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Exploring Isosceles and Equilateral Triangles: Theorems and Corollaries

Discover the properties and theorems related to isosceles and equilateral triangles, including C.A.B., triangle congruence, and angle bisector concepts. Learn how to find values of variables and measure angles in these triangle types.

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Exploring Isosceles and Equilateral Triangles: Theorems and Corollaries

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  1. 4.5 - Isosceles and Equilateral Triangles

  2. vertex angle legs base base angles Isosceles Triangles • The congruent sides of an isosceles triangles are called it legs. • The third side is the base. • The two congruent sides form the vertex angle. • The other two angles are the base angles.

  3. C A B Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. A  B

  4. Given: AC  BC Prove: A  B (hint: draw an angle bisector for C) Statements Reasons C A B

  5. C AC  BC A B Theorem 4-4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

  6. C CD  AB and CD bisects AB A D B Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

  7. Y Z X • A corollary is a statement that follows immediately from a theorem. • Corollary to Theorem 4-3 • If a triangle is equilateral, then the triangle is equiangular. X  Y  Z

  8. Y XY  YZ  ZX Z X • Corollary to Theorem 4-4 • If a triangle is equiangular, then the triangle is equilateral.

  9. Find the values of the variables.

  10. Find the values of the variables.

  11. Complete each statement.

  12. Find the measure of each angle.

  13. Homeworkp. 2132 - 17, 21 - 26

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