 : 2 : 3. What is the measure of each angle? Solve each proportion. = 9 _ y 22. 6 _ 8 2m + 2 _ 3 = 12 _ 2m + 2 25. 23. x _ 14 5y _ 16 26. = 50 _ 35 = 125 _ y 24. z _ 12 = 12 27. Travel 28. Given that 5y = 25x, find the ratio of x to y in simplest form. 29. Given that 35b = 21c, find the ratio b : c in simplest form. 30. Travel Madurodam is a park in the Netherlands that contains a complete Dutch city built entirely of miniature models. One of the models of a windmill is 1.2 m tall and 0.8 m wide. The width of the actual windmill is 20 m. What is its height? Given that a __ b the following equations. 32. b _ a = 34. Sports During the 2003 NFL season, the Dallas Cowboys won 10 of their 16 regular-season games. What is their ratio of wins to losses in simplest form?, complete each of 7 33. a _ 5 31. 7a = = 5 __ = For more than 50 years, Madurodam has been Holland’s smallest city. The canal houses, market, airplanes, and windmills are all replicated on a 1 : 25 scale. Source: madurodam.nl Write a ratio expressing the slope of the line through each pair of points. 35. (-6, -4) and (21, 5), -2) and (4, 5 1 _ 37. (6 1 _ ) 38. (-6, 1) and (-2, 0) 36. (16, -5) and (6, 1) 2 2 39. This problem will prepare you for the Multi-Step TAKS Prep on page 478. A claymation film is shot on a set that is a scale model of an actual city. On the set
, a skyscraper is 1.25 in. wide and 15 in. tall. The actual skyscraper is 800 ft tall. a. Write a proportion that you can use to find the width of the actual skyscraper. b. Solve the proportion from part a. What is the width of the actual skyscraper? 458 458 Chapter 7 Similarity �� 40. Critical Thinking The ratio of the lengths of a quadrilateral’s consecutive sides is 2 : 5 : 2 : 5. The ratio of the lengths of the quadrilateral’s diagonals is 1 : 1. What type of quadrilateral is this? Explain. 41. Multi-Step One square has sides 6 cm long. Another has sides 9 cm long. Find the ratio of the areas of the squares. 42. Photography A photo shop makes prints of photographs in a variety of sizes. Every print has a length-to-width ratio of 5 : 3.5 regardless of its size. A customer wants a print that is 20 in. long. What is the width of this print? 43. Write About It What is the difference between a ratio and a proportion? 44. An 18-inch stick breaks into three pieces. The ratio of the lengths of the pieces is 1 : 4 : 5. Which of these is NOT a length of one of the pieces? 1.8 inches 3.6 inches 7.2 inches 9 inches 45. Which of the following is equivalent to 3 __ 5 = x __ y? 3 _ y = 5 _ x 3x = 5y x _ 3 y _ = 5 3 (5) = xy 46. A recipe for salad dressing calls for oil and vinegar in a ratio of 5 parts oil to 2 parts 1 _ 2 vinegar. If you use 1 1 __ cups of oil, how many cups of vinegar will you need? 4 2 1 _ 2 5 _ 8 47. Short Response Explain how to solve the proportion 36 __ 72 must assume about x in order to solve the proportion. 6 1 _ 4 = 15 __ x for x. Tell what you CHALLENGE AND EXTEND 48. The ratio of the perimeter of rectangle ABCD to the perimeter of rectangle EFGH is 4 : 7. Find x. and a + b ____ = c __ 49. Explain why a __ b d b are equivalent proportions. = c + d ____ d 50. Probability The numbers 1, 2, 3, and 6 are randomly placed in these four boxes: ___
. What is the probability that the two ratios will form a proportion? ___? 51. Express the ratio x 2 + 9x + 18 _________ x 2 - 36 in simplest form. SPIRAL REVIEW Complete each ordered pair so that it is a solution to y - 6x = -3. (Previous course) 52. (0, 54. (-4, 53. (, 3) ) ) Find each angle measure. (Lesson 3-2) 55. m∠ABD 56. m∠CDB Each set of numbers represents the side lengths of a triangle. Classify each triangle as acute, right, or obtuse. (Lesson 5-7) 57. 5, 8, 9 58. 8, 15, 20 59. 7, 24, 25 7- 1 Ratio and Proportion 459 459 �� 7-2 Use with Lesson 7-2 Activity 1 Explore the Golden Ratio In about 300 B.C.E., Euclid showed in his book Elements how to calculate the golden ratio. It is claimed that this ratio was used in many works of art and architecture to produce rectangles of pleasing proportions. The golden ratio also appears in the natural world and it is said even in the human face. If the ratio of a rectangle’s length to its width is equal to the golden ratio, it is called a golden rectangle. TEKS G.5.B Geometric patterns: use numeric and geometric patterns to make generalizations about geometric properties... ratios in similar figures.... Also G.1.A, G.2.B, G.5.A KEYWORD: MG7 Lab7 1 Construct a segment and label its endpoints A ̶̶ AP is ̶̶ PB. What are AP, PB, and AB? What and B. Place P on the segment so that longer than is the ratio of AP to PB and the ratio of AB to AP? Drag P along the segment until the ratios are equal. What is the value of the equal ratios to the nearest hundredth? 2 Construct a golden rectangle beginning with ̶̶ AB. Then construct a circle ̶̶ AB. ̶̶ AB through a square. Create with its center at A and a radius of Construct a line perpendicular to A. Where the circle and the perpendicular line intersect, label the point D. Construct perpendicular lines through B and D and label their intersection C. Hide the lines and the circle, leaving only the
segments to complete the square. 3 Find the midpoint of ̶̶ AB and label it M. Create a segment from M to C. Construct a ̶̶̶ MC. circle with its center at M and radius of Construct a ray with endpoint A through B. Where the circle and the ray intersect, label the point E. Create a line through E that is AB. Show the previously perpendicular to hidden line through D and C. Label the point of intersection of these two lines F. Hide the lines and circle and create segments to complete golden rectangle AEFD. 4 Measure ̶̶ AE, ̶̶ EF,and ̶̶ BE. Find the ratio of AE to EF and the ratio of EF to BE. Compare these ratios to those found in Step 1. What do you notice? 460 460 Chapter 7 Similarity Try This 1. Adjust your construction from Step 2 so that the side of the original square is 2 units long. Use the Pythagorean Theorem to find the length of Calculate the length of decimal rounded to the nearest thousandth. ̶̶ AE. Write the ratio of AE to EF as a fraction and as a ̶̶̶ MC. 2. Find the length of ̶̶ BE in your construction from Step 3. Write the ratio of EF to BE as a fraction and as a decimal rounded to the nearest thousandth. Compare your results to those from Try This Problem 1. What do you notice? 3. Each number in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13 …) is created by adding the two preceding numbers together. That is, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on. Investigate the ratios of the numbers in the sequence by finding the ̶̶̶ 666, 8 __ 5 = 1.6, and so on. What do you notice quotients. 1 __ 1 = 1, 2 __ 1 = 2, 3 __ 2 = 1.5, 5 __ 3 = 1. as you continue to find the quotients? Tell why each of the following is an example of the appearance of the Fibonacci sequence in nature. 4. 5. Determine whether each picture is an example of an application of the golden rectangle. Measure the length and the width of each and decide whether the ratio of the length to the width is approximately the golden ratio. 6. 7. 7-
2 Technology Lab 461 461 �� 7-2 Ratios in Similar Polygons TEKS G.5.B Geometric patterns: use... geometric patterns to make generalizations about ratios in similar figures... Objectives Identify similar polygons. Apply properties of similar polygons to solve problems. Vocabulary similar similar polygons similarity ratio Why learn this? Similar polygons are used to build models of actual objects. (See Example 3.) Figures that are similar (∼) have the same shape but not necessarily the same size. △1 is similar to △2 (△1 ∼ △2). △1 is not similar to △3 (△1 ≁ △3). Similar Polygons DEFINITION DIAGRAM STATEMENTS Also G.11.A, G.11.B Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding sides are proportional. ABCD ∼ EFGH ∠A ≅ ∠E ∠B ≅ ∠F ∠C ≅ ∠G AB ___ = BC ___ FG EF ∠D ≅ ∠H = CD ___ = DA ___ GH = 1 __ 2 HE E X A M P L E 1 Describing Similar Polygons Identify the pairs of congruent angles and corresponding sides. ∠Z ≅ ∠R and ∠Y ≅ ∠Q. By the Third Angles Theorem, ∠X ≅ ∠S. XY _ SQ = 6 _, YZ _ = 2 _ 3 9 QR = 12 _ 18 = 2 _, 3 XZ _ SR = 9 _ 13.5 = 2 _ 3 1. Identify the pairs of congruent angles and corresponding sides. 462 462 Chapter 7 Similarity �� A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of △ABC to △DEF is 3 __ 6, or 1 __ 2. The similarity ratio of △DEF to △ABC is 6 __ 3, or 2. E X A M P L E 2 Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. A rectangles PQRS and TUVW Step 1 Identify pairs of con
gruent angles. ∠P ≅ ∠T, ∠Q ≅ ∠U, ∠R ≅ ∠V, and ∠S ≅ ∠W All  of a rect. are rt.  and are ≅. Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Step 2 Compare corresponding sides., PS _ = 3 _ 4 TW = 2 _ = 4 _ 3 6 = 12 _ 16 PQ _ TU Since corresponding sides are not proportional, the rectangles are not similar. B △ABC and △DEF Step 1 Identify pairs of congruent angles. ∠A ≅ ∠D, ∠B ≅ ∠E ∠C ≅ ∠F Given Third  Thm. Step 2 Compare corresponding sides., AC _, BC _ = 4 _ = 4 _ 3 3 DF EF = 4 _ 3 Thus the similarity ratio is 4 __ 3, and △ABC ∼ △DEF. = 24 _ 18 = 20 _ 15 = 16 _ 12 AB _ DE 2. Determine if △JLM ∼ △NPS. If so, write the similarity ratio and a similarity statement. Proportions with Similar Figures When I set up a proportion, I make sure each ratio compares the figures in the same order. To find x, I wrote 10 __ = 6 __ x. 4 This will work because the first ratio compares the lengths starting with rectangle ABCD. The second ratio compares the widths, also starting with rectangle ABCD. Anna Woods Westwood High School ABCD ∼ EFGH 7- 2 Ratios in Similar Polygons 463 463 �� E X A M P L E 3 Hobby Application A Railbox boxcar can be used to transport auto parts. If the length of the actual boxcar is 50 ft, find the width of the actual boxcar to the nearest tenth of a foot. Let x be the width of the actual boxcar in feet. The rectangular model of a boxcar is similar to the rectangular boxcar, so the corresponding lengths are proportional. �� When you work with proportions, be sure the ratios compare corresponding measures. �� = width of boxcar __ width of model length of boxcar __ length of model = x _ 50 _ 7 2
7x = (50) (2) 7x = 100 x ≈ 14.3 Cross Products Prop. Simplify. Divide both sides by 7. The width of the model is approximately 14.3 ft. 3. A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch. THINK AND DISCUSS 1. If you combine the symbol for similarity with the equal sign, what symbol is formed? 2. The similarity ratio of rectangle ABCD to rectangle EFGH is 1 __ 9. How do the side lengths of rectangle ABCD compare to the corresponding side lengths of rectangle EFGH? 3. What shape(s) are always similar? 4. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of similar polygons, and a similarity statement. Then draw examples and nonexamples of similar polygons. 464 464 Chapter 7 Similarity �� 7-2 Exercises Exercises KEYWORD: MG7 7-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of similar figures in your classroom Identify the pairs of congruent angles and corresponding sides. p. 462 2. 3. 463 Multi-Step Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 4. rectangles ABCD and EFGH 5. △RMP and △UWX. 464 6. Art The town of Goodland, Kansas, claims that it has one of the world’s largest easels. It holds an enlargement of a van Gogh painting that is 24 ft wide. The original painting is 58 cm wide and 73 cm tall. If the reproduction is similar to the original, what is the height of the reproduction to the nearest foot? Independent Practice For See Exercises Example 7–8 9–10 11 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING Identify the pairs of congruent angles and corresponding sides. 7. 8. Multi-Step Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 9. △
RSQ and △UXZ 10. rectangles ABCD and JKLM 7- 2 Ratios in Similar Polygons 465 465 �� 11. Hobbies The ratio of the model car’s dimensions to the actual car’s dimensions is 1 __ 56. The model has a length of 3 in. What is the length of the actual car? 12. Square ABCD has an area of 4 m 2. Square PQRS has an area of 36 m 2. What is the similarity ratio of square ABCD to square PQRS? What is the similarity ratio of square PQRS to square ABCD? Tell whether each statement is sometimes, always, or never true. 13. Two right triangles are similar. 14. Two squares are similar. 15. A parallelogram and a trapezoid are similar. 16. If two polygons are congruent, they are also similar. 17. If two polygons are similar, they are also congruent. Monument 18. Critical Thinking Explain why any two regular polygons having the same number of sides are similar. Find the value of x. 19. ABCD ∼ EFGH 20. △MNP ∼ △ XYZ The height of the Statue of Liberty from the foundation of the pedestal to the torch is 305 ft. Her index finger measures 8 ft, and the fingernail is 13 in. by 10 in. Source: libertystatepark.org 21. Estimation The Statue of Liberty’s hand is 16.4 ft long. Assume that your own body is similar to that of the Statue of Liberty and estimate the length of the Statue of Liberty’s nose. (Hint : Use a ruler to measure your own hand and nose. Then set up a proportion.) 22. Write the definition of similar polygons as two conditional statements. 23. JKLM ∼ NOPQ. If m∠K = 75°, name two 75° angles in NOPQ. 24. A dining room is 18 ft long and 14 ft wide. On a blueprint for the house, the dining room is 3.5 in. long. To the nearest tenth of an inch, what is the width of the dining room on the blueprint? 25. Write About
It Two similar polygons have a similarity ratio of 1 : 1. What can you say about the two polygons? Explain. 26. This problem will prepare you for the Multi-Step TAKS Prep on page 478. A stage set consists of a painted backdrop with some wooden flats in front of it. One of the flats shows a tree that has a similarity ratio of 1 __ 2 to an actual tree. To give an illusion of distance, the backdrop includes a small painted tree that has a similarity ratio of 1 __ 10 to the tree on the flat. a. The tree on the backdrop is 0.9 ft tall. What is the height of the tree on the flat? b. What is the height of the actual tree? c. Find the similarity ratio of the tree on the backdrop to the actual tree. 466 466 Chapter 7 Similarity �� 27. Which value of y makes the two rectangles similar? 3 8.2 25.2 28.8 28. △CGL ∼ △MPS. The similarity ratio of △CGL to △MPS is 5 __. What is the length of 2 8 ̶̶ PS? 50 12 75 29. Short Response Explain why 1.5, 2.5, 3.5 and 6, 10, 12 cannot be corresponding sides of similar triangles. CHALLENGE AND EXTEND 30. Architecture An architect is designing a building that is 200 ft long and 140 ft wide. She builds a model so that the similarity ratio of the model to the building is 1 ___ 500. What is the length and width of the model in inches? 31. Write a paragraph proof. Given: ̶̶ QR ǁ ̶̶ ST Prove: △PQR ∼ △PST 32. In the figure, D is the midpoint of ̶̶ AC. a. Find AC, DC, and DB. b. Use your results from part a to help you explain why △ABC ∼ △CDB. 33. A golden rectangle has the following property: If a square is cut from one end of the rectangle, the rectangle that remains is similar to the original rectangle. a. Rectangle ABCD is a golden rectangle. Write a similarity statement for rectangle ABCD and rectangle BCFE. b. Write a proportion using the corresponding sides of these rectangles. � � � � � � � � �� c. Solve the
proportion for ℓ. (Hint : Use the Quadratic Formula.) d. The value of ℓ is known as the golden ratio. Use a calculator to find ℓ to the nearest tenth. SPIRAL REVIEW 34. There are four runners in a 200-meter race. Assuming there are no ties, in how many different orders can the runners finish the race? (Previous course) ̶̶ ̶̶ QP, m∠QPT = 45°, and QR ≅ In kite PQRS, m∠RST = 20°. Find each angle measure. (Lesson 6-6) ̶̶ PS ≅ ̶̶ RS, 36. m∠PST 35. m∠QTR 37. m∠TPS _ x = Complete each of the following equations, given that 4 40. x _ y = 39. 10 _ y = 38. 10x = _ y 10. (Lesson 7-1) 7- 2 Ratios in Similar Polygons 467 467 �� 7-3 Predict Triangle Similarity Relationships In Chapter 4, you found shortcuts for determining that two triangles are congruent. Now you will use geometry software to find ways to determine that triangles are similar. Use with Lesson 7-3 Activity 1 TEKS G.11.A Similarity and the geometry of shape: use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. Also G.2.A, G.3.B, G.9.B KEYWORD: MG7 Lab7 1 Construct △ABC. Construct ̶̶ DE longer than ̶̶ DE around ̶̶ DE around E by any of the sides of △ABC. Rotate D by rotation ∠BAC. Rotate rotation ∠ABC. Label the intersection point of the two rotated segments as F. 2 Measure angles to confirm that ∠BAC ≅ ∠EDF and ∠ABC ≅ ∠DEF. Drag a vertex of △ABC ̶̶ or an endpoint of DE to show that the two triangles have two pairs of congruent angles. 3 Measure the side lengths of both triangles. Divide each side length of △ABC by the corresponding side length of △DEF. Compare the resulting ratios. What do you notice? Try This 1. What theorem guarantees that the third pair of angles in
the triangles are also congruent? 2. Will the ratios of corresponding sides found in Step 3 always be equal? Drag ̶̶ DE to investigate this question. State a a vertex of △ABC or an endpoint of conjecture based on your results. Activity 2 1 Construct a new △ABC. Create P in the interior of the triangle. Create △DEF by enlarging △ABC around P by a multiple of 2 using the Dilation command. Drag P outside of △ABC to separate the triangles. 468 468 Chapter 7 Similarity 2 Measure the side lengths of △DEF to confirm that each side is twice as long as the corresponding side of △ABC. Drag a vertex of △ABC to verify that this relationship is true. 3 Measure the angles of both triangles. What do you notice? Try This 3. Did the construction of the triangles with three pairs of sides in the same ratio guarantee that the corresponding angles would be congruent? State a conjecture based on these results. 4. Compare your conjecture to the SSS Congruence Theorem from Chapter 4. How are they similar and how are they different? Activity 3 1 Construct a different △ABC. Create P in the ̶̶ AB and ̶̶ AC interior of the triangle. Expand around P by a multiple of 2 using the Dilation command. Create an angle congruent to ∠BAC with sides that are each twice as long as ̶̶ AB and ̶̶ AC. 2 Use a segment to create the third side of a new triangle and label it △DEF. Drag P outside of △ABC to separate the triangles. 3 Measure each side length and determine the relationship between corresponding sides of △ABC and △DEF. 4 Measure the angles of both triangles. What do you notice? Try This 5. Tell whether △ABC is similar to △DEF. Explain your reasoning. 6. Write a conjecture based on the activity. What congruency theorem is related to your conjecture? 7- 3 Technology Lab 469 469 7-3 Triangle Similarity: AA, SSS, and SAS TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.5.B, G.11.A Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems. Who uses this? Engineers use similar triangles when designing buildings, such
as the Pyramid Building in San Diego, California. (See Example 5.) There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent. Postulate 7-3-1 Angle-Angle (AA) Similarity POSTULATE HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. △ABC ∼ △DEF E X A M P L E 1 Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. ̶̶ SR, ∠P ≅ ∠R, and ∠T ≅ ∠S by Since the Alternate Interior Angles Theorem. Therefore △PQT ∼ △RQS by AA ∼. ̶̶ PT ǁ 1. Explain why the triangles are similar and write a similarity statement. Theorem 7-3-2 Side-Side-Side (SSS) Similarity THEOREM HYPOTHESIS CONCLUSION If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. △ABC ∼ △DEF You will prove Theorem 7-3-2 in Exercise 38. 470 470 Chapter 7 Similarity �� Theorem 7-3-3 Side-Angle-Side (SAS) Similarity THEOREM HYPOTHESIS CONCLUSION 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. △ABC ∼ △DEF ∠B ≅ ∠E You will prove Theorem 7-3-3 in Exercise 39. E X A M P L E 2 Verifying Triangle Similarity Verify that the triangles are similar. A △PQR and △PRS QR = 2 _ = 4 _ _, 3 6 RS PQ = 6 _ = 2 _ _ 3 9 PR Therefore △PQR ∼ △PRS by SSS ∼., PR _ = 2 _ = 4 _ 3 6 PS B △JKL and △JMN ∠J ≅ ∠J by the Reflexive Property of �
�. JK = 3 _ = 1 _ _ 3 9 JM Therefore △JKL ∼ △JMN by SAS ∼. JL = 1 _ = 2 _ _, 3 6 JN 2. Verify that △TXU ∼ △VXW. E X A M P L E 3 Finding Lengths in Similar Triangles Explain why △ABC ∼ △DBE and then find BE. Step 1 Prove triangles are similar. ̶̶ AC ǁ ̶̶ ED, ∠A ≅ ∠D, and ∠C ≅ ∠E As shown by the Alternate Interior Angles Theorem. Therefore △ABC ∼ △DBE by AA ∼. Step 2 Find BE. = BC _ AB _ BE DB = 54 _ 36 _ 54 BE 36 (BE) = 54 2 36 (BE) = 2916 Corr. sides are proportional. Substitute 36 for AB, 54 for DB, and 54 for BC. Cross Products Prop. Simplify. BE = 81 Divide both sides by 36. 3. Explain why △RSV ∼ △RTU and then find RT. 7- 3 Triangle Similarity: AA, SSS, and SAS 471 471 �� E X A M P L E 4 Writing Proofs with Similar Triangles Given: A is the midpoint of D is the midpoint of ̶̶ BC. ̶̶ BE. Prove: △BDA ∼ △BEC Proof: Statements ̶̶ BC. ̶̶ BE. 1. A is the mdpt. of D is the mdpt. of ̶̶ ̶̶ DE BA ≅ ̶̶ BD ≅ ̶̶ AC, 2. 3. BA = AC, BD = DE Reasons 1. Given 2. Def. of mdpt. 3. Def. of ≅ seg. 4. BC = BA + AC, BE = BD + DE 4. Seg. Add. Post. 5. BC = BA + BA, BE = BD + BD 5. Subst. Prop. 6. BC = 2BA, BE = 2BD 7. BC _ = 2, BE _ BD BA 8. BC _ = BE _ BA BD 9. ∠B ≅ ∠B = 2 10. △BDA ∼ △BEC 6. Simplify. 7. Div. Prop. of
= 8. Trans. Prop. of = 9. Reflex. Prop. of ≅ 10. SAS ∼ Steps 8, 9 ̶̶ JK. 4. Given: M is the midpoint of ̶̶ KL, N is the midpoint of and P is the midpoint of ̶̶ JL. Prove: △JKL ∼ △NPM (Hint : Use the Triangle Midsegment Theorem and SSS ∼.) E X A M P L E 5 Engineering Application The photo shows a gable roof. △ABC ∼ △FBG and then find BF to the nearest tenth of a foot. ̶̶ AC ǁ ̶̶ FG. Use similar triangles to prove � � � �� Step 1 Prove the triangles are similar. ̶̶ ̶̶ FG AC ǁ ∠BFG ≅ ∠BAC ∠B ≅ ∠B Given Corr.  Thm. Reflex. Prop. of ≅ Therefore △ABC ∼ △FBG by AA ∼. 472 472 Chapter 7 Similarity �� Step 2 Find BF. = BF _ BA _ FG AC 17 _ = BF _ 6.5 24 17 (6.5) = 24 (BF) 110.5 = 24 (BF) 4.6 ft ≈ BF Corr. sides are proportional. Substitute the given values. Cross Products Prop. Simplify. Divide both sides by 24. 5. What if…? If AB = 4x, AC = 5x, and BF = 4, find FG. You learned in Chapter 2 that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles. Properties of Similarity Reflexive Property of Similarity △ABC ∼ △ABC (Reflex. Prop. of ∼) Symmetric Property of Similarity If △ABC ∼ △DEF, then △DEF ∼ △ABC. (Sym. Prop. of ∼) Transitive Property of Similarity If △ABC ∼ △DEF and △DEF ∼ △XYZ, then △ABC ∼ △XYZ. (Trans. Prop. of ∼) THINK AND DISCUSS 1. What additional information, if any, would you you need in order to show that △ABC ∼ △DEF by the AA
Similarity Postulate? 2. What additional information, if any, would you need in order to show that △ABC ∼ △DEF by the SAS Similarity Theorem? 3. Do corresponding sides of similar triangles need to be proportional and congruent? Explain. 4. GET ORGANIZED Copy and complete the graphic organizer. If possible, write a congruence or similarity theorem or postulate in each section of the table. Include a marked diagram for each. 7- 3 Triangle Similarity: AA, SSS, and SAS 473 473 �� 7-3 Exercises Exercises KEYWORD: MG7 7-3 KEYWORD: MG7 Parent GUIDED PRACTICE Explain why the triangles are similar and write a similarity statement. p. 470 1. 2 Verify that the triangles are similar. p. 471 3. △DEF and △JKL 4. △MNP and △MRQ Multi-Step Explain why the triangles are similar and then find each length. p. 471 5. AB 6. WY. Given: ⎯  MN ǁ ̶̶ KL p. 472 Prove: △JMN ∼ △JKL 8. Given: SQ = 2QP, TR = 2RP Prove: △PQR ∼ △PST 9. The coordinates of A, B, and C are A(0, 0), B(2, 6), and C(8, -2). What theorem or postulate justifies the statement △ABC ∼ △DEF, if the coordinates of D and E are twice the coordinates of B and C. 472 10. Surveying In order to measure the distance AB across the meteorite crater, a surveyor at S locates points A, B, C, and D as shown. What is AB to the nearest meter? nearest kilometer? 474 474 Chapter 7 Similarity ��ge07sec07l03003aaAB533 m733 m800 m586 m644 mCSADB PRACTICE AND PROBLEM SOLVING Explain why the triangles are similar and write a similarity statement. 11. 12. Verify that the given triangles are similar. 13. △KLM and △KNL 14. △U
VW and △XYZ Independent Practice For See Exercises Example 11–12 13–14 15–16 17–18 19 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 Multi-Step Explain why the triangles are similar and then find each length. 15. AB 16. PS 17. Given: CD = 3AC, CE = 3BC 18. Given: PR _ MR = QR _ NR Prove: △ABC ∼ △DEC Prove: ∠1 ≅ ∠2 19. Photography The picture shows a person taking a pinhole photograph of himself. Light entering the opening reflects his image on the wall, forming similar triangles. What is the height of the image to the nearest tenth of an inch? Draw △JKL and △MNP. Determine if you can conclude that △JKL ∼ △MNP based on the given information. If so, which postulate or theorem justifies your response? = KL _ = KL _ NP NP = KL _ NP 20. ∠K ≅ ∠N, 22. ∠J ≅ ∠M, JK _ MN JK _ MN JL _ MP JL _ MP 21. = Find the value of x. 23. 24. 7- 3 Triangle Similarity: AA, SSS, and SAS 475 475 ge07sec07l03004aAB5 ft 5 in.4 ft 6 in.15 in.�� 25. This problem will prepare you for the Multi-Step TAKS Prep on page 478. The set for an animated film includes three small triangles that represent pyramids. a. Which pyramids are similar? Why? b. What is the similarity ratio of the similar pyramids? �� 26. Critical Thinking △ABC is not similar to △DEF, and △DEF is not similar to △XYZ. Could △ABC be similar to △XYZ? Why or why not? Make a sketch to support your answer. 27. Recreation To play shuffleboard, two teams take turns sliding disks on a court. The dimensions of the scoring area for a standard shuffleboard court are shown. What are JK
and MN? 28. Prove the Transitive Property of Similarity. Given: △ABC ∼ △DEF, △DEF ∼ △XYZ Prove: △ABC ∼ △XYZ 29. 29. Draw and label △PQR and △STU such that PQ ___ ST = QR ___ TU Meteorology but △PQR is NOT similar to △STU. 30. 30. Given: △KNJ is isosceles with ∠N as the vertex angle. ∠H ≅ ∠L Prove: △GHJ ∼ △MLK This satellite image shows Hurricane Lili as it moves across the Gulf of Mexico. In October 2002, an estimated 500,000 people evacuated in advance of Lili’s hitting Texas. 31. Meteorology Satellite photography makes it possible to measure the diameter of a hurricane. The figure shows that a camera’s aperture YX is 35 mm and its focal length WZ is 50 mm. The satellite W holding the camera is 150 mi above the hurricane, centered at C. a. Why is △XYZ ∼ △ABZ? What assumption must you make about the position of the camera in order to make this conclusion? b. What other triangles in the figure must be similar? Why? c. Find the diameter AB of the hurricane. 32. /////ERROR ANALYSIS///// Which solution for the value of y is incorrect? Explain the error. 33. Write About It Two isosceles triangles have congruent vertex angles. Explain why the two triangles must be similar. 476 476 Chapter 7 Similarity �� 34. What is the length of ̶̶ TU? 36 40 48 90 35. Which dimensions guarantee that △BCD ∼ △FGH? FG = 11.6, GH = 8.4 FG = 12, GH = 14 FG = 11.4, GH = 11.4 FG = 10.5, GH = 14.5 36. ABCD ∼ EFGH.
Which similarity postulate or theorem lets you conclude that △BCD ∼ △FGH? AA SSS SAS None of these 37. Gridded Response If 6, 8, and 12 and 15, 20, and x are the lengths of the corresponding sides of two similar triangles, what is the value of x? CHALLENGE AND EXTEND 38. Prove the SSS Similarity Theorem. = BC _ EF = AC _ DF Prove: △ABC ∼ △DEF Given: AB _ DE (Hint : Assume that AB < DE and choose point X on ̶̶ DF so that ⎯  XY ǁ ̶̶ ̶̶ DX. DE so that ̶̶ EF. Show that △DXY ∼ △DEF ̶̶ AB ≅ Then choose point Y on and that △ABC ≅ △DXY.) 39. Prove the SAS Similarity Theorem. Given: ∠B ≅ ∠E, AB _ DE = BC _ EF Prove: △ABC ∼ △DEF (Hint : Assume that AB < DE and choose point X on ̶̶ EF so that ∠EXY ≅ ∠EDF. Show that △XEY ∼ △DEF ̶̶ DE so that ̶̶ EX ≅ ̶̶ BA. Then choose point Y on and that △ABC ≅ △XEF.) 40. Given △ABC ∼ △XYZ, m∠A = 50°, m∠X = (2x + 5y) °, m∠Z = (5x + y) °, and that m∠B = (102 - x) °, find m∠Z. SPIRAL REVIEW 41. Jessika’s scores in her last six rounds of golf were 96, 99, 105, 105, 94, and 107. What score must Jessika make on her next round to make her mean score 100? (Previous course) Position each figure in the coordinate plane and give possible coordinates of each vertex. (Lesson 4-7) 42. a right triangle with leg lengths of 4 units and 2 units 43. a rectangle with length 2k and width k Solve each proportion. Check your answer. (Lesson 7-1) 44. 2x _ 10 = 25 _ 10y = 35 _ 25 5y _ 450
45. 46. b - 5 _ 28 = 7 _ b - 5 7- 3 Triangle Similarity: AA, SSS, and SAS 477 477 �� SECTION 7A Similarity Relationships Lights! Camera! Action! Lorenzo, Maria, Sam, and Tia are working on a video project for their history class. They decide to film a scene where the characters in the scene are on a train arriving at a town. Since Lorenzo collects model trains, they decide to use one of his trains and to build a set behind it. To create the set, they use a film technique called forced perspective. They want to use small objects to create an illusion of great distance in a very small space. 1. Lorenzo’s model train is 1 __ 87 the size of the original train. He measures the engine of the model train and finds that it is 2 1 __ 2 in. tall. What is the height of the real engine to the nearest foot? 2. The closest building to the train needs to be made using the same scale as the train. Maria and Sam estimate that the height of an actual station is 20 ft. How tall would they need to build their model of the train station to the nearest 1 __ 4 in.? 3. To give depth to their scene, they want to construct partial buildings behind the train station. Lorenzo decided to build a restaurant. If the height of the restaurant is actually 24 ft, how tall would they need to build their model of the restaurant to the nearest inch? 4. The other buildings on the set will have triangular roofs. Which of the roofs are similar to each other? Why? 478 478 Chapter 7 Similarity �� SECTION 7A Quiz for Lessons 7-1 Through 7-3 7-1 Ratio and Proportion Write a ratio expressing the slope of each line. 1. ℓ 3. n 2. m 4. x-axis Solve each proportion. y = 12 _ _ 5. 9 6 7. 16 _ 24 8. 2 _ 3y = 20 _ t y _ 24 = 9. An architect’s model for a building is 1.4 m long and 0.8 m wide. The actual building is 240 m wide. What is the length of the building? 7-2 Ratios in Similar Polygons Determine whether the
two polygons are similar. If so, write the similarity ratio and a similarity statement. 10. rectangles ABCD and WXYZ 11. △JMR and △KNP 12. Leonardo da Vinci’s famous portrait the Mona Lisa is 30 in. long and 21 in. wide. Janelle has a refrigerator magnet of the painting that is 3.5 cm wide. What is the length of the magnet? 7-3 Triangle Similarity: AA, SSS, and SAS 13. Given: ABCD Prove: △EDG ∼ △FBG 14. Given: MQ = 1 __ 3 MN, MR = 1 __ 3 MP Prove: △MQR ∼ △MNP 15. A geologist wants to measure the length XY of a rock formation. To do so, she locates points U, V, X, Y, and Z as shown. What is XY? Ready to Go On? 479 479 �� 7-4 Investigate Angle Bisectors of a Triangle In a triangle, an angle bisector divides the opposite side into two segments. You will use geometry software to explore the relationships between these segments. Use with Lesson 7-4 Activity 1 TEKS G.5.B Geometric patterns: use... geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures... Also G.2.A, G.3.B, G.9.B KEYWORD: MG7 Lab7 1 Construct △ABC. Bisect ∠BAC and create the point of intersection of the angle bisector and ̶̶ BC. Label the intersection D. 2 Measure ̶̶ AB, ̶̶ AC, ̶̶ BD, and measurements to write ratios. What are the results? Drag a vertex of △ABC and examine the ratios again. What do you notice? ̶̶ CD. Use these Try This 1. Choose Tabulate and create a table using the four lengths and the ratios from Step 2. Drag a vertex of △ABC and add the new measurements to the table. What conjecture can you make about the segments created by an angle bisector? 2. Write a proportion based on your conjecture. Activity 2 1 Construct △DEF. Create the incenter of the triangle and label it I. Hide the angle
bisectors of ̶̶ ∠E and ∠F. Find the point of intersection of EF and the bisector of ∠D. Label the intersection G. 2 Find DI, DG, and the perimeter of △DEF. ̶̶ 3 Divide the length of DI by the length of DG. ̶̶ ̶̶ Add the lengths of DF. Then divide DE and this sum by the perimeter of △DEF. Compare the two quotients. Drag a vertex of △DEF and examine the quotients again. What do you notice? 4 Write a proportion based on your quotients. What conjecture can you make about this relationship? Try This 3. Show the hidden angle bisector of ∠E or ∠F. Confirm that your conjecture is true for this bisector. Drag a vertex of △DEF and observe the results. 4. Choose Tabulate and create a table with the measurements you used in your proportion in Step 4. 480 480 Chapter 7 Similarity 7-4 Applying Properties of Similar Triangles TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.2.A, G.3.B, G.5.B, G.9.B, G.11.A Objectives Use properties of similar triangles to find segment lengths. Apply proportionality and triangle angle bisector theorems. Who uses this? Artists use similarity and proportionality to give paintings an illusion of depth. (See Example 3.) Artists use mathematical techniques to make two-dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller and closer objects look larger. Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings. Theorem 7-4-1 Triangle Proportionality Theorem THEOREM HYPOTHESIS CONCLUSION If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. AE _ EB = AF _ FC ̶̶ EF ǁ ̶̶ BC You can use a compass-and-straightedge construction to verify this theorem. Although the construction is not a proof, it should help convince you that the theorem is true. After you have completed the construction, use a ruler to measure ̶̶ AF, and ̶̶ EB, ̶̶ AE, ̶̶ FC to see that AE ___

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