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LESSON 4–6

LESSON 4–6. Isosceles and Equilateral Triangles. Five-Minute Check (over Lesson 4–5) TEKS Then/Now New Vocabulary Theorems: Isosceles Triangle Example 1: Congruent Segments and Angles Proof: Isosceles-Triangle Theorem Corollaries: Equilateral Triangle Example 2: Find Missing Measures

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LESSON 4–6

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  1. LESSON 4–6 Isosceles and Equilateral Triangles

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

  3. 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. 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. 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. 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. 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, mid-segments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(F) TEKS

  8. You identified isosceles and equilateral triangles. • Use properties of isosceles triangles. • Use properties of equilateral triangles. Then/Now

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

  10. Concept

  11. ___ 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

  12. ___ 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

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

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

  15. Concept

  16. Concept

  17. 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

  18. 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

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

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

  21. 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

  22. 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

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

  24. 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

  25. 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 Example 4

  26. 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. EX || OG 5. NG  NE 5. Midpoint Theorem 6. Given Apply Triangle Congruence Proof: Example 4

  27. 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

  28. 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

  29. 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

  30. LESSON 4–6 Isosceles and Equilateral Triangles

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