Find the measure of angle 𝐴.
Let’s have a closer look at the
figure. There is a triangle 𝐴𝐵𝐶, in
which one of the interior angles, angle 𝐵, has been marked with a small square,
indicating that it is a right angle. We are asked to determine the
measure of angle 𝐴, which is one of the other interior angles in this triangle. The measure of the final interior
angle in the triangle, angle 𝐶, is also unknown.
However, we can work this out if we
consider the position of angle 𝐶 relative to the angle marked as 40 degrees. These angles are nonadjacent angles
formed by the intersection of two straight lines and are therefore vertically
opposite angles. Vertically opposite angles are
congruent. And so we can deduce that the
measure of angle 𝐶 is also 40 degrees.
We now know the measures of two of
the interior angles of triangle 𝐴𝐵𝐶. And we wish to calculate the
measure of the third. We can recall that the sum of the
interior angle measures in any triangle is 180 degrees. So we can form the equation the
measure of angle 𝐴 plus 90 degrees plus 40 degrees equals 180 degrees. Simplifying on the left-hand side
gives the measure of angle 𝐴 plus 130 degrees equals 180 degrees. And hence, the measure of angle 𝐴
is equal to 180 degrees minus 130 degrees, which is 50 degrees.