SOLUTION: I have a triangleABC that is bisected by a lineAD. I am supposed to prove the triangle is isosceles. I have seen many different proofs about isosceles and congruent triangles, bu
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Question 252012: I have a triangleABC that is bisected by a lineAD. I am supposed to prove the triangle is isosceles. I have seen many different proofs about isosceles and congruent triangles, but none that is like This: Given angle BAD is congruent to CAD and given AD is perpendicular to BC, prove ABC is isosceles Found 2 solutions by richwmiller, solver91311:Answer by richwmiller(17219) (Show Source):
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So we have three triangles
Two right triangles BDA and CDA and one to be proven is isosceles ABC.
What do we know?
BD=DC because AD bisects the triangle ABC
Angle BDA and CDA are right angles because BC is perpendicular to AD
So we have two legs of one right triangle congruent to two legs of another right triangle.
Since angle BAD is congruent to angle CAD so we have two angles of triangle BDA and CDA
Therefore the other angles must also be congruent and so the two triangles are congruent and there the third sides are congruent.
But since side AB is congruent to side AC then triangle ABC must be isosceles.
