Given that 𝐵𝐶 equals 𝐴𝐷, 𝐴𝐶 equals 𝐴𝐸, and the measure of angle 𝐶𝐴𝐵 equals 68 degrees, find the measure of angle 𝐸𝐴𝐷.
In this question, we can see that we have two triangles. And we’re asked to find a measure of angle 𝐸𝐴𝐷. So let’s compare our triangles and see if we have enough information to allow us to find the missing angle. Starting with the line 𝐵𝐶 in triangle 𝐴𝐶𝐵, we’re told that it’s equal to the line 𝐴𝐷 in triangle 𝐸𝐴𝐷. We’re also told that there’s another pair of equivalent sides. Side 𝐴𝐶 in triangle 𝐴𝐶𝐵 is equivalent to side 𝐴𝐸 in triangle 𝐸𝐴𝐷. We’re not given any equivalences for the third side in each of the triangles. But let’s have a look at the angles.
We can see that the measure of angle 𝐶𝐵𝐴 is a 90-degree angle. And we can also see another 90-degree angle. That’s the angle 𝐸𝐷𝐴. And since both of these angles are 90 degrees, then we have two equivalent angles. So looking at three pieces of information on the triangles, we have information regarding two sides and another piece of information about one of the angles. We can conclude that our two triangles 𝐴𝐶𝐵 and 𝐸𝐴𝐷 are congruent using the SSA congruency criterion; that’s side, side, angle. Notice that it wouldn’t be the SAS criterion, since the angle isn’t the included angle between the two sides.
Let’s see if using the fact that these triangles are congruent will help us find our missing angle. So looking at our triangles, then the angle that we want to find out, that is, angle 𝐸𝐴𝐷, must be equal to the angle 𝐴𝐶𝐵 in triangle 𝐴𝐶𝐵. We’re not given an angle measurement for this angle. But let’s see if we can work it out. We’re going to use the fact that the angles in a triangle add up to 180 degrees. So the measure of angle 𝐴𝐶𝐵 is equal to 180 degrees subtract 68 degrees subtract 90 degrees. So this simplifies to 22 degrees. So now, we know that the measure of angle 𝐴𝐶𝐵 is 22 degrees. Then the equivalent angle 𝐸𝐴𝐷 is also 22 degrees. So our final answer is the measure of angle 𝐸𝐴𝐷 is 22 degrees.