1. 4.7 – Isosceles Triangles Geometry Ms. Rinaldi

2. Isosceles Triangles Remember that a triangle is isosceles if it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs. The angle formed by the legs is the vertex angle. The third side is the base of the isosceles triangle. The two angles adjacent to the base are called base angles.

3. Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. If, then

4. Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. If, then

5. EXAMPLE 1 Apply the Base Angles Theorem SOLUTION In DEF, DE DF. Name two congruent angles. DE DF, so by the Base Angles Theorem, E F.

6. EXAMPLE 2 Apply the Base Angles Theorem In. Name two congruent angles. P RQ

7. EXAMPLE 3 Apply the Base Angles Theorem Copy and complete the statement. 1. If HG HK, then ? ?. If KHJ KJH, then ? ?. 2.

8. EXAMPLE 4 Apply the Base Angles Theorem P R Q (30)° Find the measures of the angles. SOLUTION Since a triangle has 180°, 180 – 30 = 150° for the other two angles. Since the opposite sides are congruent, angles Q and P must be congruent. 150/2 = 75° each.

9. EXAMPLE 5 Apply the Base Angles Theorem P R Q (48)° Find the measures of the angles.

10. EXAMPLE 6 Apply the Base Angles Theorem P R Q (62)° Find the measures of the angles.

11. EXAMPLE 7 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. P RQ (20x-4)° (12x+20)° SOLUTION Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4 20 = 8x – 4 24 = 8x 3 = x Plugging back in, And since there must be 180 degrees in the triangle,

12. EXAMPLE 8 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. P R Q (11x+8)°(5x+50)°

13. EXAMPLE 9 Apply the Base Angles Theorem Find the value of x. Then find the length of the labeled sides. P R Q (80)° SOLUTION Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40 4x = 40 x = 10 7x 3x+40 Plugging back in, QR = 7(10)= 70 PR = 3(10) + 40 = 70

14. EXAMPLE 10 Apply the Base Angles Theorem Find the value of x. Then find the length of the labeled sides. P R Q (50)° 10x – 2 5x+3