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Congruent Polygons and Corresponding Parts

Learn about congruent polygons and how to identify and use corresponding parts to prove congruence. Also explore real-world examples and theorems related to triangle congruence.

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Congruent Polygons and Corresponding Parts

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–2) Then/Now New Vocabulary Key Concept: Definition of Congruent Polygons Example 1: Identify Corresponding Congruent Parts Example 2: Use Corresponding Parts of Congruent Triangles Theorem 4.3: Third Angles Theorem Example 3: Real-World Example: Use the Third Angles Theorem Example 4: Prove that Two Triangles are Congruent Theorem 4.4: Properties of Triangle Congruence Lesson Menu

  3. A B C D Find m1. A. 115 B. 105 C. 75 D. 65 5-Minute Check 1

  4. A B C D Find m2. A. 75 B. 72 C. 57 D. 40 5-Minute Check 2

  5. A B C D Find m3. A. 75 B. 72 C. 57 D. 40 5-Minute Check 3

  6. A B C D Find m4. A. 18 B. 28 C. 50 D. 75 5-Minute Check 4

  7. A B C D Find m5. A. 70 B. 90 C. 122 D. 140 5-Minute Check 5

  8. A B C D One angle in an isosceles triangle has a measure of 80°. What is the measure of one of the other two angles? A. 35 B. 40 C. 50 D. 100 5-Minute Check 5

  9. You identified and used congruent angles. (Lesson 1–4) • Name and use corresponding parts of congruent polygons. • Prove triangles congruent using the definition of congruence. Then/Now

  10. congruent • congruent polygons • corresponding parts Vocabulary

  11. Concept 1

  12. Angles: Sides: Identify Corresponding Congruent Parts Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ. Example 1

  13. A B C D A. B. C. D. The support beams on the fence form congruent triangles. In the figure ΔABC ΔDEF,which of the following congruence statements directly matches corresponding angles or sides? Example 1

  14. Use Corresponding Parts of Congruent Triangles In the diagram, ΔITP ΔNGO. Find the values of x and y. O  P CPCTC mO = mP Definition of congruence 6y – 14 = 40 Substitution Example 2

  15. CPCTC Use Corresponding Parts of Congruent Triangles 6y = 54Add 14 to each side. y= 9Divide each side by 6. NG= ITDefinition of congruence x – 2y = 7.5 Substitution x – 2(9) = 7.5 y = 9 x – 18 = 7.5 Simplify. x= 25.5Add 18 to each side. Answer:x = 25.5, y = 9 Example 2

  16. A B C D In the diagram, ΔFHJ ΔHFG. Find the values of x and y. A.x = 4.5, y = 2.75 B.x = 2.75, y = 4.5 C.x = 1.8, y = 19 D.x = 4.5, y = 5.5 Example 2

  17. Concept 2

  18. Use the Third Angles Theorem ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If J  K and mJ = 72, find mJIH. ΔJIK  ΔJIH Congruent Triangles mKJI + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem H  K, I  I andJ  J CPCTC Example 3

  19. Use the Third Angles Theorem 72 + 72 + mJIK = 180 Substitution 144 + mJIK = 180 Simplify. mJIK = 36 Subtract 144 from each side. mJIH = 36 Third Angles Theorem Answer:mJIH = 36 Example 3

  20. A B C D TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM  ΔNJL, KLM  KML and mKML = 47.5, find mLNJ. A. 85 B. 45 C. 47.5 D. 95 Example 3

  21. Prove That Two Triangles are Congruent Write a two-column proof. Prove:ΔLMNΔPON Example 4

  22. Statements Reasons 1. Given 1. 2. LNM  PNO 2. Vertical Angles Theorem 3. M  O 3. Third Angles Theorem 4. ΔLMNΔPON 4. CPCTC Prove That Two Triangles are Congruent Proof: Example 4

  23. Statements Reasons 1. Given 1. 2. Reflexive Property of Congruence 2. 3.Q  O, NPQ  PNO 3. Given 4. _________________ 4.QNP  ONP ? 5.ΔQNPΔOPN 5. Definition of Congruent Polygons Find the missing information in the following proof. Prove:ΔQNPΔOPN Proof: Example 4

  24. A B C D A. CPCTC B. Vertical Angles Theorem C. Third Angle Theorem D. Definition of Congruent Angles Example 4

  25. Concept 3

  26. End of the Lesson

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