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Special Isosceles Triangle Properties - Problem 2
The altitude of an isosceles triangle is the angle bisector of the vertex it involves. So, given the measure of a base angle, it is possible to find the measure of the half angle.
By the triangle angle sum theorem, the sum of the three angles is 180 °. Since the base angles of an isosceles triangle are congruent, the third angle's measure is 180° - twice the measure of the given base angle. Then, since the altitude bisects this third angle, the angle formed by the altitude and one of the legs is half of this value.
Let’s look at another problem where we have to take what we know about isosceles triangles and this segment going from the vertex to the base. It says if angle A is 40 degrees, find angle ABD.
Well let’s start by drawing in what we know. We know that this angle right here is 40 degrees. Our theorem about isosceles triangle says that if these two legs are congruent, then the base angles must be congruent. So I’m going to write that angle C is 40 degrees.
If I look at this segment right here, it appears as though it’s a median. I know that because this point bisects the base. So what I can say is that this must also be the angle bisector because this line segment does three things; it bisects the vertex angle, it bisects the base also known as a median and it’s an altitude, but the altitude doesn’t really matter as much here. Actually it does Mr. McCall, since this is an altitude, we can say that these three angles must sum to 180 degrees which means that this angle right here must be 50 degrees.
Another way to look at this, let’s say you forgot about the altitude is to say triangle angle sum. Well if these two are both 40, we know that the sum of these must be 180, so this large vertex angle must be 100 degrees. Since this angle is bisected, we know that each of these must be 50 degrees. So there’s two ways to do that, one using the altitude and one using your triangle angle sum.