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Question Video: Finding the Measure of the Interior Angle of a Triangle Mathematics

In the given figure, find the measure of the interior angle ∠𝐴.

02:02

Video Transcript

In the given figure, find the measure of the interior angle 𝐴.

The figure shows a triangle, in which one of the interior angle measures is marked as 50 degrees and one of the exterior angle measures is marked as 105 degrees. The measure of the interior angle 𝐴, which we’ve been asked to calculate, has been labeled as π‘₯. To answer this question, we need to recall the relationship between the measure of any exterior angle in a triangle and the sum of the measures of the two opposite interior angles. It’s this. The measure of any exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. This is derived from the fact that the sum of the measures of the angles in a triangle is 180 degrees and so is the sum of the measures of angles on a straight line.

Hence, the third angle in the triangle, which we’ll call 𝑏, is equal to 180 minus π‘₯ plus 𝑦, using angles in a triangle, and also equal to 180 minus 𝑧, using angles in a straight line. The value of 𝑧 is therefore equal to the value of π‘₯ plus 𝑦. We can use this result to form an equation for our triangle. Summing the two interior angle measures gives π‘₯ plus 50 degrees. And we equate this to the measure of the opposite exterior angle, which is 105 degrees. π‘₯ is therefore equal to 105 degrees minus 50 degrees, which is 55 degrees. Remember, π‘₯ represents the measure of angle 𝐴. So we’ve found that the measure of the interior angle 𝐴 is 55 degrees.

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