Key Concept Congruent Figures

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Iris Martin
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4-1 ongruent igures ommon ore State Standards Prepares for G-SRT.

.. criteria for triangles to solve problems and prove relationships

P 1, P 3, P 4, P 7 Objective To recognize congruent figures and

You ve almost finished, except for a few pieces of the sky.

ow did you figure out where to place the pieces? aving trouble?

1 TTI PRTIS 2 3 esson Vocabulary congruent polygons ongruent figures

When two figures are congruent, you can slide, flip, or turn one so

1 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 To 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 TTI PRTIS 2 3 esson Vocabulary congruent 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 Turn lip ssential Understanding You can determine whether two figures are congruent by comparing their corresponding parts. 218 hapter 4 ongruent Triangles

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

The congruence statement IK O tells you which parts correspond. Problem 1 inding ongruent Parts If IK O, what are the congruent corresponding parts? Sides: I I K O K O ngles: I K O Got It?

In the congruent triangles, corresponds to, so you know that. You can find m by first finding m. ultiple hoice The wings of an SR-71 lackbird aircraft suggest congruent triangles.

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? The congruence statement IK O tells you which parts correspond. Problem 1 inding ongruent Parts 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? K 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 The wings of an SR-71 lackbird aircraft suggest congruent triangles. What is mj? Use the Triangle ngle- Sum Theorem to write an equation involving m. m = 180 Solve for m. and are corresponding parts of congruent triangles, so. m = 180 m = 75 m = m = 75 The 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 Triangles 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.

3 Problem 3 inding ongruent Triangles 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 Triangle ngle-sum Theorem: The sum of the measures of the angles in a triangle is 180. The next theorem follows from the Triangle ngle-sum Theorem. Theorem 4-1 Third ngles Theorem Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If... and Then... Proof Proof of Theorem 4-1: Third ngles Theorem 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 Thm. 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 Triangles

4 Proof Problem 4 Proving Triangles 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) Third ngles Theorem 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: T OR a. T? b. R? 3. Given: UK a. U? b.? c.? o you URST? TTI PRTIS 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 TI, I and P. a. What is the relationship between and T? b. If m = 52 and m P = 36, what is m T? esson 4-1 ongruent igures 221

5 Practice and Problem-Solving xercises TTI PRTIS 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 king post truss attic frame truss 9. The 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. 10.? 11. K? 12.? 13.? K 14. K? 15.? 16.? 17. K? 18.? 19. K? 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. 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. TRK and TUK 31. SPQ and TUV R T K U S 5 Q 7 6 P V 4 8 T 7 See Problem 3. U 222 hapter 4 ongruent Triangles

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. 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 = 3z + 2, = z = 7a + 5, = 5a Think 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

7 45. Given: PR } TQ, PR TQ, PS QS, PQ bisects RT Proof Prove: PRS QTS P 46. Writing The 225 cards in Tracy s sports card collection are rectangles of three different sizes. ow could Tracy quickly sort the cards? T S R hallenge oordinate Geometry The 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? PROR TSK pply What You ve earned ook back at the information given on page 217 about how amal located the points in the diagram. The diagram is shown again below. X TTI PRTIS P 1 Y W S a. opy and label the diagram. Include all the given information in your diagram. b. Which angles do you know to be congruent? xplain. c. Which sides do you know to be congruent? xplain. d. an you conclude that YX using the definition of congruent triangles? If not, what additional information would you need? 224 hapter 4 ongruent Triangles

To recognize congruent figures and their corresponding parts

To recognize congruent figures and their corresponding parts 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