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Congruent Triangles Chapter 5 Objectives • Identify corresponding parts of congruent triangles. • Show triangles are congruent using the SSS, SAS, and ASA Congruence Postulates, and the AAS and HL Congruence Theorems. • Use angle bisectors and perpendicular bisectors to compute angle measures and segment lengths in situations involving triangles. • Reflect figures over lines and use reflections to discover lines of symmetry in a figure. Essential Questions • How does the important property of congruence relate to triangles? • How can you identify the corresponding parts of congruent triangles? Sections • 5.1 Congruence and Triangles • 5.2 Proving Triangles are Congruent: SSS and SAS • 5.3 Proving Triangles are Congruent: ASA and AAS • 5.4 Hypotenuse-Leg Congruence Theorem: HL • 5.5 Using Congruent Triangles • 5.6 Angle Bisectors and Perpendicular Bisectors • 5.7 Reflections and Symmetry Congruence and Triangles Section 5.1 Objectives: • Identify congruent triangles and corresponding parts. Key Vocabulary • • • • Congruent Congruent Figures Corresponding Parts CPCTC Congruence If two geometric figures or polygons have exactly the same shape and size, they are congruent. • Congruent Figures • Not Congruent • While positioned differently, figures 1, 2, and 3 are exactly the same shape and size. • Figures 4 and 5 are exactly the same shape, but not the same size. • Figures 5 and 6 are the same size, but not exactly the same shape. Two figures are congruent if they are the same size and same shape. Congruent figures can be rotations of one another. Congruent figures can be reflections of one another. Corresponding Parts • If two polygons are congruent, then all parts of one polygon are congruent to the corresponding parts (or matching parts) of the other polygon. • Corresponding parts include corresponding angles and corresponding sides. Corresponding Parts • To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. • In a congruence statement, the order of the vertices indicates the corresponding parts. Example: Congruent polygons and Corresponding Parts Definition of Congruent Polygons • Two polygons are congruent if and only if their corresponding parts are congruent. Then: • Example Given: Congruent polygons and Corresponding Parts • When you write a congruence statement such as ABC DEF, you are also stating which parts are congruent. • Therefore, valid congruence statements for congruent polygons list corresponding vertices in the same order. • Given the valid congruence statement ∆ABC≅∆DEF • Other valid congruence statements; ∆BCA≅∆EFD or ∆CBA≅∆FED or ∆CAB≅∆FDE • Invalid congruence statements; ∆ABC≅∆FED or ∆CAB≅∆DFE Identifying Corresponding Congruent Parts C Z A B X Y ∆ABC is congruent to ∆XYZ ∆ABC is congruent to ∆XYZ C Z A B X Corresponding parts of these triangles are congruent. Y ∆ABC is congruent to ∆XYZ C Z A B X Corresponding parts of these triangles are congruent. Corresponding parts are angles and sides that “match.” Y ∆ABC is congruent to ∆XYZ C Z A B X Corresponding parts of these triangles are congruent. A X Y ∆ABC is congruent to ∆XYZ C Z A B X Corresponding parts of these triangles are congruent. B Y Y ∆ABC is congruent to ∆XYZ C Z A B X Corresponding parts of these triangles are congruent. C Z Y ∆ABC is congruent to ∆XYZ C Z A B X Y Corresponding parts of these triangles are congruent. AB XY ∆ABC is congruent to ∆XYZ C Z A B X Y Corresponding parts of these triangles are congruent. BC YZ ∆ABC is congruent to ∆XYZ C Z A B X Y Corresponding parts of these triangles are congruent. AC XZ ∆DEF is congruent to ∆QRS F Q D E R S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. D Q S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. E R S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. F S S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. DE QR S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. DF QS S ∆DEF is congruent to ∆QRS F Q D E R Corresponding parts of these triangles are congruent. FE SR S Example 1 Given that JKL RST, list all corresponding congruent parts. SOLUTION The order of the letters in the names of the triangles shows which parts correspond. Corresponding Angles Corresponding Sides ∆JKL ∆RST, so J R. ∆JKL ∆RST, so JK RS. ∆JKL ∆RST, so K S. ∆JKL ∆RST, so KL ST. ∆JKL ∆RST, so L T. ∆JKL ∆RST, so JL RT. Example 2 The two triangles are congruent. a. Identify all corresponding congruent parts. b. Write a congruence statement. SOLUTION a. Corresponding Angles Corresponding Sides A F AB FD B D BC DE C E AC FE b. List the letters in the triangle names so that the corresponding angles match. One possible congruence statement is ∆ABC ∆FDE. Your Turn: Given STU YXZ, list all corresponding congruent parts. ANSWER ST YX; TU XZ; SU YZ; S Y; T X; U Z Your Turn: Which congruence statement is correct? Why? A. JKL MNP B. JKL NMP C. JKL NPM ANSWER B; This statement matches up the corresponding vertices in order. Practice Time! Your Turn 1) Are these shapes congruent? Explain. 1) Are these shapes congruent? Explain. These shapes are congruent because they are both parallelograms of equal size. 2) Are these shapes congruent? Explain. 2) Are these shapes congruent? Explain. These shapes are not congruent because they are different sizes. 3) Are these shapes congruent? Explain. 3) Are these shapes congruent? Explain. These shapes are congruent because they are the same size. 4) ∆BAD is congruent to ∆THE Name all corresponding parts. D E B A T H 4) ∆BAD is congruent to ∆THE Name all corresponding parts. D E A B T ANGLES B A D H SIDES T BA H AD E DB TH HE ET 5) ∆FGH is congruent to ∆JKL Name all corresponding parts. F J H G K L 5) ∆FGH is congruent to ∆JKL Name all corresponding parts. F J G H K ANGLES F H G L SIDES J FG L GH K HF JK KL LJ 6) ∆QRS is congruent to ∆BRX Name all corresponding parts. S R B Q X 6) ∆QRS is congruent to ∆BRX Name all corresponding parts. S R B Q X ANGLES Q S R SIDES B QR X QS R SR BR BX XR 7) ∆EFG is congruent to ∆FGH Name all corresponding parts. E H G F 7) ∆EFG is congruent to ∆FGH Name all corresponding parts. E H G F ANGLES E F G SIDES H EF F EG G GF HF HG GF Stands for Corresponding Parts of Congruent Triangles are Congruent Definition CPCTC • The bi-conditional phrase “if and only if” in the congruent polygon definition means that both the conditional and its converse are true. • Therefore, definition of CPCTC is; If the corresponding parts of two triangles are congruent, then the two triangles are congruent. AND If two triangles are congruent, then the corresponding parts of the two triangles are congruent. CPCTC Practice O If CAT DOG, then A ___ CPCTC because ________. C O Add markings! D G A T CPCTC Practice If FJH QRS, thenJH RS ___ CPCTC and F Q ___ because _______. If XYZ ABC, then ZX CA ___ and Y B ___ because CPCTC _______. Example 3: Naming Congruent Corresponding Parts Given: ∆PQR ∆STW Identify all pairs of corresponding congruent parts. Angles: P S, Q T, R W Sides: PQ ST, QR TW, PR SW Your Turn If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Angles: L E, M F, N G, P H Sides: LM EF, MN FG, NP GH, LP EH Example 4 E Use the two triangles at the right. a. Identify all corresponding congruent parts. b. Determine whether the triangles are congruent. If they are congruent, write a congruence statement. F D SOLUTION a. Corresponding Angles Corresponding Sides D G DE GE DEF GEF DF GF DFE GFE EF EF G Example 4 E F D G b. All three sets of corresponding angles are congruent and all three sets of corresponding sides are congruent, so the two triangles are congruent. A congruence statement is DEF GEF. Example 5 In the figure, HG || LK. Determine whether the triangles are congruent. If so, write a congruence statement. SOLUTION Start by labeling any information you can conclude from the figure. You can list the following angles congruent. HJG KJL H K G L Vertical angles are congruent. Alternate Interior Angles Theorem Alternate Interior Angles Theorem The congruent sides are marked on the diagram, so HJ KJ, HG KL, and JG JL. Since all corresponding parts are congruent, HJG KJL. Your Turn: In the figure, XY || ZW. Determine whether the two triangles are congruent. If they are, write a congruence statement. ANSWER yes; Sample answer: XVY ZVW Example 6 In the diagram, PQR XYZ. a. Find the length of XZ. b. Find mQ. SOLUTION a. Because XZ PR, you know that XZ = PR = 10. b. Because Q Y, you know that mQ = mY = 95°. Your Turn: Given ∆ABC ∆DEF, find the length of DF and mB. ANSWER 3; 28° Example 7A: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find the value of x. BCA and BCD are rt. s. BCA BCD mBCA = mBCD (2x – 16)° = 90° 2x = 106 x = 53 Def. of lines. Rt. Thm. Def. of s Substitute values for mBCA and mBCD. Add 16 to both sides. Divide both sides by 2. Example 7B: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find mDBC. mABC + mBCA + mA = 180° mABC + 90 + 49.3 = 180 mABC + 139.3 = 180 ∆ Sum Thm. Substitute values for mBCA and mA. Simplify. mABC = 40.7 DBC ABC Subtract 139.3 from both sides. Corr. s of ∆s are . mDBC = mABC Def. of s. mDBC 40.7° Trans. Prop. of = Your Turn Given: ∆ABC ∆DEF Find the value of x. AB DE Corr. sides of ∆s are . AB = DE Def. of parts. 2x – 2 = 6 2x = 8 x=4 Substitute values for AB and DE. Add 2 to both sides. Divide both sides by 2. Your Turn Given: ∆ABC ∆DEF Find mF. mEFD + mDEF + mFDE = 180° ABC DEF ∆ Sum Thm. Corr. s of ∆ are . mABC = mDEF Def. of s. mDEF = 53° Transitive Prop. of =. mEFD + 53 + 90 = 180 mF + 143 = 180 mF = 37° Substitute values for mDEF and mFDE. Simplify. Subtract 143 from both sides.