Understanding Isosceles and Equilateral Triangles in Geometry

Exploring the properties and theorems related to isosceles and equilateral triangles in geometry. Discover concepts like the congruence of legs and angles, the relationship between sides and angles, and the bisector of the vertex angle in an isosceles triangle. Explore examples with detailed solutions to enhance your understanding of these triangle types.

• Geometry
• Isosceles Triangle
• Equilateral Triangle
• Theorems
• Triangle Properties

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1. SECTION 4.5 ISOSCELES SECTION 4.5 ISOSCELES & EQUILATERAL & EQUILATERAL TRIANGLES TRIANGLES Geometry Geometry

2. Isosceles Triangle Isosceles Triangle Legs are congruent Legs are congruent Vertex Angle Vertex Angle Legs Legs Base Angles Base Angles

3. Theorem 4-3 (Isosceles Triangle Thm): If two sides of a triangle are congruent, then the angles opposite those sides are congruent. c b a

5. Theorem 4 Theorem 4- -4 (Converse of Isosceles Triangle 4 (Converse of Isosceles Triangle Thm If two angles of a triangle are congruent, then the sides opposite the angles are congruent. Thm): ):

6. Theorem 4 Theorem 4- -5 5 The bisector of the vertex angle of an isosceles The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. triangle is the perpendicular bisector of the base. A A B B C C

7. Ex. 2) Ex. 2) Find the values of x and y. Find the values of x and y. M y x 63 L N

8. Corollary: Corollary: a statement that follows immediately from a theorem a statement that follows immediately from a theorem Corollary to Corollary to Thm Thm 4 4- -3: 3: If a triangle is equilateral, then it is equiangular. If a triangle is equilateral, then it is equiangular. Corollary to Corollary to Thm Thm 4 4- -4: 4: If a triangle is equiangular, then it is equilateral. If a triangle is equiangular, then it is equilateral.

9. Ex. 3) Ex. 3) BAC m = 38 C If find A A m B B C C M M

10. Ex. 4) Ex. 4) = A A 23 m BAM m CAM If find m BMA B B C C M M

11. Ex. 5) Ex. 5) = 2 x A A and x 3 = 20 m BAC If find x. B m B B C C M M

12. Ex. 6) Find the missing angles and sides. Ex. 6) Find the missing angles and sides.