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 𝑥. 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 𝑥 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 𝑥 plus 𝑦. We can use this result to form an
equation for our triangle. Summing the two interior angle
measures gives 𝑥 plus 50 degrees. And we equate this to the measure
of the opposite exterior angle, which is 105 degrees. 𝑥 is therefore equal to 105
degrees minus 50 degrees, which is 55 degrees. Remember, 𝑥 represents the measure
of angle 𝐴. So we’ve found that the measure of
the interior angle 𝐴 is 55 degrees.
