Question Video: Determining Which Inequality Applies to the Internal and External Angles of a Triangle Mathematics

Video Transcript

Which of the following inequalities is correct? (A) The measure of angle 𝐴𝐵𝐶 is less than the measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐴𝐶𝐵. Or (B) the measure of angle 𝐴𝐵𝐶 is greater than that of angle 𝐵𝐴𝐶 which is greater than the measure of angle 𝐴𝐶𝐵. (C) The measure of angle 𝐵𝐴𝐶 is greater than that of 𝐴𝐵𝐶 which is greater than that of 𝐴𝐶𝐵. Or option (D), the measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐴𝐵𝐶 is less than that of 𝐴𝐶𝐵. Or finally, the measure of angle 𝐴𝐶𝐵 is less than the measure of angle 𝐴𝐵𝐶 which is less than the measure of angle 𝐵𝐴𝐶.

We note that each of the five options involves the measures of the three angles 𝐴𝐵𝐶, 𝐵𝐴𝐶, and 𝐴𝐶𝐵 and that we’re given the measure of one of these. That’s angle 𝐴𝐶𝐵, which is 29 degrees. And so to answer the question, we need to find the measures of the two interior angles 𝐵𝐴𝐶 and 𝐴𝐵𝐶.

We can find the measure of angle 𝐵𝐴𝐶 by noting that the measures of the angles that make a straight line sum to 180 degrees. And so we have 109 degrees plus the measure of angle 𝐵𝐴𝐶 equals 180 degrees. Subtracting 109 degrees from both sides gives us the measure of angle 𝐵𝐴𝐶 equals 71 degrees. Next, we can find the measure of angle 𝐴𝐵𝐶 by recalling that the measure of the interior angles of a triangle sum to 180 degrees. Thus, we have 71 degrees plus the measure of angle 𝐴𝐵𝐶 plus 29 degrees equals 180 degrees.

Now, subtracting 71 and 29 degrees from both sides, we have the measure of angle 𝐴𝐵𝐶 equals 80 degrees. Since 80 is greater than 71, which in turn is greater than 29, we see that the measure of angle 𝐴𝐵𝐶 is greater than the measure of angle 𝐵𝐴𝐶 which is greater than the measure of angle 𝐴𝐶𝐵. And this corresponds to option (B), so option (B) is correct.