1. Isosceles and Equilateral Triangles LESSON 4–6

2. Lesson Menu 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

3. Over Lesson 4–5 5-Minute Check 1 A.ΔVXY B.ΔVZY C.ΔWYX D.ΔZYW Refer to the figure. Complete the congruence statement. ΔWXY  Δ_____ by ASA. ?

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

5. Over Lesson 4–5 5-Minute Check 3 A.ΔWXZ B.ΔVWX C.ΔWVX D.ΔYVX Refer to the figure. Complete the congruence statement. ΔVWZ  Δ_____ by SSS. ?

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

7. TEKS 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)

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

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

10. Concept

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

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

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

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

15. 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. Example 2 Find Missing Measures A. Find m  R. Triangle Sum Theorem m  Q = 60, m  P = m  R Simplify. Subtract 60 from each side. Divide each side by 2. Answer: m  R = 60

18. 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. Example 2 Find Missing Measures B. Find PR. Answer: PR = 5 cm

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

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

21. Example 3 Find Missing Values ALGEBRA Find the value of each variable. 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°.

22. Example 3 Find Missing Values m  DFE= 60Definition of equilateral triangle 4x – 8 = 60Substitution 4x= 68Add 8 to each side. x= 17Divide 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= FEDefinition of equilateral triangle 6y + 3= 8y – 5Substitution 3= 2y – 5Subtract 6y from each side. 8= 2yAdd 5 to each side.

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

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

25. Example 4 Apply Triangle Congruence Given: HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX || OG. Prove:ΔENX is equilateral. ___

26. Example 4 Apply Triangle Congruence Proof: ReasonsStatements 1.Given1.HEXAGO is a regular polygon. 5.Midpoint Theorem 5.NG  NE 6.Given 6.EX || OG 2.Given 2.ΔONG is equilateral. 3.Definition of a regular hexagon 3. EX  XA  AG  GO  OH  HE 4.Given 4.N is the midpoint of GE.

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

28. Example 4 Proof: ReasonsStatements 1.Given1.HEXAGO is a regular hexagon. 2.Given 2.  NHE   HEN   NAG   AGN 3.Definition of regular hexagon 4.ASA 3.HE  EX  XA  AG  GO  OH 4.ΔHNE  ΔANG ___ Given: HEXAGO is a regular hexagon.  NHE   HEN   NAG   AGN Prove: HN  EN  AN  GN ___

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

30. Isosceles and Equilateral Triangles LESSON 4–6