1. Sect. 4.6 Isosceles, Equilateral, and Right Triangles
Goal Using Properties of Isosceles Triangles
Goal Using Properties of Right Triangles

2. Isosceles Triangle Base Angles – angles adjacent to the base
Using Properties of Isosceles Triangles
Isosceles Triangle
Base Angles – angles adjacent to the base
Vertex angle - angle opposite the base

3. Theorem 4.6 Base Angles Theorem
Using Properties of Isosceles Triangles
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Summary - In other words if you have two congruent sides, you have two congruent base angles.
Theorem 4.6 Base Angles Theorem
If then

4. Theorem 4-7 Converse of the Base Angles Theorem
Using Properties of Isosceles Triangles
Theorem 4-7 Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Summary - If you have two congruent angles, then you have two congruent legs.
                       
If then

5. Equilateral Triangle – special type of Isosceles triangle
Using Properties of Isosceles Triangles
Equilateral Triangle – special type of Isosceles triangle
Corollary to Theorem 4.6
If a triangle is equilateral, then it is equiangular.
Corollary to Theorem 4.7
If a triangle is equiangular, then it is equilateral.

6. Find the value of x Find the value of y
Using Properties of Isosceles Triangles
Find the value of x
Find the value of y

7. Find the value of x Find the value of y
Using Properties of Isosceles Triangles
Find the value of x
Find the value of y
Beware! – Do not expect all diagrams to be drawn to scale. The above diagram may be shown as:

8. Find the value of x Find the value of y
Using Properties of Isosceles Triangles
Find the value of x
Find the value of y

9. Theorem 4.8 Hypotenuse –Leg Congruence Theorem (HL)
Using Properties of Right triangles
Theorem 4.8 Hypotenuse –Leg Congruence Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.
                                          

10. HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!
Using Properties of Right triangles
HL (Hypotenuse - Leg) is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate. 
HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!

11. Given:
Prove: EFG  EGH

12. Given: ADC is isosceles with base ;
Using Properties of Right triangles
Given: ADC is isosceles with base ;
Prove:

13. Methods of Proving Triangles Congruent
SSS
If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
SAS
If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
ASA
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
AAS
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL
If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.
Methods of Proving Triangles Congruent
Using Properties of Right triangles