1. 4.6 Isosceles and Equilateral

2. CCSS Content Standards G.CO.10 Prove theorems about triangles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others.

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

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

5. Concept

6. 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. ___

7. 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. ___

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

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

10. Concept

11. 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... Answer: m  R = 60

12. 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

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

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

15. 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°.

16. 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.

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

18. 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.

19. 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. ___

20. 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.

21. 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

22. 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 ___

23. 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 ?