Congruent Triangles • Name and label corresponding parts of congruent triangles. • Identify congruence transformations. Vocabulary • congruent triangles • congruence transformations are triangles used in bridges? In 1930, construction started on the West End Bridge in Pittsburgh, Pennsylvania. The arch of the bridge is trussed, not solid. Steel rods are arranged in a triangular web that lends structure and stability to the bridge. CORRESPONDING PARTS OF CONGRUENT TRIANGLES Triangles that are the same size and shape are congruent triangles . Each triangle has three angles and three sides. If all six of the corresponding parts of two triangles are congruent, then the triangles are congruent. B F C G A E ← ← If nABC is congruent to nEFG, the vertices of the two triangles correspond in the same order as the letters naming the triangles. ← ← ← ← nABC > nEFG This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles. /A > /E /B > /F /C > /G AB w w>E wF w B C>F ww wG w A C>E ww wG w The corresponding sides and angles can be determined from any congruence statement by following the order of the letters. Definition of Congruent Triangles (CPCTC) Study Tip Congruent Parts In congruent triangles, congruent sides are opposite congruent angles. Two triangles are congruent if and only if their corresponding parts are congruent. CPCTC stands for corresponding parts of congruent triangles are congruent. “If and only if” is used to show that both the conditional and its converse are true. 192 Chapter 4 Congruent Triangles Aaron Haupt Example 1 Corresponding Congruent Parts FURNITURE DESIGN The seat and legs of this stool form two triangles. Suppose the measures in inches are QR 5 12, RS 5 23, QS 5 24, RT 5 12, TV 5 24, and RV 5 23. a. Name the corresponding congruent angles and sides. /Q > /T /QRS > /TRV /S > /V QR w w>T wR w R RV wS w>w w Q T R S Q wS w>T wV w V b. Name the congruent triangles. nQRS > nTRV Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive. Properties of Triangle Congruence Theorem 4.4 Congruence of triangles is reflexive, symmetric, and transitive. Reflexive nJKL ù nJKL K Transitive If nJKL ù nPQR, and nPQR ù nXYZ, then nJKL ù nXYZ K L J Symmetric If nJKL ù nPQR, then nPQR ù nJKL. K L J Q K L J Q R Y L P J R P Z X You will prove the symmetric and reflexive parts of Theorem 4.4 in Exercises 33 and 35, respectively. Proof Theorem 4.4 (Transitive) Given: nABC > nDEF B E H nDEF > nGHI Prove: nABC > nGHI Proof: Statements 1. nABC > nDEF C A 2. /A > /D, /B > /E, /C > /F F I G D Reasons 1. Given 2. CPCTC AB DF w w>D wE w, B wC w>E wF w, A wC w>w w 3. nDEF > nGHI 3. Given 4. /D > /G, /E > /H, /F > /I 4. CPCTC DE GIw w w>G wH w, E wF w>H wIw, D wF w>w 5. /A > /G, /B > /H, /C > /I 5. Congruence of angles is transitive. B>G GIw 6. w Aw wH w, B wC w>H wIw, A wC w>w 6. Congruence of segments is transitive. 7. nABC > nGHI 7. Def. of > ns www.geometryonline.com/extra_examples Lesson 4-3 Congruent Triangles 193 Private Collection/Bridgeman Art Library Study Tip Naming Congruent Triangles IDENTIFY CONGRUENCE TRANSFORMATIONS In the figures below, nABC is congruent to nDEF. If you slide nDEF up and to the right, nDEF is still congruent to nABC. E' B There are six ways to name each pair of congruent triangles. E slide A D' D F' C F The congruency does not change whether you turn nDEF or flip nDEF. nABC is still congruent to nDEF. E D' Study Tip E D Transformations Not all of the transformations preserve congruence. Only transformations that do not change the size or shape of the triangle are congruence transformations. flip D' turn F D F' F F' E' E' If you slide, flip, or turn a triangle, the size and shape do not change. These three transformations are called congruence transformations . Example 2 Transformations in the Coordinate Plane COORDINATE GEOMETRY The vertices of nCDE are C(25, 7), D(28, 6), and E(23, 3). The vertices of nC9D9E9 are C9(5, 7), D9(8, 6), and E9(3, 3). a. Verify that nCDE > nC9D9E9. Use the Distance Formula to find the length of each side in the triangles. (25)]2w 2 7)2 1 (6w [28 2w DC 5 Ïw 5 Ï9w 1 1 or Ïw 10 1 (6w DE 5 Ïw [28 2w (23)]2w 2 3)2 5 Ï25 1 9 or Ïw 34 w 1 (7w CE 5 Ïw [25 2w (23)]2w 2 3)2 5 Ï4w 1 16 or Ïw 20 D C 8 y C' D' 4 E –8 –4 E' O 4 8x D9C9 5 Ï(8 2 5w )2 1 (6w 2 7)2 w 5 Ï9w 1 1 or Ïw 10 D9E9 5 Ï(8 2 3w )2 1 (6w 2 3)2 w 5 Ï25 1 9 or Ï34 w w C9E9 5 Ï(5 2 3w )2 1 (7w 2 3)2 w 5 Ï4w 1 16 or Ï20 w By the definition of congruence, w DC D9wC DE C9wE w>w w9w, w w>D w9wE w9w, and C wE w>w w9w. Use a protractor to measure the angles of the triangles. You will find that the measures are the same. D9wE CE C9wE In conclusion, because D wC w>D w9wC w9w, D wE w>w w9w, and w w>w w9w, /D > /D9, /C > /C9, and /E > /E9, nDCE > nD9C9E9. b. Name the congruence transformation for nCDE and nC9D9E9. nC9D9E9 is a flip of nCDE. 194 Chapter 4 Congruent Triangles Concept Check Guided Practice 1. Explain how slides, flips, and turns preserve congruence. 2. OPEN ENDED Draw a pair of congruent triangles and label the congruent sides and angles. Identify the congruent triangles in each figure. D 3. A 4. H K F C B J T 5. If nWXZ > nSTJ, name the congruent angles and congruent sides. B 6. QUILTING In the quilt design, assume that angles and segments that appear to be congruent are congruent. Indicate which triangles are congruent. M A G E 7. The coordinates of the vertices of nQRT and nQ9R9T9 are Q(24, 3), Q9(4, 3), R(24, 22), R9(4, 22), T(21, 22), and T9(1, 22). Verify that nQRT > nQ9R9T9. Then name the congruence transformation. Application 8. GARDENING This garden lattice will be covered with morning glories in the summer. Wesley wants to save two triangular areas for artwork. If nGHJ > nKLP, name the corresponding congruent angles and sides. L N J H F C K D H G L J K P Practice and Apply For Exercises See Examples 9–22, 27–35 23–26 1 Identify the congruent triangles in each figure. 9. 10. K S L F 2 J Extra Practice See page 761. V H C R P 11. T 12. Z F E S W G Q V H Name the congruent angles and sides for each pair of congruent triangles. 13. nTUV > nXYZ 14. nCDG > nRSW 15. nBCF > nDGH 16. nADG > nHKL Lesson 4-3 Congruent Triangles 195 Assume that segments and angles that appear to be congruent in the numbered triangles are congruent. Indicate which triangles are congruent. 17. 18. 19. 5 9 2 1 5 4 3 6 7 1 9 8 6 10 7 8 1 2 10 2 3 11 4 12 12 7 3 8 6 11 4 10 14 13 17 18 16 15 19 20 9 5 20. All of the small triangles in the figure at the right are congruent. Name three larger congruent triangles. S V B U F E T A C D 21. MOSAICS The picture at the left is the center of a Roman mosaic. Because the four triangles connect to a square, they have at least one side congruent to a side in another triangle. What else do you need to know to conclude that the four triangles are congruent? Verify that each of the following preserves congruence and name the congruence transformation. 22. nPQV > nP9Q9V9 23. nMNP > nM9N9P9 P M y Q' Q N y P' P V Mosaics A mosaic is composed of glass, marble, or ceramic pieces often arranged in a pattern. The pieces, or tesserae, are set in cement. Mosaics are used to decorate walls, floors, and gardens. O V' x P' M' O N' x 24. nGHF > nG9H9F9 25. nJKL > nJ9K9L9 H y J H' y L L' F Source: www.dimosaico.com G F' O G' x K' O x K J' Determine whether each statement is true or false. Draw an example or counterexample for each. 26. Two triangles with corresponding congruent angles are congruent. 27. Two triangles with angles and sides congruent are congruent. 28. UMBRELLAS Umbrellas usually have eight congruent triangular sections with ribs of equal length. Are the statements nJAD > nIAE and nJAD > nEAI both correct? Explain. F A North Carolina Museum of Art, Raleigh. Gift of Mr. & Mrs. Gordon Hanes J I E 196 Chapter 4 Congruent Triangles D B C G ALGEBRA For Exercises 29 and 30, use the following information. nQRS > nGHJ, RS 5 12, QR 5 10, QS 5 6, and HJ 5 2x 2 4. 29. Draw and label a figure to show the congruent triangles. 30. Find x. ALGEBRA For Exercises 31 and 32, use the following information. nJKL > nDEF, m/J 5 36, m/E 5 64, and m/F 5 3x 1 52. 31. Draw and label a figure to show the congruent triangles. 32. Find x. 33. PROOF The statements below can be used to prove that congruence of triangles is symmetric. Use the statements to construct a correct flow proof. Provide the reasons for each statement. Given: nRST > nXYZ Prove: nXYZ > nRST Flow Proof: R S /R > /X, /S > /Y, /T > /Z, XY > RS, YZ > ST, XZ > RT RS > XY, ST > YZ, RT > XZ 34. PROOF Y T /X > /R, /Y > /S, /Z > /T, ? X Z nRST > nXYZ nXYZ > nRST ? ? ? Copy the flow proof and provide the reasons for each statement. B>C CB DC Given: A ww wD w, A wD w>w w, A wD w'w w, Aw B'w BC BC CD w w, A wD w\w w, A wB w\w w Prove: nACD > nCAB B C 4 2 3 1 A D Flow Proof: AB > CD AD > CB AC > CA AD ' DC AB ' BC AD || BC AB || CD a. ? b. ? c. ? d. ? f. ? i. ? k. ? /D is a rt. /. /B is a rt. /. /1 > /4 /2 > /3 e. ? g. ? j. ? l. ? /D > /B h. ? n ACD > nCAB m. ? 35. PROOF Write a flow proof to prove Congruence of triangles is reflexive. (Theorem 4.4) 36. CRITICAL THINKING nRST is isosceles with RS = RT, M, N, and P are midpoints of their sides, /S > /MPS, and N wP w>M wP w. What else do you need to know to prove that nSMP > nTNP? www.geometryonline.com/self_check_quiz R M S N P T Lesson 4-3 Congruent Triangles 197 37. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. Why are triangles used in bridges? Include the following in your answer: • whether the shape of the triangle matters, and • whether the triangles appear congruent. Standardized Test Practice 38. Determine which statement is true given nABC > nXYZ. A B ZX wC w>w w 39. ALGEBRA A B A XZ wC w>w w A B>w YZ ww w C D cannot be determined D 185 Ïw Find the length of w DF w if D(25, 4) and F(3, 27). Ïw5 B 13 Ïw 57 Ïw C Maintain Your Skills Mixed Review Find x. (Lesson 4-2) 40. 41. 40˚ x˚ 42. x˚ x˚ 30˚ 115˚ x˚ 100˚ 42˚ Find x and the measure of each side of the triangle. (Lesson 4-1) 43. nBCD is isosceles with B wC w>C wD w, BC 5 2x 1 4, BD 5 x 1 2, and CD 5 10. 44. Triangle HKT is equilateral with HK 5 x 1 7 and HT 5 4x 2 8. Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson 3-4) 3 45. contains (0, 3) and (4, 23) 46. m 5 }}, y-intercept 5 8 4 47. parallel to y 5 24x 1 1; 48. m 5 24, contains (23, 2) contains (23, 1) Getting Ready for the Next Lesson PREREQUISITE SKILL Find the distance between each pair of points. (To review the Distance Formula, see Lesson 1-4.) 49. (21, 7), (1, 6) 50. (8, 2), (4, 22) 51. (3, 5), (5, 2) P ractice Quiz 1 Lessons 4-1 through 4-3 1. Identify the isosceles triangles in the figure, if w Fw H and w Dw G are congruent perpendicular bisectors. (Lesson 4-1) ALGEBRA nABC is equilateral with AB 5 2x, BC 5 4x 2 7, and AC 5 x 1 3.5. (Lesson 4-1) 2. Find x. 3. Find the measure of each side. 4. Find the measure of each numbered angle. (Lesson 4-2) F D J H 1 3 50˚ 2 21˚ 70˚ 5. If nMNP > nJKL, name the corresponding congruent angles and sides. (Lesson 4-3) 198 Chapter 4 Congruent Triangles G
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