Isosceles and Equilateral Triangles

1. Isosceles and Equilateral Triangles • Recall that an isosceles triangle is a triangle with at least two congruent sides • The two congruent sides of an isosceles triangle are called the legs • The base of an isosceles triangle is the third side • Note that the base is not necessarily on the bottom! • The two angles at the base are called the base angles • The vertex angle is the angle between the two congruent sides

2. Isosceles and Equilateral Triangles Investigation Base Angles in an Isosceles Triangle • Step 1: • Draw an angle and label it C. This will be the vertex angle of your isosceles triangle • Step 2: • Measure the same distance from point C on both rays of the angle, and label the points A and B • Step 3: • Draw segment AB, the base of your isosceles triangle • Step 4 • Use your protractor to compare the measures of the base angles. What do you notice?

3. A B C D E F Isosceles and Equilateral Triangles Theorem 4.3  Base Angles Theorem • If two sides of a triangle are congruent, then the angles opposite them are congruent • If AB  AC then BC Theorem 4.4  Converse of the Base Angles Theorem • If two angles of a triangle are congruent, then the sides opposite them are congruent • If EF then DE  DF • How long are the legs of the isosceles triangle at right? 3x 2x + 3

4. Isosceles and Equilateral Triangles • Equilateral triangles are also isosceles, since at least two sides are congruent • What does the isosceles triangle conjecture tell you about the angles in an equilateral triangle? Theorem 4.5  Equilateral Theorem • If a triangle is equilateral, then it is equiangular • All angles of an equilateral triangle are congruent • This means each angle in an equilateral triangle is 60° Theorem 4.6  Equiangular Theorem • If a triangle is equiangular, then it is equilateral