6-1 The Polygon ngle-sum Theorems Vocabulary Review 1. Underline the correct word to complete the sentence. In a convex polygon, no point on the lines containing the sides of the polygon is in the interior / exterior of the polygon.. Cross out the polygon that is NOT convex. Vocabulary Builder regular polygon (noun) REG yuh lur PHL ih gahn efinition: regular polygon is a polygon that is both equilateral and equiangular. Example: n equilateral triangle is a regular polygon with three congruent sides and three congruent angles. Use Your Vocabulary Underline the correct word(s) to complete each sentence. 3. The sides of a regular polygon are congruent / scalene. 4. right triangle is / is not a regular polygon. 5. n isosceles triangle is / is not always a regular polygon. Write equiangular, equilateral, or regular to identify each hexagon. Use each word once. 6. 7. 10 10 10 10 10 10 8. 10 10 10 10 10 10 equilateral regular equiangular Chapter 6 146
Theorem 6-1 Polygon ngle-sum Theorem and Corollary Theorem 6-1 The sum of the measures of the interior angles of an n-gon is (n )180. (n )180 Corollary The measure of each interior angle of a regular n-gon is n. 9. When n 5 1, the polygon is a(n) 9. 10. When n 5, the polygon is a(n) 9. triangle quadrilateral Problem 1 Finding a Polygon ngle Sum Got It? What is the sum of the interior angle measures of a 17-gon? 11. Use the justifications below to find the sum. 1. raw diagonals from vertex to check your answer. sum 5 Q n R180 Polygon ngle-sum Theorem 5 Q 17 R180 Substitute for n. 5 15? 180 Subtract. 5 700 Simplify. 13. The sum of the interior angle measures of a 17-gon is 700. Problem Using the Polygon ngle-sum Theorem Got It? What is the measure of each interior angle in a regular nonagon? Underline the correct word or number to complete each sentence. 14. The interior angles in a regular polygon are congruent / different. 15. regular nonagon has 7 / 8 / 9 congruent sides. 16. Use the Corollary to the Polygon ngle-sum Theorem to find the measure of each interior angle in a regular nonagon. Measure of an angle 5 5 5 Q Q 9 7 140 9 R180 R180 17. The measure of each interior angle in a regular nonagon is 140. 9 147 Lesson 6-1
Problem 3 Using the Polygon ngle-sum Theorem Got It? What is mlg in quadrilateral EFGH? 18. Use the Polygon ngle-sum Theorem to find m/g for n 5 4. m/e 1 m/f 1 m/g 1 m/h 5 (n )180 m/e 1 m/f 1 m/g 1 m/h 5 Q 4 R180 F E G 10 85 53 H 85 1 10 1 m/g 1 53 5? 180 m/g 1 58 5 m/g 5 19. m/g in quadrilateral EFGH is 10. 360 10 Theorem 6- Polygon Exterior ngle-sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. 0. In the pentagon below, m/1 1 m/ 1 m/3 1 m/4 1 m/5 5 360. 3 4 5 1 Use the Polygon Exterior ngle-sum Theorem to find each measure. 1. 10. 81 56 90 75 87 7 66 73 10 1 81 1 7 1 87 5 360 90 1 56 1 75 1 73 1 66 5 360 Problem 4 Finding an Exterior ngle Measure Got It? What is the measure of an exterior angle of a regular nonagon? Underline the correct number or word to complete each sentence. 3. Since the nonagon is regular, its interior angles are congruent / right. 4. The exterior angles are complements / supplements of the interior angles. 5. Since the nonagon is regular, its exterior angles are congruent / right. 6. The sum of the measures of the exterior angles of a polygon is 180 / 360. 7. regular nonagon has 7 / 9 / 1 sides. Chapter 6 148
8. What is the measure of an exterior angle of a regular nonagon? Explain. 40. Explanations may vary. Sample: ivide 360 by 9, the number of sides in a nonagon. Lesson Check o you UNERSTN? Error nalysis Your friend says that she measured an interior angle of a regular polygon as 130. Explain why this result is impossible. 9. Use indirect reasoning to find a contradiction. ssume temporarily that a regular n-gon has a 1308 interior angle. angle sum 5 130? n regular n-gon has n congruent angles. angle sum 5 Q n R180 Polygon ngle-sum Theorem 130n 130n 50n 5 Q n R180 Use the Transitive Property of Equality. 5 180n 360 Use the istributive Property. 5 360 Subtract 180n from each side. n 5 7. ivide each side by 50. n 7. The number of sides in a polygon is a whole number $ 3. 30. Explain why your friend s result is impossible. nswers may vary. Sample: If each interior angle measures 130, then the regular polygon would have 7. sides. This is impossible because the number of sides is an integer greater than. So, each angle cannot be 130. Math Success Check off the vocabulary words that you understand. equilateral polygon equiangular polygon regular polygon Rate how well you can find angle measures of polygons. Need to review 0 4 6 8 10 Now I get it! 149 Lesson 6-1
6- Properties of Parallelograms Vocabulary Review 1. Supplementary angles are two angles whose measures sum to 180.. Suppose /X and /Y are supplementary. If m/x 5 75, then m/y 5 105. Underline the correct word to complete each sentence. 3. linear pair is complementary / supplementary. 4. /FB and /EF at the right are complementary / supplementary. B C 60 F 10 E Vocabulary Builder consecutive (adjective) kun SEK yoo tiv efinition: Consecutive items follow one after another in uninterrupted order. Math Usage: Consecutive angles of a polygon share a common side. Examples: The numbers 3,, 1, 0, 1,, 3,... are consecutive integers. Non-Example: The letters, B, C, F, P,... are NOT consecutive letters of the alphabet. Use Your Vocabulary Use the diagram at the right. raw a line from each angle in Column to a consecutive angle in Column B. Column Column B 5. / /F 6. /C /E 7. / / Write the next two consecutive months in each sequence. 8. January, February, March, pril, May, June 9. ecember, November, October, September, ugust, July F E B C Chapter 6 150
Theorems 6-3, 6-4, 6-5, 6-6 Theorem 6-3 If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem 6-4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem 6-5 If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem 6-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other. Use the diagram at the right for Exercises 10 1. 10. Mark parallelogram BC to model Theorem 6-3 and Theorem 6-5. E 11. E > CE 1. BE > E B C Problem 1 Using Consecutive ngles Got It? Suppose you adjust the lamp so that mls is 86. What is mlr in ~PQRS? Underline the correct word or number to complete each statement. 13. /R and /S are adjacent / consecutive angles, so they are supplementary. 14. m/r 1 m/s 5 90 / 180 15. Now find m/r. mlr 1 86 5 180 mlr 5 180 86 mlr 5 94 16. m/r 5 94. Problem Using Properties of Parallelograms in a Proof Got It? Use the diagram at the right. Given: ~BC, K > MK 17. Circle the classification of nkm. equilateral isosceles right 18. Complete the proof. The reasons are given. Statements Reasons 1) K > MK 1) Given Prove: /BC > /CM ) /B > lcm ) ngles opposite congruent sides of a triangle are congruent. 3) /BC > lb 3) Opposite angles of a parallelogram are congruent. B K P Q C 64 M S R 4) /BC > lcm 4) Transitive Property of Congruence 151 Lesson 6-
Problem 3 Using lgebra to Find Lengths Got It? Find the values of x and y in ~PQRS at the right. What are PR and SQ? P Q 19. Circle the reason PT > TR and ST > TQ. iagonals of a Opposite sides of PR is the parallelogram a parallelogram perpendicular bisect each other. are congruent. bisector of QS. S 3y 7 x 1 T y x R 0. Cross out the equation that is NOT true. 3(x 1 1) 7 5 x y 5 x 1 1 3y 7 5 x 1 1 3y 7 5 x 1. Find the value of x.. Find the value of y. 3(x 1 1) 7 5 x 3x 1 3 7 5 x 3x 4 5 x x 5 4 y 5 x 1 1 y 5 4 1 1 y 5 5 3. Find PT. 4. Find ST. PT 5 3 5 7 ST 5 4 1 1 PT 5 15 7 ST 5 5 PT 5 8 5. Find PR. 6. Find SQ. PR 5 ( 8 ) SQ 5 ( 5 ) PR 5 16 SQ 5 10 7. Explain why you do not need to find TR and TQ after finding PT and ST. nswers may vary. Sample: The diagonals of a parallelogram bisect each other, so PT 5 TR and ST 5 TQ. Theorem 6-7 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Use the diagram at the right for Exercises 8 and 9. 8. If * B ) 6 * C ) 6 * EF ) and C > CE, then B >. 9. Mark the diagram to show your answer to Exercise 8. F C E B F Chapter 6 15
Problem 4 Using Parallel lines and Transversals Got It? In the figure at the right, * E ) n * BF ) n * CG ) n * H ). If EF 5 FG 5 GH 5 6 and 5 15, what is C? 30. You know that the parallel lines cut off congruent segments on transversal. 31. By Theorem 6-7, the parallel lines also cut off congruent segments on. 3. 5 B 1 BC 1 C by the Segment ddition Postulate. 1 33. B 5 BC 5 C, so 5 3? C. Then C 5 3?. 1 34. You know that 5 15, so C 5 3? 15 5 5. * ) * EH ) E B C F G H Lesson Check o you UNERSTN? Error nalysis Your classmate says that QV 5 10. Explain why the statement may not be correct. 35. Place a in the box if you are given the information. Place an if you are not given the information. three lines cut by two transversals P R T Q S V 5 cm three parallel lines cut by two transversals congruent segments on one transversal 36. What needs to be true for QV to equal 10? The lines cut by transversals need to be parallel. 37. Explain why your classmate s statement may not be correct. nswers may vary. Sample: The lines may not be parallel. Math Success Check off the vocabulary words that you understand. parallelogram opposite sides opposite angles consecutive angles Rate how well you understand parallelograms. Need to review 0 4 6 8 10 Now I get it! 153 Lesson 6-
6-3 Proving That a Quadrilateral Is a Parallelogram Vocabulary Review 1. oes a pentagon have opposite sides? Yes / No. oes an n-gon have opposite sides if n is an odd number? Yes / No raw a line from each side in Column to the opposite side in Column B. Column Column B 3. B BC 4. C C B Vocabulary Builder parallelogram (noun) pa ruh LEL uh gram efinition: parallelogram is a quadrilateral with two pairs of opposite sides parallel. Opposite sides may include arrows to show the sides are parallel. Related Words: square, rectangle, rhombus Use Your Vocabulary Write P if the statement describes a parallelogram or NP if it does not. NP 5. octagon NP 6. five congruent sides P 7. regular quadrilateral Write P if the figure appears to be a parallelogram or NP if it does not. P 8. NP 9. P 10. P parallelogram Q S R Chapter 6 154
Theorems 6-8 through 6-1 Theorem 6-8 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-9 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem 6-10 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6-11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-1 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Use the diagram at the right and Theorems 6-8 through 6 1 for Exercises 11 16. B 11. If B > C, and BC >, then BC is a ~. 1. If m/ 1 m/b 5 180 and m/ 1 m/ 5 180, then BC is a ~. C 13. If / > / C and / B > /, then BC is a ~. 14. If E > CE and BE > E, then BC is a ~. 15. If BC > and BC 6, then BC is a ~. 16. If C > B and C 6 B, then BC is a ~. Problem 1 Finding Values for Parallelograms Got It? Use the diagram at the right. For what values of x and y must EFGH be a parallelogram? 17. Circle the equation you can use to find the value of y. Underline the equation you can use to find the value of x. y 1 10 5 3y y 1 10 5 4x 1 13 ( y 1 10) 1 (3y ) 5 180 18. Find y. 19. Find x. (y 1 10) 1 (3y ) 5 180 4y 1 8 5 180 4y 5 17 y 5 43 0. What equation could you use to find the value of x first? y 1 10 5 4x 1 13 43 1 10 5 4x 1 13 40 5 4x 10 5 x (4x 1 13) 1 (1x 1 7) 5 180 H E (3y ) (4x 13) (y 10) (1x 7) G F 1. EFGH must be a parallelogram for x 5 10 and y 5 43. 155 Lesson 6-3
Problem eciding Whether a Quadrilateral Is a Parallelogram Got It? Can you prove that the quadrilateral is a parallelogram based on the given information? Explain. E Given: EF > G, E 6 FG Prove: EFG is a parallelogram.. Circle the angles that are consecutive with /G. / /E /F 3. Underline the correct word to complete the sentence. Same-side interior angles formed by parallel lines cut by a transversal are complementary / congruent / supplementary. G F 4. Circle the interior angles on the same side of transversal G. Underline the interior angles on the same side of transversal EF. / /E /F /G 5. Can you prove EFG is a parallelogram? Explain. No. Explanations may vary. Sample: n angle of EFG is not supplementary to both of its consectuive angles, so Theorem 6 9 does not apply. Problem 3 Identifying Parallelograms Got It? Reasoning truck sits on the platform of a vehicle lift. Two moving arms raise the platform. What is the maximum height that the vehicle lift can elevate the truck? Explain. Q 6 ft P 6 ft 6 ft R 6 ft 6. o the lengths of the opposite sides change as the truck is lifted? Yes / No 7. The least and greatest possible angle measures for /P and /Q are 0 and 90. 8. The greatest possible height is when m/p and m/q are 90. 9. What is the maximum height that the vehicle lift can elevate the truck? Explain. 6 ft. Explanations may vary. Sample: The maximum height occurs S Q 6 ft P 6 ft 6 ft R 6 ft S when the angles are 90 and PQRS is a rectangle. Chapter 6 156
Lesson Check o you UNERSTN? Compare and Contrast How is Theorem 6-11 in this lesson different from Theorem 6-6 in the previous lesson? In what situations should you use each theorem? Explain. For each theorem, circle the hypothesis and underline the conclusion. 30. Theorem 6-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other. 31. Theorem 6-11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. raw a line from each statement in Column to the corresponding diagram in Column B. Column 3. quadrilateral is a parallelogram. Column B 33. The diagonals of a quadrilateral bisect each other. 34. Circle the word that describes how Theorem 6-6 and Theorem 6-11 are related. contrapositive converse inverse 35. In which situations should you use each theorem? Explain. nswers may vary. Sample: Use Theorem 6-6 to use properties of parallelograms. Use Theorem 6-11 to prove a quadrilateral is a parallelogram. Math Success Check off the vocabulary words that you understand. diagonal parallelogram quadrilateral Rate how well you can prove that a quadrilateral is a parallelogram. Need to review 0 4 6 8 10 Now I get it! 157 Lesson 6-3
6-4 Properties of Rhombuses, Rectangles, and Squares Vocabulary Review 1. Circle the segments that are diagonals. G C H GC BF E EG EF H G B F C E. Is a diagonal ever a line or a ray? Yes / No 3. The diagonals of quadrilateral JKLM are JL and KM. Vocabulary Builder rhombus (noun) RHM bus rhombus efinition: rhombus is a parallelogram with four congruent sides. Main Idea: rhombus has four congruent sides but not necessarily four right angles. Examples: diamond, square Use Your Vocabulary Complete each statement with always, sometimes, or never. 4. rhombus is 9 a parallelogram. 5. parallelogram is 9 a rhombus. 6. rectangle is 9 a rhombus. 7. square is 9 a rhombus. 8. rhombus is 9 a square. 9. rhombus is 9 a hexagon. always sometimes sometimes always sometimes never Chapter 6 158
Key Concept Special Parallelograms rhombus is a parallelogram rectangle is a parallelogram square is a parallelogram with four congruent sides. with four right angles. with four congruent sides and four right angles. 10. Write the words rectangles, rhombuses, and squares in the Venn diagram below to show that one special parallelogram has the properties of the other two. Special Parallelograms rhombuses squares rectangles Problem 1 Classifying Special Parallograms Got It? Is ~EFGH a rhombus, a rectangle, or a square? Explain. 11. Circle the number of sides marked congruent in the diagram. E 1 3 4 H F 1. re any of the angles right angles? Yes / No 13. Is ~EFGH a rhombus, a rectangle, or a square? Explain. Rhombus. Explanations may vary. Sample: ~EFGH has four congruent sides and no right angles. Theorems 6-13 and 6-14 Theorem 6-13 If a parallelogram is a rhombus, then its diagonals are perpendicular. Theorem 6-14 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Use the diagram at the right for Exercises 14 18. 14. If BC is a rhombus, then C ' B. 15. If BC is a rhombus, then C bisects / B and / BC. 16. If BC is a rhombus, then /1 > / > / 5 > / 6. 17. If BC is a rhombus, then B bisects / C and / BC. B 8 7 G 1 3 4 5 6 C 18. If BC is a rhombus, then /3 > / 4 > / 7 > / 8. 159 Lesson 6-4
Problem Finding ngle Measures Got It? What are the measures of the numbered angles in rhombus PQRS? 19. Circle the word that describes npqr and nrsp. equilateral isosceles right 0. Circle the congruent angles in npqr. Underline the congruent angles in nrsp. /1 / /3 /4 /Q /S P Q 104 1 3 4 S R 1. m/1 1 m/ 1 104 5 180. m/1 1 m/ 5 76 3. m/1 5 38 4. Each diagonal of a rhombus 9 a pair of opposite angles. bisects 5. Circle the angles in rhombus PQRS that are congruent. /1 / /3 /4 6. m/1 5 38, m/ 5 38, m/3 5 38, and m/4 5 38. Theorem 6-15 Theorem 6-15 If a parallelogram is a rectangle, then its diagonals are congruent. 7. If RSTU is a rectangle, then RT > SU. Problem 3 Finding iagonal Length Got It? If LN 5 4x 17 and MO 5 x 1 13, what are the lengths of the diagonals of rectangle LMNO? Underline the correct word to complete each sentence. 8. LMNO is a rectangle / rhombus. 9. The diagonals of this figure are congruent / parallel. 30. Complete. LN 5 MO, so 4x 17 5 x 1 13. 31. Write and solve an equation to find the 3. Use the value of x to find the length value of x. of LN. 4x 17 5 x 1 13 x 17 5 13 x 5 30 x 5 15 4x 17 5 4(15) 17 5 60 17 5 43 33. The diagonals of a rectangle are congruent, so the length of each diagonal is 43. M L P N O Chapter 6 160
Lesson Check o you UNERSTN? Error nalysis Your class needs to find the value of x for which ~EFG is a rectangle. classmate s work is shown below. What is the error? Explain. (9x 6) G x + 8 = 9x - 6 14 = 7x = x E (x 8) F Write T for true or F for false. F T 34. If a parallelogram is a rectangle, then each diagonal bisects a pair of opposite angles. 35. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. 36. If EFG is a rectangle, m/ 5 m/ E 5 m/ F 5 m/ G. 37. m/f 5 90. 38. What is the error? Explain. nswers may vary. Sample: The diagonals of a rhombus bisect a pair of opposite angles, but the diagonals of a rectangle do not. The expressions should be added and set equal to 90. 39. Find the value of x for which ~EFG is a rectangle. 40. The value of x for which ~EFG is a rectangle is 8. Math Success Check off the vocabulary words that you understand. parallelogram rhombus rectangle square diagonal Rate how well you can find angles and diagonals of special parallelograms. Need to review x 1 8 1 9x 6 5 90 11x 1 5 90 11x 5 88 x 5 8 0 4 6 8 10 Now I get it! 161 Lesson 6-4
6-5 Conditions for Rhombuses, Rectangles, and Squares Vocabulary Review 1. quadrilateral is a polygon with 4 sides.. Cross out the figure that is NOT a quadrilateral. Vocabulary Builder diagonal (noun) dy G uh nul diagonals efinition: diagonal is a segment with endpoints at two nonadjacent vertices of a polygon. Word Origin: The word diagonal comes from the Greek prefix dia-, which means through, and gonia, which means angle or corner. Use Your Vocabulary 3. Circle the polygon that has no diagonal. triangle quadrilateral pentagon hexagon 4. Circle the polygon that has two diagonals. triangle quadrilateral pentagon hexagon 5. raw the diagonals from one vertex in each figure. 6. Write the number of diagonals you drew in each of the figures above. pentagon: hexagon: 3 heptagon: 4 Chapter 6 16
Theorems 6-16, 6-17, and 6-18 Theorem 6-16 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6-17 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. 7. Insert a right angle symbol in the parallelogram at the right to illustrate Theorem 6-16. Insert congruent marks to illustrate Theorem 6-17. Use the diagram from Exercise 7 to complete Exercises 8 and 9. 8. If BC is a parallelogram and C ' B, then BC is a rhombus. B 1 3 4 C 9. If BC is a parallelogram, /1 > l, and /3 > l4, then BC is a rhombus. Theorem 6-18 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 10. Insert congruent marks and right angle symbols in the parallelogram to the right to illustrate Theorem 6-18. 11. Use the diagram from Exercise 10 to complete the statement. B C If BC is a parallelogram, and B > is a rectangle. C then BC 1. Circle the parallelogram that has diagonals that are both perpendicular and congruent. parallelogram rectangle rhombus square Problem 1 Identifying Special Parallelograms Got It? parallelogram has angle measures of 0, 160, 0, and 160. Can you conclude that it is a rhombus, a rectangle, or a square? Explain. 13. raw a parallelogram in the box below. Label the angles with their measures. Use a protractor to help you make accurate angle measurements. Parallelograms may vary. Sample is given. 0 160 160 0 163 Lesson 6-5
Underline the correct word or words to complete each sentence. 14. You do / do not know the lengths of the sides of the parallelogram. 15. You do / do not know the lengths of the diagonals. 16. The angles of a rectangle are all acute / obtuse / right angles. 17. The angles of a square are all acute / obtuse / right angles. 18. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. No. Explanations may vary. Sample: The parallelogram cannot be a rectangle or square because it does not have four right angles. There is not enough information to tell whether it is a rhombus. Problem Using Properties of Special Parallelograms Got It? For what value of y is ~EFG a rectangle? G 19. For ~EFG to be a parallelogram, the diagonals must 9 each other. bisect 0. EG 5 ( 5y 1 3 ) 1. F 5 ( 7y 5 ) 5 10y 1 6 5 14y 10 E 5y 3 4 7y 5 4 F. For ~EFG to be a rectangle, the diagonals must be 9. 3. Now write an equation and solve for y. 10y 1 6 5 14y 10 10y 14y 510 6 4y 516 y 5 4 4. ~EFG is a rectangle for y 5 4. Problem 3 Using Properties of Parallelograms Got It? Suppose you are on the volunteer building team at the right. You are helping to lay out a square play area. How can you use properties of diagonals to locate the four corners? 5. You can cut two pieces of rope that will be the diagonals of the square play area. Cut them the same length because a parallelogram is a 9 if the diagonals are congruent. rectangle congruent Chapter 6 164
6. You join the two pieces of rope at their midpoints because a quadrilateral is a 9 if the diagonals bisect each other. 7. You move so the diagonals are perpendicular because a parallelogram is a 9 if the diagonals are perpendicular. 8. Explain why the polygon is a square when you pull the ropes taut. parallelogram rhombus nswers may vary. Sample: Congruent diagonals bisect each other, so you formed a rectangle. iagonals are perpendicular, so you formed a rhombus. rectangle that is a rhombus is a square. Lesson Check o you UNERSTN? Name all of the special parallelograms that have each property.. iagonals are perpendicular. B. iagonals are congruent. C. iagonals are angle bisectors.. iagonals bisect each other. E. iagonals are perpendicular bisectors of each other. 9. Place a in the box if the parallelogram has the property. Place an if it does not. Property Rectangle Rhombus Square B Math Success Need to review C E Check off the vocabulary words that you understand. rhombus rectangle square diagonal Rate how well you can use properties of parallelograms. 0 4 6 8 10 Now I get it! 165 Lesson 6-5
6-6 Trapezoids and Kites Vocabulary Review Underline the correct word to complete each sentence. 1. n isosceles triangle always has two / three congruent sides.. n equilateral triangle is also a(n) isosceles / right triangle. 3. Cross out the length(s) that can NOT be side lengths of an isosceles triangle. 3, 4, 5 8, 8, 10 3.6, 5, 3.6 7, 11, 11 Vocabulary Builder trapezoid (noun) TRP ih zoyd trapezoid base Related Words: base, leg leg base angles leg efinition: trapezoid is a quadrilateral with exactly one pair of parallel sides. Main Idea: The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles. Use Your Vocabulary 4. Cross out the figure that is NOT a trapezoid. base 5. Circle the figure(s) than can be divided into two trapezoids. Then divide each figure that you circled into two trapezoids. Chapter 6 166
Theorems 6-19, 6-0, and 6-1 Theorem 6-19 If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. Theorem 6-0 If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent. 6. If TRP is an isosceles trapezoid with bases R and TP, then /T > / P and /R > /. R 7. Use Theorem 6-19 and your answers to Exercise 6 to draw congruence marks on the trapezoid at the right. T P 8. If BC is an isosceles trapezoid, then C > B. B C 9. If BC is an isosceles trapezoid and B 5 5 cm, then C 5 5 cm. 5 cm 5 cm 10. Use Theorem 6-0 and your answer to Exercises 8 and 9 to label the diagram at the right. Theorem 6-1 Trapezoid Midsegment Theorem If a quadrilateral is a trapezoid, then (1) the midsegment is parallel to the bases, and () the length of the midsegment is half the sum of the lengths of the bases. 11. If TRP is a trapezoid with midsegment MN, then (1) MN 6 TP 6 R () MN 5 1 Q TP 1 R R M T R N P Problem Finding ngle Measures in Isosceles Trapezoids Got It? fan has 15 angles meeting at the center. What are the measures of the base angles of the congruent isosceles trapezoids in its second ring? Use the diagram at the right for Exercises 1 16. 1. Circle the number of isosceles triangles in each wedge. Underline the number of isosceles trapezoids in each wedge. one two three four 13. a 5 360 4 15 5 4 14. b 5 180 4 5 78 15. c 5 180 78 5 10 a b c d 16. d 5 180 10 5 78 17. The measures of the base angles of the isosceles trapezoids are 10 and 78. 167 Lesson 6-6
Problem 3 Using the Midsegment Theorem Q 10 R Got It? lgebra MN is the midsegment of trapezoid PQRS. What is x? What is MN? M x 11 N 18. The value of x is found below. Write a reason for each step. P 8x 1 S MN 5 1 (QR 1 PS) Trapezoid Midsegment Theorem x 1 11 5 1 f10 1 (8x 1)g Substitute. x 1 11 5 1 (8x ) Simplify. x 1 11 5 4x 1 istributive Property x 1 1 5 4x dd 1 to each side. 1 5 x Subtract x from each side. 6 5 x ivide each side by. 19. Use the value of x to find MN. MN 5 x 1 11 5 (6) 1 11 5 1 1 11 5 3 kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent. Problem 4 Theorem 6- Theorem 6- If a quadrilateral is a kite, then its diagonals are perpendicular. B 0. If BC is a kite, then C ' B. 1. Use Theorem 6- and Exercise 0 to draw congruence marks and right angle symbol(s) on the kite at the right. Finding ngle Measures in Kites Got It? Quadrilateral KLMN is a kite. What are ml1, ml, and ml3?. iagonals of a kite are perpendicular, so m/1 5 90. 3. nknm > nklm by SSS, so m/3 5 m/nkm 5 36. 4. m/ 5 m/1 m/ 3 by the Triangle Exterior ngle Theorem. K L 3 1 36 N C M 5. Solve for m/. ml 5 90 36 5 54 Chapter 6 168
Lesson Check o you UNERSTN? Compare and Contrast How is a kite similar to a rhombus? How is it different? Explain. 6. Place a in the box if the description fits the figure. Place an if it does not. Kite escription Rhombus Quadrilateral Perpendicular diagonals Each diagonal bisects a pair of opposite angles. Congruent opposite sides Two pairs of congruent consecutive sides Two pairs of congruent opposite angles Supplementary consecutive angles 7. How is a kite similar to a rhombus? How is it different? Explain. nswers may vary. Sample: Similar: Both are quadrilaterals with perpendicular diagonals, two pairs of congruent consecutive sides and one pair of congruent opposite angles. ifferent: Kites have no congruent opposite sides, only one pair of congruent opposite angles, and no supplementary consecutive angles. Math Success Check off the vocabulary words that you understand. trapezoid kite base leg midsegment Rate how well you can use properties of trapezoids and kites. Need to review 0 4 6 8 10 Now I get it! 169 Lesson 6-6
6-7 Polygons in the Coordinate Plane Vocabulary Review 1. raw a line from each item in Column to the corresponding part of the coordinate plane in Column B. Column origin Quadrant I Quadrant II Quadrant III Quadrant IV x-axis y-axis Column B y x Vocabulary Builder classify (verb) KLS uh fy efinition: To classify is to organize by category or type. Math Usage: You can classify figures by their properties. Related Words: classification (noun), classified (adjective) Example: Rectangles, squares, and rhombuses are classified as parallelograms. Use Your Vocabulary Complete each statement with the correct word from the list. Use each word only once. classification classified classify. Trapezoids are 9 as quadrilaterals. classified 3. Taxonomy is a system of 9 in biology. classification 4. Schools 9 children by age. classify Chapter 6 170
Key Concept Formulas on the Coordinate Plane istance Formula Midpoint Formula Slope Formula Formula d (x x 1 ) (y y 1 ) x 1 x y 1 y! M 1, m y y 1 x x 1 To determine whether To determine To determine whether When to Use It sides are congruent diagonals are congruent the coordinates of the midpoint of a side whether diagonals bisect each other opposite sides are parallel diagonals are perpendicular sides are perpendicular ecide when to use each formula. Write for istance Formula, M for Midpoint Formula, or S for Slope Formula. M 5. You want to know whether diagonals bisect each other. S 6. You want to find whether opposite sides of a quadrilateral are parallel. 7. You want to know whether sides of a polygon are congruent. Problem 1 Classifying a Triangle Got It? kef has vertices (0, 0), E(1, 4), and F(5, ). Is kef scalene, isosceles, or equilateral? 8. Graph nef on the coordinate plane at the right. y Use the istance Formula to find the length of each side. 4 E 9. EF 5 a5 1 Å b 1 a 4 b 5 16 1 Ä 5 Ä 0 10. E 5 a1 b Å 4 O 4 0 1 a4 b 11. F 5 a5 0 b Å 5 1 1 16 5 5 1 4 Ä Ä 5 17 Ä 5 9 Ä 1. What type of triangle is nef? Explain. nswers may vary. Sample: F x 1 a b 0 0 No side lengths are equal, so nef is scalene. 171 Lesson 6-7
Problem Classifying a Parallelogram Got It? ~MNPQ has vertices M(0, 1), N(1, 4), P(, 5), and Q(3, ). Is ~MNPQ a rectangle? Explain. 13. Find MP and NQ to determine whether the diagonals MP and NQ are congruent. MP 5 Å a 0 b 1 a5 1 b NQ 5 a3 1 b 1 a 4 b Å 5 4 1 16 5 16 1 4 Ä Ä 5 0 Ä 5 0 Ä 14. Is ~MNPQ a rectangle? Explain. Yes. Explanations may vary. Sample: The diagonals are congruent. Problem 3 Classifying a Quadrilateral Got It? n isosceles trapezoid has vertices (0, 0), B(, 4), C(6, 4), and (8, 0). What special quadrilateral is formed by connecting the midpoints of the sides of BC? 15. raw the trapezoid on the coordinate plane at the right. 16. Find the coordinates of the midpoints of each side. B a 0 1, 0 1 4 b 5 Q 1, R C BC Q 6 1 8, 4 1 0 R 5 (7, ) Q 1 6, 4 1 4 R 5 (4, 4) Q 0 1 8 17. raw the midpoints on the trapezoid and connect them. Judging by appearance, what type of special quadrilateral did you draw? Circle the most precise answer. kite parallelogram rhombus trapezoid 4 y B O 4 6 8 18. To verify your answer to Exercise 17, find the slopes of the segments. connecting midpoints of B and BC: 3 connecting midpoints of BC and C: 3 connecting midpoints of C and : 3 connecting midpoints of and B: 3 19. re the slopes of opposite segments equal? Yes / No 0. re consecutive segments perpendicular? Yes / No C, 0 1 0 R 5 (4, 0) x 1. The special quadrilateral is a 9. rhombus Chapter 6 17
Lesson Check o you UNERSTN? Error nalysis student says that the quadrilateral with vertices (1, ), E(0, 7), F(5, 6), and G(7, 0) is a rhombus because its diagonals are perpendicular. What is the student s error? 6 y E F. raw EFG on the coordinate plane at the right. 3. Underline the correct words to complete Theorem 6-16. If the diagonals of a parallelogram / polygon are perpendicular, then the parallelogram / polygon is a rhombus. 4. Check whether EFG is a parallelogram. slope of E: 7 5 5 slope of FG: 0 6 0 1 7 5 5 4 G O 4 6 8 3 x slope of G: 0 7 1 5 1 3 slope of EF: 6 5 7 0 5 1 5 5. re both pairs of opposite sides parallel? Yes / No 6. Find the slope of diagonal F. 7. Find the slope of diagonal EG. m 5 y y 1 x x 1 m 5 6 5 1 m 5 4 4 m 5 1 8. re the diagonals perpendicular? Yes / No 9. Explain the student s error. nswers may vary. Sample: lthough the quadrilateral has perpendicular diagonals, it is not a parallelogram, so it cannot be a rhombus. Math Success Check off the vocabulary words that you understand. m 5 y y 1 x x 1 m 5 0 7 7 0 m 5 7 7 m 51 distance midpoint slope Rate how well you can classify quadrilaterals in the coordinate plane. Need to review 0 4 6 8 10 Now I get it! 173 Lesson 6-7
6-8 pplying Coordinate Geometry Vocabulary Review Write T for true or F for false. T F 1. The vertex of an angle is the endpoint of two rays.. When you name angles using three points, the vertex gets named first. T 3. polygon has the same number of sides and vertices. B 4. Circle the vertex of the largest angle in nbc at the right. 5. Circle the figure that has the greatest number of vertices. C hexagon kite rectangle trapezoid Vocabulary Builder coordinates (noun) koh WR din its efinition: Coordinates are numbers or letters that specify the location of an object. Math Usage: The coordinates of a point on a plane are an ordered dpair of numbers. Main Idea: The first coordinate of an ordered pair is the x-coordinate. The second is the y-coordinate. Use Your Vocabulary raw a line from each point in Column to its coordinates in Column B. Column Column B 6. (1, 3) 7. B (1, 3) 8. C (3, 1) 9. (3, 1) 4 coordinates ( 1, 3) x- coordinate y- coordinate 4 B y O 4 C 4 x Chapter 6 174
Problem 1 Naming Coordinates Got It? RECT is a rectangle with height a and length b. The y-axis bisects EC and RT. What are the coordinates of the vertices of RECT? E y C 10. Use the information in the problem to mark all segments that are congruent to OT. 11. Rectangle RECT has length b, R O so RT 5 b and RO 5 OT 5 b. 1. The coordinates of O are ( 0, 0), so the coordinates of T are ( b, 0), and the coordinates of R are ( b, 0). 13. Rectangle RECT has height a, so TC 5 RE 5 a. 14. The coordinates of C are ( b, a ), so the coordinates of E are ( b, a ). 15. Why is it helpful that one side of rectangle RECT is on the x-axis and the figure is centered on the y-axis. nswers may vary. Sample: The only variables the coordinates contain are a and b. T x Problem Using Variable Coordinates Got It? Reasoning The diagram at the right shows a general parallelogram with a vertex at the origin and one side along the x-axis. Explain why the x-coordinate of B is the sum of a and b. 16. Complete the diagram. 17. Complete the reasoning model below. Think Opposite sides of a parallelogram are congruent. The x-coordinate is the sum of the lengths in the brackets. 18. Explain why the x-coordinate of B is the sum of a 1 b. O y b O BC C (b, c) Write a a The x-coordinate of B is b a a b. nswers may vary. Sample: Since opposite sides of a parallelogram are congruent, BC 5 a. The x-coordinate of C is b, so the (a, 0) B(a b, c) x x-coordinate of B is a 1 b. 175 Lesson 6-8
You can use coordinate geometry and algebra to prove theorems in geometry. This kind of proof is called a coordinate proof. Problem 3 Planning a Coordinate Proof Got It? Plan a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1). 19. Underline the correct words to complete Theorem 5-1. If a segment joins the vertices / midpoints of two sides of a triangle, then the segment is perpendicular / parallel to the third side, and is half its length. 0. Write the coordinates of the vertices of nbc on the grid below. Use multiples of to name the coordinates. y B( b, c ) F E O C( 0, 0 ) ( a, 0 ) x 1. Reasoning Why should you make the coordinates of and B multiples of? nswers may vary. Sample: Make the coordinates of and B multiples of so the coordinates of the midpoints will not be fractions.. Complete the Given and Prove. Given: E is the 9 of B and F is the 9 of BC. Prove: EF 6 C, and EF 5 1 C midpoint midpoint 3. Circle the formula you need to use to prove EF 6 C. Underline the formula you need to use to prove EF 5 1 C. istance Formula Midpoint Formula Slope Formula Underline the correct word to complete each sentence. 4. If the slopes of EF and C are equal, then EF and C are congruent / parallel. 5. If you know the lengths of EF and C, then you can add / compare them. 6. Write three steps you must do before writing the plan for a coordinate proof. ccept reasonable answers. Sample: raw and label a figure, state the Given and Prove, and determine the formulas you will need. Chapter 6 176
Lesson Check o you UNERSTN? Error nalysis classmate says the endpoints of the midsegment of the trapezoid at the right are Q b, c R and Q d 1 a, c R. What is your classmate s error? Explain. 7. What is the Midpoint Formula? M 5 a x 1 1 x, y 1 1 y b 8. Find the midpoint of each segment to find the endpoints of MN. O y R(b, c) (d, c) M N x P(a, 0) OR Q 0 1 b, 0 1 c R 5 Q b, c R 5 (b, c) P Q d 1 a, c 1 0 R 5 Q (d 1 a), c R 5 (d 1 a, c) 9. The endpoints of the midsegment are ( b, c ) and ( d 1 a, c ). 30. How are the endpoints that your classmate found different from the endpoints that you found in Exercise 8? nswers may vary. Sample. Each coordinate of my classmate s endpoints is half the corresponding coordinate of the endpoints that I found. 31. What is your classmate s error? Explain. nswers may vary. Sample: My classmate either divided the sum of the coordinates by 4 or used the Midpoint Formula with R(b, c), (d, c), and P(a, 0). Math Success Check off the vocabulary words that you understand. coordinate geometry coordinate proof variable coordinates Rate how well you can use properties of special figures. Need to review 0 4 6 8 10 Now I get it! 177 Lesson 6-8
6-9 Proofs Using Coordinate Geometry Vocabulary Review 1. Circle the Midpoint Formula for a segment in the coordinate plane. Underline the istance Formula for a segment in the coordinate plane. M 5 x 1 1 x, y 1 1 y d 5 "(x x 1 ) 1 (y y 1 ) m 5 y y 1 x x 1. Circle the Midpoint Formula for a segment on a number line. Underline the istance Formula for a segment on a number line. M 5 x 1 1 x d 5 x 1 x m 5 x 1 x Vocabulary Builder variable (noun) VEHR ee uh bul Related Words: vary (verb), variable (adjective) efinition: variable is a symbol (usually a letter) that represents one or more numbers. Math Usage: variable represents an unknown number in equations and inequalities. Use Your Vocabulary Underline the correct word to complete each sentence. 3. n interest rate that can change is a variable / vary interest rate. 4. You can variable / vary your appearance by changing your hair color. 5. The amount of daylight variables / varies from summer to winter. 6. Circle the variable(s) in each expression below. 3n 4 1 x p p 7. Cross out the expressions that do NOT contain a variable. x and y are often used as variables. 1 m 36 4 (? 3) 9a 4a 8 (15 4 3) 4 y Chapter 6 178
Problem 1 Writing a Coordinate Proof Got It? Reasoning You want to prove that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. What is the advantage of using coordinates O(0, 0), E(0, b), and F(a, 0) rather than O(0, 0), E(0, b), and F(a, 0)? 8. Label each triangle. y E(0, b ) y E( 0, b) M M O (0, 0) F(a, 0 ) x O (0, 0) F( a, 0) x 9. Use the Midpoint Formula M 5 x 1 1 x M in each triangle., y 1 1 y to find the coordinates of Fisrt Triangle Second Triangle a a 1 0 0 1 b, b 5 a a b a 1 0, b a, 0 1 b b 5 ( a, b) 10. Use the istance Formula, d 5 "(x x 1 ) 1 (y y 1 ) and your answers to Exercise 9 to verify that EM 5 FM 5 OM for the first triangle. EM FM OM É a0 a b 1 ab b b 5 É a a b 1 a b b a 5 Å 4 1 b 4 11. Use the istance Formula, d 5 "(x x 1 ) 1 (y y 1 ) and your answers to Exercise 9 to verify that EM 5 FM 5 OM for the second triangle. EM FM OM "(0 a) 1 (b b) 5 "a 1 b É aa a b 1 a0 b b 5 É a a b a 5 Å 4 1 b 4 1 a b b "(a a) 1 (0 b) 5 "a 1 b É a0 a b 1 a0 b b 5 É a a b 1 a b b a 5 Å 4 1 b 4 "(0 a) 1 (0 b) 5 "a 1 b 1. Which set of coordinates is easier to use? Explain. nswers may vary. Sample: Coordinates O(0, 0), E(0, b), and F(a, 0) are easier to use because I don t have fractions in the istance Formula. 179 Lesson 6-9
Problem Writing a Coordinate Proof Got It? Write a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1). Given: E is the midpoint of B and F is the midpoint of BC Prove: EF 6 C, EF 5 1 C Use the diagram at the right. 13. Label the coordinates of point C. 14. Reasoning Why should you make the coordinates of and B multiples of? O y F( b, c ) B( b, c ) E( a + b, c ) C( 0, 0 ) ( a, 0 ) x nswers may vary. Sample: Make the coordinates of and B multiples of so the coordinates of the midpoints are not fractions. 15. Label the coordinates of and B in the diagram. 16. Use the Midpoint Formula to find the coordinates of E and F. Label the coordinates in the diagram. coordinates of E coordinates of F a a + b, 0 + c 0 + b 0 b 5 ( a 1 b, c ) a, + c b 5 ( b, c ) 17. Use the Slope Formula to determine whether EF 6 C. c c slope of EF 5 5 0 a 1 b b 0 0 slope of C 5 5 0 a 0 18. Is EF 6 C? Explain. Yes. Explanations may vary but should state that EF n C because the slopes of EF and C are equal. 19. Use the istance Formula to determine whether EF 5 1 C. EF 5 Å ( a 1 b b ) 1 ( c c ) 5 Î (a) 1 (0) 5 a C 5 Å ( a 0 ) 1 ( 0 0 ) 5 Î (a) 1 (0) 5 a 0. 1 C 5 1? a 5 a 5 EF Chapter 6 180
Lesson Check o you know HOW? Use coordinate geometry to prove that the diagonals of a rectangle are congruent. 1. raw rectangle PQRS with P at (0, 0).. Label Q(a, 0 ), R( a, b), and S( 0, b ). S y (0, b) R (a, b) 3. Complete the Given and Prove statements. Given: PQRS is a rectangle. Prove: PR > QS 4. Use the istance Formula to find the length of eatch diagonal. P (0, 0) x Q (a, 0) PR 5 ( a 0 ) 1 ( b 0 ) 5 Ä "a 1 b QS 5 ( a 0 ) 1 ( 0 b ) 5 Ä "a 1 b 5. PR 5 QS, so PR > QS. Lesson Check Math Success o you UNERSTN? Error nalysis Your classmate places a trapezoid on the coordinate plane. What is the error? 6. Check whether the coordinates are for an isosceles trapezoid. OP 5 (b 0 ) 1 (c 0 ) 5 Ä QR 5 (a a b ) 1 (0 c ) 5 Ä 7. oes the trapezoid look like an isosceles triangle? Yes / No 8. escribe your classmate s error. "b 1 c "b 1 c nswers may vary. Sample: The x-coordinate of Q is for an isosceles trapezoid. Check off the vocabulary words that you understand. y O P(b, c) Q(a b, c) proof theorem coordinate plane coordinate geometry Rate how well you can prove theorems using coordinate geometry. R(a, 0) x Need to review 0 4 6 8 10 Now I get it! 181 Lesson 6-9