# Question Video: Finding the Measure of Base Angles in Isosceles Triangles Mathematics • 11th Grade

Find 𝑚∠𝐶𝐵𝐴 and 𝑚∠𝐷𝐴𝐶.

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### Video Transcript

Find the measure of angle 𝐶𝐵𝐴 and the measure of angle 𝐷𝐴𝐶.

The first angle measure that we need to find is that of angle 𝐶𝐵𝐴. This is part of the larger triangle 𝐴𝐵𝐶.

We observe that triangle 𝐴𝐵𝐶 has two congruent line segments marked: 𝐴𝐶 and 𝐵𝐶. That means that this is an isosceles triangle, as an isosceles triangle is defined as a triangle that has two congruent sides. We can then use the isosceles triangle theorem, which states that in an isosceles triangle, the angles opposite the congruent sides are congruent. In triangle 𝐴𝐵𝐶, these two congruent base angles will be angle 𝐶𝐴𝐵 and angle 𝐶𝐵𝐴. If we defined these angles to have a measure of 𝑥 degrees, then we can write that 𝑥 degrees plus 𝑥 degrees plus 38 degrees equals 180 degrees.

Remember, we can set these three angle measures equal to 180 degrees because the sum of the internal measures in a triangle is 180 degrees. We can then simplify this by collecting the like terms and then subtracting 38 degrees from both sides, which gives us that two 𝑥 degrees is equal to 142 degrees. Dividing through by two, we have that 𝑥 is equal to 71 degrees. So now we know the measure of both the base angles in triangle 𝐴𝐵𝐶. They are both 71 degrees. And that’s the answer for the first angle, since the measure of angle 𝐶𝐵𝐴 is 71 degrees.

Now let’s consider the second angle measure, which is the measure of angle 𝐷𝐴𝐶. Observe that we have this angle 𝐷𝐴𝐶 and the angle 𝐷𝐴𝐵, which together form the larger angle 𝐶𝐴𝐵 whose measure we have already calculated as 71 degrees. If we knew the measure of angle 𝐷𝐴𝐵, we could calculate the measure of angle 𝐷𝐴𝐶.

Let’s take a closer look at triangle 𝐴𝐵𝐷. It has three congruent sides. And that means that triangle 𝐴𝐵𝐷 is an equilateral triangle. And we know that an equilateral triangle has all three angles congruent, which means they are all 60 degrees. So, the measure of angle 𝐷𝐴𝐶 is equal to 71 degrees minus 60 degrees, which is 11 degrees.

Therefore, we can give the answers to both parts of the question. The measure of angle 𝐶𝐵𝐴 is 71 degrees, and the measure of angle 𝐷𝐴𝐶 is 11 degrees.