Question Video: Finding the Measure of an Angle in an Isosceles Triangle given the Measure of Its Vertex Angle Mathematics

Video Transcript

In the displayed model of a house, what angle does the roof make with the horizontal, given that triangle 𝐴𝐸𝐵 is isosceles?

It’s probably most useful to begin this question by highlighting the triangle 𝐴𝐸𝐵 that we want to consider. This triangle forms the roof of this house, and we’re told that it’s isosceles. We should recall that in an isosceles triangle, we have two sides equal in length and two base angles are equal. So in this diagram, the side 𝐴𝐸 is equal to the side 𝐴𝐵 and the angle 𝐴𝐸𝐵 is equal to the angle 𝐴𝐵𝐸.

Now that we’ve had a look at the diagram, let’s focus on what we’re asked, to find the angle that the roof makes with the horizontal. That means that we’re really looking for the angle created by the slope of the roof and the horizontal axis. Either of the two base angles would give us the answer for this. So let’s see if we can work out one of these angles, angle 𝐴𝐵𝐸.

In order to do this, we’ll need to remember an important fact about the angles in a triangle. And that is that the angles in a triangle add up to 180 degrees. This means that we can write that angle 𝐵𝐴𝐸 plus angle 𝐴𝐸𝐵 plus angle 𝐴𝐵𝐸 is equal to 180 degrees. We’re given that angle 𝐵𝐴𝐸 is 107 degrees. So if we subtract 107 degrees from both sides of this equation, we get that angle 𝐴𝐸𝐵 plus angle 𝐴𝐵𝐸 is equal to 73 degrees. As we have an isosceles triangle, we know that our two base angles are equal. So angle 𝐴𝐸𝐵 is equal to angle 𝐴𝐵𝐸.

We could think of this then that two times angle 𝐴𝐵𝐸 is 73 degrees. And so to find angle 𝐴𝐵𝐸, we must divide both sides of this equation by two, which means that the measure of angle 𝐴𝐵𝐸 is 36.5 degrees as a decimal. We can therefore give our answer that the angle that the roof makes with the horizontal is 36.5 degrees.