SOLUTION: Please help. I have such a hard time with proofs. I need to prove that an angle bisector of an equilateral triangle is perpendicular to the opposite side. how would i do this?

Question 197375: Please help. I have such a hard time with proofs. I need to prove that an angle bisector of an equilateral triangle is perpendicular to the opposite side. how would i do this?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

You know that an equilateral triangle is also an equiangular triangle. You know that the sum of the interior angles of any triangle is 180 degrees, so if there are three equal angles, each must measure 60 degrees.

When you construct the angle bisector of one of the angles of an equilateral triangle, you form two new triangles. Consider one of these new triangles: One of the angles is (by definition of bisector) half the measure of the bisected angle, so 30 degrees. Another angle in this new triangle is one of the angles of the original equilateral triangle, hence it measures 60 degrees. That leaves exactly 90 degrees for the measure of the angle formed by the bisector and the opposite side. Therefore, the bisector is perpendicular to the opposite side.

Q.E.D.

John

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