In the figure below, given that 𝐴𝐵𝐶𝐷 is a parallelogram, determine the measure of angle 𝐻𝐷𝐶.
The angle that we’re looking for, angle 𝐻𝐷𝐶, is the angle marked in orange. It is the interior angle of a triangle that has been created by connecting a vertex of the parallelogram to one of the other sides. We’ve been given the size of one of the angles in a parallelogram, 91 degrees, and the size of the angle that the line 𝐷𝐻 makes with the side 𝐵𝐶, 132 degrees.
We haven’t been given any of the other angles in the triangle. So let’s see if we can calculate them. Angle 𝐷𝐻𝐶 is on a straight line with the angle of 132 degrees. And therefore their sum must be 180 degrees. This means we can calculate angle 𝐷𝐻𝐶 by subtracting 132 degrees from 180 degrees. It’s 48 degrees.
So now we know one of the angles inside the triangle 𝐷𝐻𝐶. Let’s see if we can calculate the other angle. Angle 𝐷𝐶𝐻 is one of the interior angles not just in the triangle, but also in the parallelogram 𝐴𝐵𝐶𝐷. So we need to use properties of the angles in parallelograms to help find it.
One of the key properties of angles in parallelograms is that consecutive angles, i.e. those next to each other, are supplementary, which means they sum to 180 degrees. Therefore, angle 𝐷𝐶𝐻 can be found by subtracting 91 degrees from 180 degrees. Angle 𝐷𝐶𝐻 is 89 degrees.
Now we know both of the other two angles in the triangle 𝐷𝐻𝐶, which means we can find the angle we were asked to calculate. The angle sum in a triangle is always one 180 degrees. So we find the final, angle 𝐻𝐷𝐶, by subtracting 89 degrees and 48 degrees from 180 degrees. The measure of angle 𝐻𝐷𝐶 is 43 degrees.