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Isosceles Triangles - Problem 2

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Recall that the sum of a linear pair of angles is 180°. As a result, the interior angle of a triangle given an exterior angle is 180° minus the measure of the exterior angle.

Also recall that an isosceles triangle has two congruent sides and two congruent base angles. As a result, if one base angle is known, it is possible to solve for the measure of the vertex angle using the triangle angle sum theorem. Since the value of the two base angles are known (since they are congruent, they have the same measure), the value of the vertex angle is 180° minus two times the value of the base angle.

If we look at this problem right here, we see that we have an isosceles triangle. What you’re probably thinking at home is, “Mr. McCall that is not an isosceles triangle.” At which point I’d say, thanks for your opinion, I know I can’t draw, but two key things. One it’s not drawn to scale and two you can never assume anything just based on a drawing. What we do know from this picture is that we have an isosceles triangle with sides that are congruent and base angles that are congruent.

So I’m going to go ahead and I’m going to mark that these two angles must be congruent. So the question how can use just one number, 120 degrees and find out what x is? The first step is to say 120 degrees and this angle form a linear pair which means they sum to 180 degrees. This is 120, I know this angle right here has to be 60 degrees.

I know that these two base angles must be congruent which means this angle right here must also be 60 degrees, and I can see right there that 60 plus 60 plus x has to be 180, which means x must be 60 degrees.

So I was able to solve that using just mental Math, now one other key thing, notice that all three angles are 60 degrees, giving us an equilateral triangle. Something I like to throw on my true and false section of the test for this section is an equilateral triangle and isosceles triangle? And the answer is yes, because an equilateral triangle has at least two sides that are congruent, so remember that when you’re taking your quiz.

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