Applying Isosceles Triangle Theorems
This page demonstrates how to apply the isosceles triangle theorems in various geometric problems.
Example A: Determining if sides are congruent based on angle measurements
Given: ∠ABC ≅ ∠ACB
Question: Is AB congruent to CB?
Answer: Yes, according to the Converse of the Isosceles Triangle Theorem, if two angles are congruent, the opposite sides are also congruent.
Example B: Determining if angles are congruent based on side measurements
Given: AD ≅ DE
Question: Is ∠A congruent to ∠DEA?
Answer: Yes, according to the Isosceles Triangle Theorem, if two sides are congruent, the angles opposite those sides are also congruent.
Example: In a triangle WVS, if ∠WVS ≅ ∠S and TR ≅ TS, we can conclude that ∠W ≅ ∠S byIsoscelesTriangleTheorem and TR ≅ TS byConverseofIsoscelesTriangleTheorem.
Theorem 22: Angle Bisector Theorem for Isosceles Triangles
This theorem states that if a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.
In mathematical notation:
If AC ≅ BC and ∠ACD ≅ ∠BCD, then CD ⊥ AB and AD ≅ BD
Highlight: The angle bisector theorem for isosceles triangles combines the concepts of angle bisectors and perpendicular bisectors, making it a powerful tool for solving complex geometry problems.
