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
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.