Video Transcript
In the two triangles shown, the
measure of angle 𝐷𝐹𝐸 equals 30 degrees and the measure of angle 𝐷𝐸𝐹 equals 42
degrees. What is the measure of angle
𝐴?
Before we begin answering the
question, let’s have a quick look at the diagram that we’re given. We have this larger triangle 𝐴𝐵𝐶
on the outside and then we have a smaller triangle, triangle 𝐷𝐸𝐹, within it. We should also notice the markings
on the line indicating that we have two pairs of parallel lines.
Now that we’ve had a look at the
diagram, it would be a good idea to fill in the angle information that we’ve been
given. The measure of angle 𝐷𝐹𝐸 is 30
degrees, and the measure of angle 𝐷𝐸𝐹 is 42 degrees. We are asked to find the measure of
angle 𝐴.
Once we’ve established where angle
𝐴 is, we might be a little confused. After all, angle 𝐴 is part of the
larger triangle 𝐴𝐵𝐶, and the angles that were given are part of the smaller
triangle 𝐷𝐸𝐹. A sensible approach to this might
be to think about what would happen if these pairs of angles were actually the
same. In other words, are these two
triangles similar?
If we want to check if triangle
𝐴𝐵𝐶 is similar to triangle 𝐷𝐸𝐹, then we need to check if the rule AA
applies. The AA rule means that if we find
that two pairs of corresponding angles are equal, then the two triangles are
similar. When we look at the diagram, we can
see that were only given the information about angle 𝐷𝐹𝐸 and angle 𝐷𝐸𝐹. So let’s see if we can use these
parallel lines to help us find some equal angles.
If we take the first pair of
parallel lines 𝐴𝐵 and 𝐷𝐸 and then we consider that the line 𝐶𝐵 is a
transversal of these lines, which means that angle 𝐷𝐸𝐹 and 𝐴𝐵𝐶 would be
corresponding angles. Therefore, they’ll be equal. And they’ll both be an angle of 42
degrees.
We can apply the same reasoning to
the second pair of parallel lines: 𝐴𝐶 and 𝐷𝐹. Once again, we have this
transversal of the line 𝐵𝐶. So angle 𝐴𝐶𝐵 is equal to angle
𝐷𝐹𝐸 as these angles are corresponding. And so this angle at 𝐴𝐶𝐵 must
also be 30 degrees.
We have now established that we
have two pairs of angles equal, which demonstrates that the AA rule applies. And so the two triangles are
similar. Although we didn’t need to prove
that these two triangles are similar, we now know that angle 𝐷 must be the same
size as angle 𝐴.
In order to work out either the
measure of angle 𝐴 or the measure of angle 𝐷, we’ll need to recall that the angles
in a triangle add up to 180 degrees. In order to find the measure of
angle 𝐴 then, we can add together the other two angles of 30 degrees and 42 degrees
and subtract that from 180 degrees. Adding 32 degrees and 42 degrees
gives us 72 degrees. And subtracting that from 180
degrees will give us 108 degrees.
Therefore, we can give our answer
that the measure of angle 𝐴 is 108 degrees.
