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Triangle Congruence Theorems: Isosceles & Equilateral Examples | Real-World Applications

Learn and apply the properties of isosceles and equilateral triangles through examples and discover the Triangle Congruence Theorems. Test your knowledge with practical exercises and explore nature's geometric formations.

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Triangle Congruence Theorems: Isosceles & Equilateral Examples | Real-World Applications

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–5) Then/Now New Vocabulary Theorems: Isosceles Triangle Example 1: Congruent Segments and Angles Corollaries: Equilateral Triangle Example 2: Find Missing Measures Example 3: Find Missing Values Example 4: Real-World Example: Apply Triangle Congruence Lesson Menu

  3. A B C D Refer to the figure. Complete the congruence statement.ΔWXY  Δ_____ by ASA. ? A.ΔVXY B.ΔVZY C.ΔWYX D.ΔZYW 5-Minute Check 1

  4. A B C D Refer to the figure. Complete the congruence statement. ΔWYZ  Δ_____ by AAS. ? A.ΔVYX B.ΔZYW C.ΔZYV D.ΔWYZ 5-Minute Check 2

  5. A B C D Refer to the figure. Complete the congruence statement. ΔVWZ  Δ_____ by SSS. ? A.ΔWXZ B.ΔVWX C.ΔWVX D.ΔYVX 5-Minute Check 3

  6. A B C D What congruence statement is needed to use AAS to prove ΔCAT ΔDOG? A. C  D B. A  O C. A  G D. T  G 5-Minute Check 4

  7. You identified isosceles and equilateral triangles. (Lesson 4–1) • Use properties of isosceles triangles. • Use properties of equilateral triangles. Then/Now

  8. legs of an isosceles triangle • vertex angle • base angles Vocabulary

  9. Concept

  10. ___ BCA is opposite BA and A is opposite BC, so BCA  A. ___ Congruent Segments and Angles A. Name two unmarked congruent angles. Answer: BCAand A Example 1

  11. ___ BC is opposite D and BD is opposite BCD, so BC  BD. ___ ___ ___ ___ Answer: BC BD Congruent Segments and Angles B. Name two unmarked congruent segments. Example 1

  12. A B C D A. Which statement correctly names two congruent angles? A.PJM PMJ B.JMK JKM C.KJP JKP D.PML PLK Example 1a

  13. A B C D A.JP PL B.PM PJ C.JK MK D.PM PK B. Which statement correctly names two congruent segments? Example 1b

  14. Concept

  15. Since QP = QR, QP  QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR. Find Missing Measures A. Find mR. Triangle Sum Theorem mQ = 60, mP = mR Simplify. Subtract 60 from each side. Answer:mR = 60 Divide each side by 2. Example 2

  16. Find Missing Measures B. Find PR. Since all three angles measure 60, the triangle is equiangular. Because an equiangular triangle is also equilateral, QP = QR = PR. Since QP = 5, PR = 5 by substitution. Answer:PR = 5 cm Example 2

  17. A B C D A. Find mT. A. 30° B. 45° C. 60° D. 65° Example 2a

  18. A B C D B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7 Example 2b

  19. Since E = F, DE  FE by the Converse of the Isosceles Triangle Theorem. DF  FE, so all of the sides of the triangle are congruent. The triangle is equilateral. Each angle of an equilateral triangle measures 60°. Find Missing Values ALGEBRA Find the value of each variable. Example 3

  20. Find Missing Values mDFE = 60 Definition of equilateral triangle 4x – 8 = 60 Substitution 4x = 68 Add 8 to each side. x = 17 Divide each side by 4. The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal. DF = FE Definition of equilateral triangle 6y + 3 = 8y – 5 Substitution 3 = 2y – 5 Subtract 6y from each side. 8 = 2y Add 5 to each side. Example 3

  21. Find Missing Values 4 = y Divide each side by 2. Answer:x = 17, y = 4 Example 3

  22. A B C D Find the value of each variable. A.x = 20, y = 8 B.x = 20, y = 7 C.x = 30, y = 8 D.x = 30, y = 7 Example 3

  23. Given:HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX || OG. Prove:ΔENX is equilateral. ___ Apply Triangle Congruence NATURE Many geometric figures can be found in nature. Some honeycombs are shaped like a regular hexagon. That is, each of the six sides and interior angle measures are the same. Example 4

  24. Statements Reasons 1. HEXAGO is a regular polygon. 1. Given 2. Given 2. ΔONG is equilateral. 3. Definition of a regular hexagon 3. EX  XA  AG  GO  OH  HE 4. N is the midpoint of GE 4. Given 6. Given 6. EX || OG 5. NG  NE 5. Midpoint Theorem Apply Triangle Congruence Proof: Example 4

  25. Statements Reasons 7. Alternate Exterior Angles Theorem 7. NEX  NGO 8. SAS 8. ΔONG ΔENX 9. Definition of Equilateral Triangle 9. OG NO  GN 10. CPCTC 10. NO NX, GN  EN 11. Substitution 11. XE NX  EN 12. Definition of Equilateral Triangle 12.ΔENX is equilateral. Apply Triangle Congruence Proof: Example 4

  26. Given: HEXAGO is a regular hexagon.NHE  HEN  NAG  AGN Prove: HN EN  AN  GN ___ ___ ___ ___ Proof: Statements Reasons 3. HE  EX  XA  AG  GO  OH 1. HEXAGO is a regular hexagon. 1. Given 2. NHEHENNAGAGN 2. Given 3. Definition of regular hexagon 4. ΔHNE ΔANG 4. ASA Example 4

  27. A B C D Proof: Statements Reasons 5. HN  AN, EN NG 5. ___________ ? 6. HN  EN, AN  GN 6. Converse of Isosceles Triangle Theorem 7. HN  EN  AN  GN 7. Substitution A. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC D. Transitive Property Example 4

  28. End of the Lesson

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