Video Transcript
Find the measure of angle
𝐶𝐴𝐷.
In the figure, we can see a number
of different triangles. And the angle measure that we need
to determine, 𝐶𝐴𝐷, is in the middle triangle. Even if we notice that the angle
measure of 𝐵𝐴𝐸 is given as a right angle, that still won’t be enough information
to help us immediately work out the measure of angle 𝐶𝐴𝐷. Because we would also need to know
the measure of the third angle at this vertex, angle 𝐵𝐴𝐶.
Now, given that we have a few
pieces of information about congruent sides and angles, it might be worth
considering if we have any congruent triangles here. We can note that line segment 𝐴𝐶
and line segment 𝐴𝐷 are marked as congruent. That would indicate that this
central triangle, 𝐴𝐶𝐷, is an isosceles triangle.
We can note that the other two
triangles don’t appear to be isosceles. In fact, both these other triangles
are right triangles, because the measures of angle 𝐴𝐵𝐶 and 𝐴𝐸𝐷 are both marked
on the diagram as right angles of 90 degrees. And in these triangles, we have
another pair of congruent line segments, since 𝐵𝐶 and 𝐸𝐷 are marked as
congruent. So let’s consider what we have
written.
We have that the hypotenuse in each
triangle is congruent, since 𝐴𝐶 and 𝐴𝐷 are the longest sides in each triangle
and are congruent. Each triangle also has a right
angle, and there is another pair of congruent sides. So this is enough to demonstrate
that triangles 𝐴𝐵𝐶 and 𝐴𝐸𝐷 are congruent by the RHS, or right
angle-hypotenuse-side, congruence criterion.
This will now allow us to work out
some more angle measures. Angles 𝐵𝐴𝐶 and 𝐸𝐴𝐷 in the two
triangles are corresponding. And because these are congruent
triangles, then corresponding angles are congruent. They will both have a measure of 27
degrees.
So now we can use the fact that
angle 𝐵𝐴𝐸 has a measure of 90 degrees. To calculate the measure of angle
𝐶𝐴𝐷, we subtract the two angle measures of 27 degrees from 90 degrees, which
gives us the answer that the measure of angle 𝐶𝐴𝐷 is 36 degrees.
