To recognize congruent figures and their corresponding parts

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1 4-1 ongruent igures ontent Standard Prepares for G.SR.5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. Objective o recognize congruent figures and their corresponding parts You are working on a puzzle. You ve almost finished, except for a few pieces of the sky. Place the remaining pieces in the puzzle. ow did you figure out where to place the pieces? aving trouble? ow can tracing pieces 1, 2, and 3 help? 1 I PRIS 2 3 esson Vocabulary congruent c polygons ongruent figures have the same size and shape. When two figures are congruent, you can slide, flip, or turn one so that it fits exactly on the other one, as shown below. In this lesson, you will learn how to determine if geometric figures are congruent. Slide urn lip ssential Understanding You can determine whether two figures are congruent by comparing their corresponding parts. 218 hapter 4 ongruent riangles

Key oncept ongruent igures efinition ongruent polygons have congruent corresponding parts their matching sides and angles.

xample G G G G G ow do you know which sides and angles correspond? he congruence statement IK O tells you which parts correspond.

I O Problem 2 Using ongruent Parts You know two angle measures in. ow can they help?

ultiple hoice he wings of an SR-71 lackbird aircraft suggest congruent triangles. What is m?

2 Key oncept ongruent igures efinition ongruent polygons have congruent corresponding parts their matching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order. xample G G G G G ow do you know which sides and angles correspond? he congruence statement IK O tells you which parts correspond. Problem 1 inding ongruent Parts K If IK O, what are the congruent corresponding parts? Sides: I I K O K O ngles: I K O Got It? 1. If WYS KV, what are the congruent corresponding parts? I O Problem 2 Using ongruent Parts You know two angle measures in. ow can they help? In the congruent triangles, corresponds to, so you know that. You can find m by first finding m. ultiple hoice he wings of an SR-71 lackbird aircraft suggest congruent triangles. What is m? Use the riangle ngle- Sum heorem to write an equation involving m. m Solve for m. and are corresponding parts of congruent triangles, so. m m 75 m m 75 he correct answer is. Got It? 2. Suppose that WYS KV. If m W 62 and m Y 35, what is m V? xplain. esson 4-1 ongruent igures 219

Problem 3 inding ongruent riangles ow do you determine whether two triangles are congruent? ompare each pair of corresponding parts.

4 6 4 6 by the definition of congruent triangles. Got It? 3. Is? ustify your answer.

heorem 4-1 hird ngles heorem heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If... and hen.

3 Problem 3 inding ongruent riangles ow do you determine whether two triangles are congruent? ompare each pair of corresponding parts. If all six pairs are congruent, then the triangles are congruent. re the triangles congruent? ustify your answer. Given 4 6, Given Vertical angles are congruent by the definition of congruent triangles. Got It? 3. Is? ustify your answer. Recall the riangle ngle-sum heorem: he sum of the measures of the angles in a triangle is 180. he next theorem follows from the riangle ngle-sum heorem. heorem 4-1 hird ngles heorem heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If... and hen... Proof Proof of heorem 4-1: hird ngles heorem Given:, Prove: Statements Reasons 1), 1) Given 2) m m, m m 2) ef. of 3) m m m 180, m m m 180 3) ngle-sum hm. 4) m m m m m m 4) Subst. Prop. 5) m m m m m m 5) Subst. Prop. 6) m m 6) Subtraction Prop. of 7) 7) ef. of 220 hapter 4 ongruent riangles

4 Proof Problem 4 Proving riangles ongruent You know four pairs of congruent parts. What else do you need to prove the triangles congruent? You need a third pair of congruent sides and a third pair of congruent angles. Given: O, O, O, O Prove: O Statements Reasons 1) O, O 1) Given 2) 2) Reflexive Property of O 3) O, O 3) Given 4) O 4) hird ngles heorem 5) O 5) efinition of triangles Got It? 4. Given:,,, Prove: esson heck o you know OW? omplete the following statements. 1. Given: QXR Y a. QX b. Y 2. Given: OR a. b. R 3. Given: UK a. U b. c. o you URS? I PRIS 5. Open-nded When do you think you might need to know that things are congruent in your everyday life? 6. If each angle in one triangle is congruent to its corresponding angle in another triangle, are the two triangles congruent? xplain. 7. rror nalysis Walter sketched the diagram below. e claims it shows that the two polygons are congruent. What information is missing to support his claim? 4. In P and I, I and P. a. What is the relationship between and? b. If m 52 and m P 36, what is m? esson 4-1 ongruent igures 221

5 Practice and Problem-Solving xercises I PRIS Practice 8. onstruction uilders use the king post truss (below left) for the top of a simple See Problem 1. structure. In this truss,. ist the congruent corresponding parts. G I 9. he attic frame truss (above right) provides open space in the center for storage. In this truss, G I. ist the congruent corresponding parts. K. omplete the congruence statements K K K K K POY SI. ist each of the following. 20. four pairs of congruent sides 21. four pairs of congruent angles t an archeological site, the remains of two ancient step pyramids are congruent. If G, find each of the following. (iagrams are not to scale.) See Problem G 24. m G 25. m m 29. m G 45 ft ft ft G ft or xercises 30 and 31, can you conclude that the triangles are congruent? ustify your answers. 30. RK and UK 31. SPQ and UV R K U S 5 Q 7 6 P V See Problem 3. U 222 hapter 4 ongruent riangles

6 pply 32. Given:,, Proof, Prove: 33. If, which of the following must be a correct congruence statement? See Problem Reasoning Randall says he can use the information in the figure to prove. Is he correct? xplain. lgebra. ind the measures of the given angles or the lengths of the given sides. 35. m x 10, m 2x 36. m 3y, m 6y z 2, z a 5, 5a hink bout a Plan. ind the value of x. What does it mean for two triangles to be congruent? Which angle measures do you already know? ow can you find the missing angle measure in a triangle? lgebra ind the values of the variables. 51 (x 5) x 45 4 in. 2t in. K K 6x omplete in two different ways:. 43. Open-nded Write a congruence statement for two triangles. ist the congruent sides and angles. R Z 44. Given:,,,, Proof Prove: esson 4-1 ongruent igures 223

45. Given: PR Q, PR Q, PS QS, PQ bisects R Proof Prove: PRS QS P 46. Writing he 225 cards in racy s sports card collection are rectangles of three different sizes.

7 45. Given: PR Q, PR Q, PS QS, PQ bisects R Proof Prove: PRS QS P 46. Writing he 225 cards in racy s sports card collection are rectangles of three different sizes. ow could racy quickly sort the cards? S R hallenge oordinate Geometry he vertices of G are G( 2, 1), ( 2, 3), and (1, 3). 47. K G. ind K,, and K. Q 48. If and have coordinates (3, 3) and (6, 3), how many pairs of coordinates are possible for K? ind one such pair. 49. a. ow many quadrilaterals (convex and concave) with different shapes or sizes can you make on a three-by-three geoboard? Sketch them. One is shown at the right. b. ow many quadrilaterals of each type are there? Standardized est Prep S/ 50. GS, m 66, and m S 42. What is m? 51. he measure of one angle in a triangle is 80. he other two angles are congruent. What is the measure of each? 52. Given (2, 6) and ( 3, 6), what is? 53. What is the number of feet in the perimeter of a square with side length 7 ft? ixed Review Write an equation for the line perpendicular to the given line that contains P. See esson P(2, 7); y 3 2 x P(1, 1); y 4x 3 ind the distance between the points. If necessary, round to the nearest tenth. 56. (0, 0), (4, 3) 57. X(11, 24), Y( 7, 24) 58. (1, 12), (1, 2) Get Ready! o prepare for esson 4-2, do xercises See esson 1-7. What can you conclude from each diagram? P S 61. Q R See essons 1-4 and 3-2. O U V 224 hapter 4 ongruent riangles

Key Concept Congruent Figures

Key Concept Congruent Figures 4-1 ongruent igures ommon ore State Standards Prepares for G-SRT..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective