Video Transcript
Find the measure of angle 𝐶 plus the measure of angle 𝐷 plus the measure of angle 𝐸.
In this question, we have two triangles. We have this smaller triangle, triangle 𝐴𝐵𝐶, and the larger triangle, 𝐷𝐸𝐹. We’re asked to work out the sum of the measure of three angles: angle 𝐶, angle 𝐷, and angle 𝐸. When we’ve identified these three angles, we can see that, in fact, they’re on the two different triangles. We’re not given any angle measurements on these triangles. But if we look at this angle 𝐹, if for example we knew this angle of 𝐹, then we could work out the sum of 𝐷 plus 𝐸 plus 𝐹.
As we look at the two triangles, we might wonder if there’s a possibility of angle 𝐶 being equal to angle 𝐹. This would happen if the two triangles were similar to each other. Of course, the corresponding pairs of angles would also be equal if the two triangles were congruent, but congruent triangles are the same size and the lengths of these triangles are different. So let’s recall what it means for two triangles to be similar.
Similar triangles have corresponding angles equal and corresponding sides in proportion. Let’s begin by identifying some corresponding sides. We have 𝐴𝐵 and 𝐷𝐸. As we’re thinking about proportions, we can write this as 𝐴𝐵 over 𝐷𝐸. We could also write it as 𝐷𝐸 over 𝐴𝐵. It doesn’t matter. The next pair of corresponding sides will be 𝐵𝐶 and 𝐸𝐹. In the first proportion, we wrote the side of 𝐴𝐵, which is part of triangle 𝐴𝐵𝐶, as the numerator. So we’re going to write that side 𝐵𝐶, part of triangle 𝐴𝐵𝐶, on the numerator as well. This means that the proportion is 𝐵𝐶 over 𝐸𝐹.
The final pair of corresponding sides are 𝐴𝐶 and 𝐷𝐹. So the proportion is 𝐴𝐶 over 𝐷𝐹. Now we need to check if these two triangles are similar. That means we’ll need to check if the proportion of these corresponding sides is equal. We can do this by filling in the numerical values for the lengths of the sides that we’re given. Starting with 𝐴𝐵 over 𝐷𝐸, that will be equal to three over six. 𝐵𝐶 over 𝐸𝐹 is five over 10. And 𝐴𝐶 over 𝐸𝐹 is 6.5 over 13. So are these three proportions equal? Yes, they are because, of course, all of these fractions or proportions would simplify to one-half.
We’ve therefore proved that the side-side-side or SSS rule for similarity will apply. Therefore, triangle 𝐴𝐵𝐶 is similar to triangle 𝐷𝐸𝐹. And that will make finding the sum of the angles much more simple. Because we know that, in similar triangles, the corresponding angles are equal, then there’s a corresponding angle to this angle 𝐶. And it’s angle 𝐹. We were asked to calculate the measure of angle 𝐶 plus 𝐷 plus 𝐸. And that’s going to be equal to the measure of angle 𝐹 plus the measure of angle 𝐷 plus the measure of angle 𝐸. Remember that that’s because we know that the measure of angle 𝐶 and 𝐹 will be the same.
These three angles are all in the same triangle. We know that the angles in a triangle add up to 180 degrees. So the sum of these three angles are 180 degrees. We can give the answer then that the measure of angle 𝐶 plus the measure of angle 𝐷 plus the measure of angle 𝐸 is 180 degrees.
