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Question Video: Completing a Proof About Triangles Mathematics

In the following triangle 𝐴𝐡𝐢, if π‘šβˆ πΆ = π‘šβˆ πΆπ΄π· = 43Β° and π‘šβˆ π΅ = π‘šβˆ π΅π΄π·, find π‘šβˆ π΅π΄πΆ.

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Video Transcript

In the following triangle 𝐴𝐡𝐢, if the measure of angle 𝐢 equals the measure of angle 𝐢𝐴𝐷 equals 43 degrees and the measure of angle 𝐡 equals the measure of angle 𝐡𝐴𝐷, find the measure of angle 𝐡𝐴𝐢.

We can start this question by identifying the two pairs of congruent angle measures. We have that the measure of angle 𝐢 is equal to the measure of angle 𝐢𝐴𝐷, and those are both 43 degrees. We also have that the measure of angle 𝐡 is equal to the measure of angle 𝐡𝐴𝐷, although we aren’t given an exact measurement for those. We can then identify that the angle that we wish to calculate is that of the measure of angle 𝐡𝐴𝐢, which occurs at the vertex 𝐴 in the larger triangle 𝐴𝐡𝐢.

A property that we can apply in this question is that the sum of the measures of the interior angles in a triangle is 180 degrees. So then, if we consider the large triangle 𝐴𝐡𝐢, we can say that the measure of angle 𝐡𝐴𝐢 plus the measure of angle 𝐡 plus the measure of angle 𝐢 is equal to 180 degrees. From the diagram then, we can observe that the measure of angle 𝐡𝐴𝐢 actually consists of an angle of 43 degrees and the measure of angle 𝐡𝐴𝐷.

Then, we are given in the question that the measure of angle 𝐡 is equal to the measure of angle 𝐡𝐴𝐷. Adding in the measure of angle 𝐢, which is 43 degrees, we can add the left-hand side, and it will be equal to 180 degrees. We can then simplify this by adding the two 43 degrees, which is 86 degrees. And we know that there will be two lots of the measure of angle 𝐡𝐴𝐷. Subtracting 86 degrees from both sides, we have that two times the measure of angle 𝐡𝐴𝐷 is equal to 94 degrees. Finally, dividing through by two, we have that the measure of angle 𝐡𝐴𝐷 is 47 degrees. Now that we know the measure of this angle, we can calculate the measure of angle 𝐡𝐴𝐢. It will be equal to 43 degrees plus 47 degrees, which gives us a final answer of 90 degrees.

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