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4.6 Isosceles Triangles

4.6 Isosceles Triangles. Theorem 4.9 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4.10 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollaries

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4.6 Isosceles Triangles

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  1. 4.6 Isosceles Triangles • Theorem 4.9 Isosceles Triangle Theorem • If two sides of a triangle are congruent, then the angles opposite those sides are congruent. • Theorem 4.10 • If two angles of a triangle are congruent, then the sides opposite those angles are congruent. • Corollaries • 4.3 A triangle is equilateral if and only if it is equiangular • 4.4 Each angle of an equilateral triangle measures 60°

  2. Write a two-column proof. Given: Prove: Example 6-1a

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

  4. Write a two-column proof. Given: . Prove: Example 6-1c

  5. 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 Example 6-1d Proof:

  6. Multiple-Choice Test Item If and what is the measure of Read the Test Item CDE is isosceles with base Likewise, CBA is isosceles with Example 6-2a A. 45.5 B. 57.5 C. 68.5 D. 75

  7. Step 1The base angles of CDE are congruent. Let Example 6-2b Solve the Test Item Angle Sum Theorem Substitution Add. Subtract 120 from each side. Divide each side by 2.

  8. Step 2are vertical angles so they have equal measures. Step 3 The base angles of CBA are congruent. Example 6-2c Def. of vertical angles Substitution Angle Sum Theorem Substitution Add. Subtract 30 from each side. Divide each side by 2.

  9. Example 6-2d Answer: D

  10. Multiple-Choice Test Item If and what is the measure of A. 25 B. 35 C. 50 D. 130 Example 6-2e Answer: A

  11. Name two congruent angles. Example 6-3a Answer:

  12. Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Example 6-3b Answer:

  13. Example 6-3c a. Name two congruent angles. Answer: b. Name two congruent segments. Answer:

  14. EFG is equilateral, and bisects bisectsFindand Each angle of an equilateral triangle measures 60°. Since the angle was bisected, Example 6-4a

  15. is an exterior angle of EGJ. Example 6-4b Exterior Angle Theorem Substitution Add. Answer:

  16. EFG is equilateral, and bisects bisectsFind Example 6-4c Linear pairs are supplementary. Substitution Subtract 75 from each side. Answer: 105

  17. ABC is an equilateral triangle. bisects Example 6-4d a. Find x. Answer: 30 b. Answer: 90

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