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1 Page 1 of Congruent Figures Teks Focus TEKS (6)(C) Apply the efinition of congruence, in terms of rigi transformations, to ientify congruent figures an their corresponing sies an angles. TEKS (1)(B) Use a problem-solving moel that incorporates analyzing given information, formulating a plan or strategy, etermining a solution, justifying the solution, an evaluating the problem-solving process an the reasonableness of the solution. Aitional TEKS (1)(F), (1)(G) Vocabulary Congruent polygons polygons that have congruent corresponing sies an angles Formulate create with careful effort an purpose. You can formulate a plan or strategy to solve a problem. Reasonableness the quality of being within the realm of common sense or soun reasoning. The reasonableness of a solution is whether or not the solution makes sense. Strategy a plan or metho for solving a problem ESSENTIAL UNDERSTANDING You can etermine whether two figures are congruent by comparing their corresponing sies an angles. take note Key Concept Congruent Figures Definition Congruent polygons have congruent corresponing sies an angles. When you name congruent polygons, you must list corresponing vertices in the same orer. ABCD EFGH Example AB EF BC FG CD GH DA HE A E B F C G D H take note Theorem 4-1 Thir Angles Theorem Theorem If Then A D an B E If two angles of one triangle are congruent to two angles of another triangle, then the thir angles are congruent. C F You will prove Theorem 4-1 in Exercise 37. Page 148

2 Page 1 of 2 Problem 1 Fining Congruent Sies an Angles If HIJK LMNO, what are the congruent corresponing parts? Sies: HI LM IJ MN JK NO KH OL Angles: H L I M J N K O Plan How o you know which sies an angles correspon? The congruence statement HIJK LMNO tells you which parts correspon. Problem 2 TEKS Process Stanar (1)(B) Using Congruent Sies an Angles Multiple Choice The wings of an SR-71 Blackbir aircraft suggest congruent triangles. What is m D? A 30 B 75 C 105 D 150 Think Write Use the Triangle Angle-Sum Theorem to write an equation involving m A. m A = 180 m A = 180 Solve for m A. m A = 75 m A = m D = 75 A an D are corresponing angles of congruent triangles, so A D. The correct answer is B. Plan You know two angle measures in ΔABC. How can they help? In the congruent triangles, D correspons to A, so you know that D A. You can fin m D by first fining m A. Problem 3 TEKS Process Stanar (1)(G) Fining Congruent Triangles Are the triangles congruent? Justify your answer.

3 Page 2 of 2 AB ED Given BC DC BC = 4 = DC AC EC AC = 6 = EC A E, B DGiven BCA DCE Vertical angles are congruent. ΔABC ΔEDC by the efinition of congruent triangles. Plan How o you etermine whether two triangles are congruent? Compare each pair of corresponing parts. If all six pairs are congruent, then the triangles are congruent. PearsonTEXAS.com Page 149

4 Page 1 of 2 Problem 4 Proof Proving Triangles Congruent Given: LM LO, MN ON, M O, MLN OLN Prove: ΔLMN ΔLON Plan You know four pairs of congruent parts. What else o you nee to prove the triangles congruent? You nee a thir pair of congruent sies an a thir pair of congruent angles. Statements Reasons 1) LM LO, MN ON 1) Given 2) LN LN 2) Reflexive Property of 3) M O, MLN OLN 3) Given 4) MNL ONL 4) Thir Angles Theorem 5) ΔLMN ΔLON 5) Definition of triangles PRACTICE an APPLICATION EXERCISES 1. Apply Mathematics (1)(A) Builers use the king post truss (below left) for the top of a simple structure. In this truss, ΔABC ΔABD. List the congruent corresponing sies an angles. Scan page for a Virtual Ner tutorial vieo. 2. The attic frame truss (above right) provies open space in the center for storage. In this truss, ΔEFG ΔHIJ. List the congruent corresponing sies an angles. ΔLMC ΔBJK. Complete the congruence statements. 3. LC _?_ 4. KJ _?_ For aitional support when completing your homework, go to PearsonTEXAS.com. 5. K _?_ 6. M _?_ 7. ΔCML _?_ 8. ΔKBJ _?_ POLY SIDE. List each of the following. 9. four pairs of congruent sies 10. four pairs of congruent angles At an archeological site, the remains of two ancient step pyramis are congruent. If ABCD EFGH, fin each of the following. (Diagrams are not to scale.) 11. AD 12. GH 13. m GHE 14. m BAD 15. EF

5 Page 2 of BC 17. m DCB 18. m EFG Page 150

6 Page 1 of 2 Explain Mathematical Ieas (1)(G) For Exercises 19 an 20, can you conclue that the triangles are congruent? Justify your answers. 19. ΔTRK an ΔTUK 20. ΔSPQ an ΔTUV Proof 21. Given: AB DC, B D, AB DC, BC AD Prove: ΔABC ΔCDA 22. Evaluate Reasonableness (1)(B) Ranall says he can use the information in the figure to prove ΔBCD ΔDAB. Is he correct? Explain. Connect Mathematical Ieas (1)(F) ΔABC ΔDEF. Fin the measures of the given angles or the lengths of the given sies. 23. m A = x + 10, m D = 2x 24. m B = 3y, m E = 6y BC = 3z + 2, EF = z AC = 7a + 5, DF = 5a If ΔDEF ΔLMN, which of the following must be a correct congruence statement? A. DE LN B. FE NL C. N F D. M F Connect Mathematical Ieas (1)(F) Fin the values of the variables. 28.

7 Page 2 of Complete in two ifferent ways: ΔJLM _?_. PearsonTEXAS.com Page 151

8 Page 1 of 1 Proof 31. Given: AB AD, BC CD, AB CD, AD CB, AB CD Prove: ΔABD ΔCDB 32. Analyze Mathematical Relationships (1)(F) Write a congruence statement for two triangles. List the congruent sies an angles. Proof 33. Given: PR TQ, PR TQ, PS QS, PQ bisects RT Prove: ΔPRS ΔQTS 34. Apply Mathematics (1)(A) The 225 cars in Tracy's sports car collection are rectangles of three ifferent sizes. How coul Tracy quickly sort the cars? 35. Connect Mathematical Ieas (1)(F) The vertices of ΔGHJ are G( 2, 1), H( 2, 3), an J(1, 3), an ΔKLM ΔGHJ. If L an M have coorinates L(3, 3) an M (6, 3), how many pairs of coorinates are possible for K? Fin one such pair. 36. a. How many quarilaterals (convex an concave) with ifferent shapes or sizes can you make on a three-by-three geoboar? Sketch them. One is shown at the right. b. How many quarilaterals of each type are there? Proof 37. Given: A D, B E Prove: C F TEXAS Test Practice 38. ΔHLN ΔGST, m H = 66, an m S = 42. What is m T? 39. The measure of one angle in a triangle is 80. The other two angles are congruent. What is the measure of each? 40. What is the number of feet in the perimeter of a square with sie length 7 ft? Page 152