# Video: Finding the Relation between the Angles of a Triangle

Which inequality is satisfied by this figure? [A] 𝑚∠𝐵 < 𝑚∠𝐵𝐴𝐶 < 𝑚∠𝐶 [B] 𝑚∠𝐷𝐴𝐶 < 𝑚∠𝐵 < 𝑚∠𝐶 [C] 𝑚∠𝐵𝐴𝐶 < 𝑚∠𝐶 < 𝑚∠𝐷𝐴𝐶 [D] 𝑚∠𝐷𝐴𝐶 < 𝑚∠𝐵 < 𝑚∠𝐵𝐴𝐶 [E] 𝑚∠𝐶 < 𝑚∠𝐵 < 𝑚∠𝐵𝐴𝐶

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### Video Transcript

Which inequality is satisfied by this figure? (A) The measure of angle 𝐵 is less than the measure of angle 𝐵𝐴𝐶, which is less than the measure of angle 𝐶. (B) The measure of angle 𝐷𝐴𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐶. (C) The measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐶, which is less than the measure of angle 𝐷𝐴𝐶. Or (D) the measure of angle 𝐷𝐴𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶. And finally (E) the measure of angle 𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶.

For us to be able to order these angles, we’ll need to find the measures of a few of the missing angles. Currently, we don’t know the measure of angle 𝐶 or the measure of angle 𝐵𝐴𝐶. We should see that angle 𝐵𝐴𝐶 and angle 𝐷𝐴𝐶 make a straight line. If these two angles together make a straight line, they are supplementary angles and they’ll add together to be 180 degrees. If we plug in the measure for angle 𝐷𝐴𝐶, which we know is 92 degrees, then the measure of angle 𝐵𝐴𝐶 plus 92 degrees equals 180 degrees. And if we subtract 92 degrees from both sides of this equation, then we find that the measure of angle 𝐵𝐴𝐶 equals 88 degrees. We can add that to our figure.

And from there, we recognize that we have triangle 𝐴𝐵𝐶. And in a triangle, the three angles must add up to 180 degrees. So we say the measure of angle 𝐵 plus the measure of angle 𝐵𝐴𝐶 plus the measure of angle 𝐶 must equal 180 degrees. Angle 𝐵 is 52 degrees, angle 𝐵𝐴𝐶 is 88 degrees, and we want to find angle 𝐶. If we add 52 plus 88, we get 140 degrees. To find the measure of angle 𝐶, we then need to subtract 140 degrees from both sides of our equation to show that the measure of angle 𝐶 is 40 degrees. And then we can add that back to our figure.

What we can do now is list the angles that we know in order from least to greatest. Our smallest angle is angle 𝐶, which measures 40 degrees, followed by the measure of angle 𝐵, which is 52 degrees, followed by the measure of angle 𝐵𝐴𝐶, which is 88 degrees. And the largest of the angles we see in this figure is angle 𝐷𝐴𝐶, which is 92 degrees. Using this compound inequality, we can see which of the answer choices is true. And only option (E) list the angles in correct order, which says the measure of angle 𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶.