Triangle Congruence in Isosceles Triangle: Proving ADC ≅ ADB

Question

Given: ΔABC isosceles

and the line AD cuts the side BC.

Are ΔADC and ΔADB congruent?

And if so, according to which congruence theorem?

AAABBBCCCDDD

Step-by-Step Solution

Since we know that the triangle is isosceles, we can establish that AC=AB and that

AD=AD since it is a common side to the triangles ADC and ADB

Furthermore given that the line AD intersects side BC, we can also establish that BD=DC

Therefore, the triangles are congruent according to the SSS (side, side, side) theorem

Answer

Congruent by L.L.L.