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Congruent Triangles: Definitions, Examples, and Proofs

In this lesson, you will learn about congruent triangles, including how to identify corresponding congruent parts, use the Third Angles Theorem, and prove triangle congruence. Real-world examples and problem-solving are included.

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Congruent Triangles: Definitions, Examples, and Proofs

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  1. LESSON 4–3 Congruent Triangles

  2. Five-Minute Check (over Lesson 4–2) TEKS 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. Find m1. A. 115 B. 105 C. 75 D. 65 5-Minute Check 1

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

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

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

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

  8. 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 6

  9. Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(F), G.1(G) TEKS

  10. You identified and used congruent angles. • Name and use corresponding parts of congruent polygons. • Prove triangles congruent using the definition of congruence. Then/Now

  11. congruent polygons • corresponding parts Vocabulary

  12. Concept 1

  13. 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

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

  15. 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

  16. 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

  17. 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

  18. Concept 2

  19. 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 IJK  IKJ and mIJK = 72, find mJIH. ΔJIK  ΔJIH Congruent Triangles mIJK + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem Example 3

  20. Use the Third Angles Theorem mIJK + mIJK + mJIK = 180 Substitution 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

  21. 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

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

  23. 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

  24. 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

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

  26. Concept 3

  27. LESSON 4–3 Congruent Triangles

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