1. Isosceles Triangles

2. Leg Base Angles Base Remember– the properties of an isosceles triangle….. Vertex Angle

3. Investigating Isosceles Triangles Use a straightedge to draw an ACUTE ISOSCELES triangle-- where and is the acute vertex angle. Use scissors to cut the triangle out Then fold the triangle as shown REPEAT the procedure for an OBTUSE ISOSCELES triangle -- where and is the obtuse vertex angle. What observation can you make about the base angles?

4. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent.

5. Use ALGEBRA to find the missing measures (not drawn to scale) 1. 44 x y 30 m r

6. Use ALGEBRA to find the missing measures (not drawn to scale) 1. 44 x y 30 m r x+y+ 44 = 180 Sum x = y because the two base angles are congruent to each other b/c they are opposite congruent sides 180 = x + x + 44 136 = 2x 68=x 68 = y 68

7. Use ALGEBRA to find the missing measures (not drawn to scale) 2. 30° m r

8. Find the missing measures (not drawn to scale) 30 + r + m = 180 r is the other base angle and must be 30° b/c its opposite from a congruent side. 30 + 30 + m = 180 60 + m = 180 m = 120 2. 30° m r 120°

9. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. Given: Prove:

10. Proof of Base Angles Theorem Given:Prove: Statements 1.Label H as the midpoint of CY 2.Draw NH Reasons 1.Ruler Postulate 2.2 points determine a line 3.Def. of midpoint 4.Reflexive Prop 5.Given 6.SSS 7.CPCTC

11. Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. A R T

12. Corollary-- A corollary is a theorem that follows easily from a theorem that has already been prove. Corollary : If triangle is equilateral, then it is also equiangular. A B C Corollary : If a triangle is equiangular, then it is also equilateral. W ER