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4.7 Use Isosceles & Equilateral Triangles

This guide focuses on the properties and theorems related to isosceles and equilateral triangles. It highlights the concepts of the vertex angle, legs, base, and base angles in isosceles triangles, including the Isosceles Triangle Theorem and its converse. For equilateral triangles, it introduces the relationship between equilateral and equiangular triangles, where each angle measures 60 degrees. The guide includes examples, proofs, and problems to enhance understanding of these essential geometric concepts.

yvonne-buckner
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4.7 Use Isosceles & Equilateral Triangles

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  1. 4.7 Use Isosceles & Equilateral 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 4.7 (Base Angles Theorem) If two sides of a ∆ are ≅, then the s opposite those sides are ≅ (if AC ≅ AB, then B ≅ C). A B C The Converse is also true! 

  5. The Converse of Isosceles Triangle Theorem • Theorem 4.8 If two s of a ∆ are ≅, then the sides opposite those s are ≅.

  6. Name two congruent angles. Example 1: Answer:

  7. Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Example 1: Answer:

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

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

  10. 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 2: Proof:

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

  12. 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:

  13. Properties of Equilateral ∆s • Corollary A ∆ is equilateral iff it is equiangular. • Corollary Each  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

  18. Assignment • Geometry: Pg. 267 #3 – 30, 46

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