Question Video: Finding the Measures of the Angles of an Isosceles Triangle Mathematics • 11th Grade

Video Transcript

Find the measures of the angles of triangle 𝐴𝐵𝐶.

Let’s begin by considering the figure that we are given. We can see that there are two parallel lines marked on the diagram, the ray 𝐴𝐷 and the line segment 𝐵𝐶. And we also have two congruent line segments, which are 𝐴𝐶 and 𝐵𝐶. Let’s consider the angle 𝐷𝐴𝐶, which is created by transversal 𝐴𝐶 between the two parallel lines. The angle 𝐴𝐶𝐵 is alternate to the angle 𝐷𝐴𝐶, and so these will both have a measure of 39.5 degrees.

Next, we can return to the two congruent line segments. These are part of the triangle 𝐴𝐵𝐶, and so, by definition, this is an isosceles triangle. That’s because we can recall that an isosceles triangle has two congruent sides. And by the isosceles triangle theorem, we know that the angles opposite the congruent sides in an isosceles triangle are congruent.

In triangle 𝐴𝐵𝐶, these congruent base angles will be angle 𝐵𝐴𝐶 and angle 𝐴𝐵𝐶. As these will both have the same measure, we can define them with the same letter, for example, the letter 𝑦. Since we know that the internal angle measures in a triangle sum to 180 degrees, we can write that the three angles of 𝑦, 𝑦, and 39.5 degrees sum to give 180 degrees. And simplifying the 𝑦 plus 𝑦 gives us two 𝑦. We can then subtract 39.5 degrees from both sides to give us that two 𝑦 equals 140.5 degrees. Finally, dividing both sides by two, we have that 𝑦 equals 70.25 degrees.

We have therefore found all three of the required angle measures in triangle 𝐴𝐵𝐶. The measure of angle 𝐴𝐵𝐶 is 70.25 degrees. The measure of angle 𝐵𝐴𝐶 is also 70.25 degrees. And the first angle that we determined, the measure of angle 𝐴𝐶𝐵, is 39.5 degrees.