1. Isosceles Triangles Geometry D – Chapter 4.6

2. Definitions - Review Define an isosceles triangle. A triangle with two congruent sides. Name the parts of an isosceles triangle. Legs are the congruent sides. Vertex angle is the included angle of the legs. Base is the side opposite the vertex angle. Base angle is the included angle of the base and leg.

3. Definitions - Review is an isosceles triangle. Name each item(s): Vertex Angle Base Legs Base Angles AC AB, CB Side opposite CAB Angle opposite BC

4. Definitions - Review Define an equilateral triangle. A triangle with three congruent sides.

5. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

6. Proof - Isosceles Triangle Theorem Given: VW XW Prove:  Since every angle has a bisector, construct the angle bisector of angle W.

7. Proof - Isosceles Triangle Theorem Given: VW XW Prove:  Since every angle has a bisector, construct the angle bisector of angle W.  Given VW XW

8. Proof - Isosceles Triangle Theorem Given: VW XW Prove:  Since every angle has a bisector, construct the angle bisector of angle W.  Given VW XW  By the definition of angle bisectors  WZ is congruent with itself by the reflexive property.  By SAS,  By CPCTC,

9. Corollary 1 A triangle is equilateral if and only if it is equiangular. Corollary 2 Each angle of an equiangular triangle has a measure of 60 o.

10. Example 1 Find the measure of each angle. x + x + 30 o = 180 o 2x + 30 o = 180 o 2x = 150 o x = 75 o

11. Example 2 Find the length of each side. 3x – 6 2x 6 3x – 6 = 6 3x = 12 x = 4 EF = 6 EG = 8

12. Example 3 Find the measure of each angle. (2x – 4) + (x + 2) + (x + 2) = 180 o 4x = 180 o x = 45 o

13. Given:, B is the midpoint of AC and D is the midpoint of CE. Prove: is isosceles. Example 4  by CPCTC  AC is congruent to EC since the sides opposite of congruent angles are congruent.  Triangle ACE is isosceles by the definition of isosceles triangles.