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4.6 Isosceles Triangles

4.6 Isosceles Triangles. Objectives. Use properties of isosceles triangles Use properties of equilateral triangles. Properties of Isosceles Triangles. The  formed by the ≅ sides is called the vertex angle . The two ≅ sides are called legs . The third side is called the base .

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4.6 Isosceles Triangles

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  1. 4.6 Isosceles Triangles

  2. Objectives • Use properties of isosceles triangles • Use properties of equilateral triangles

  3. Properties of Isosceles Triangles • The  formed by the ≅ sides is called the vertex angle. • The two ≅ sides are called legs. The third side is called the base. • The two s formed by the base and the legs are called thebase angles. vertex leg leg base

  4. Isosceles Triangle Theorem • Theorem 4.9If two sides of a ∆ are ≅, then the s opposite those sides are ≅ (if AC ≅ AB, then B ≅ C). A B C

  5. Write a two-column proof. Given: Prove: Example 1:

  6. Statements Reasons 1. 1.Given 2. 2.Def. of Segments 3. ABC and BCD are isosceles triangles 3.Def. of Isosceles  4. 4.Isosceles  Theorem 5. 5.Given 6. 6.Substitution Example 1: Proof:

  7. Write a two-column proof. Given: . Prove: Your Turn:

  8. Statements Reasons 1.Given 1. 2.ADB is isosceles. 2.Def. of Isosceles Triangles 3. 3.Isosceles  Theorem 4. 4.Given 5. 5. Def. of Midpoint 6. ABC ADC 6.SAS 7. 7.CPCTC Your Turn: Proof:

  9. The Converse of Isosceles Triangle Theorem • Theorem 4.10 If two s of a ∆ are ≅, then the sides opposite those s are ≅ (if B ≅ C, then AC ≅ AB).

  10. Name two congruent angles. Example 2: Answer:

  11. Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Example 2: Answer:

  12. Your Turn: a. Name two congruent angles. Answer: b. Name two congruent segments. Answer:

  13. Properties of Equilateral ∆s • Corollary 4.3A ∆ is equilateral iff it is equiangular. • Corollary 4.4Each  of an equilateral ∆measures 60°.

  14. EFG is equilateral, and bisects bisectsFindand Each angle of an equilateral triangle measures 60°. Since the angle was bisected, Example 3a:

  15. is an exterior angle of EGJ. Example 3a: Exterior Angle Theorem Substitution Add. Answer:

  16. EFG is equilateral, and bisects bisectsFind Example 3b: Linear pairs are supplementary. Substitution Subtract 75 from each side. Answer: 105

  17. ABC is an equilateral triangle. bisects Your Turn: a. Find x. Answer: 30 b. Answer: 90

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