Question Video: Understanding the Relation between Sides and Corresponding Angles in a Triangle Mathematics • Second Year of Preparatory School

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Which of the following is the correct relation between the angles of triangle 𝐴𝐵𝐶? [A] 𝑚∠𝐴 > 𝑚∠𝐵 > 𝑚∠𝐶 [B] 𝑚∠𝐵 > 𝑚∠𝐴 > 𝑚∠𝐶 [C] 𝑚∠𝐶 > 𝑚∠𝐵 > 𝑚∠𝐴 [D] 𝑚∠𝐴 > 𝑚∠𝐶 > 𝑚∠𝐵 [E] 𝑚∠𝐶 > 𝑚∠𝐴 > 𝑚∠𝐵

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Video Transcript

Which of the following is the correct relation between the angles of triangle 𝐴𝐵𝐶? Option (A) the measure of angle 𝐴 is greater than the measure of angle 𝐵 is greater than the measure of angle 𝐶. Option (B) the measure of angle 𝐵 is greater than the measure of angle 𝐴 is greater than the measure of angle 𝐶. Option (C) the measure of angle 𝐶 is greater than the measure of angle 𝐵 is greater than the measure of angle 𝐴. Option (D) the measure of angle 𝐴 is greater than the measure of angle 𝐶 is greater than the measure of angle 𝐵. Or is it option (E) the measure of angle 𝐶 is greater than the measure of angle 𝐴 is greater than the measure of angle 𝐵?

In this question, we are given the side lengths of a triangle, 𝐴𝐵𝐶. And we are asked to use this to determine the correct relationship between the measure of its interior angles. Since we want to compare the measures of the interior angles of the triangle by using the side lengths, we can recall the angle comparison theorem in triangles. This tells us that in any triangle, the angle opposite the longer side will have measure larger than an angle that is opposite a shorter side. This means that we can compare the measures of the interior angles of a triangle by comparing the lengths of the sides opposite the angle.

In this triangle, we can see that side 𝐵𝐶 is the longest, with length 29.7. We then see that side 𝐴𝐵 is the next longest, with length 21.3. Finally, side 𝐴𝐶 is the shortest, with length 21.

We can then use this, combined with the angle comparison theorem in triangles, to compare the measures of the interior angles. Remember, the angle opposite the longer side will have larger measure. We see that angle 𝐴 is opposite the longest side, so it has the largest measure. And we see that angle 𝐶 is opposite the second-longest side, so it has the second-largest measure. Finally, we can see that angle 𝐵 is opposite the shortest side, so it has the smallest measure. This gives us that the measure of angle 𝐴 is greater than the measure of angle 𝐶 is greater than the measure of angle 𝐵, which we can see is option (D).