Sect. 4.6 Isosceles, Equilateral, and Right Triangles Goal 1 Using Properties of Isosceles Triangles Goal 2 Using Properties of Right Triangles

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

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

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

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.

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

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:

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

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.                                           

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!!!!!

Given: Prove: EFG  EGH

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

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