In the figure, 𝐴𝐵𝐶𝐷 and 𝐶𝐵𝐻𝑂 are parallelograms. Find the measure of obtuse angle 𝐴𝐵𝐻.
So here we have two parallelograms. We have 𝐴𝐵𝐶𝐷 above the line 𝐶𝐿 and we have 𝐶𝐵𝐻𝑂 below the line 𝐶𝐿. In a parallelogram, we can remember that we have two pairs of opposite sides parallel. So 𝐴𝐷 is parallel to 𝐶𝐵 and 𝐶𝐵 is parallel to 𝑂𝐻. In both parallelograms, the opposite sides are also parallel. We’re asked to find the measure of angle 𝐴𝐵𝐻. And as we’re told that it’s the obtuse angle, an angle we’re looking for is here in orange.
In order to find the obtuse angle 𝐴𝐵𝐻, we can start by finding the reflex angle of 𝐴𝐵𝐻 using what we know about the angles in a parallelogram. We can recall that opposite angles in a parallelogram are equal. So let’s focus on 𝐴𝐵𝐶𝐷. If opposite angles are equal, then the angle at 𝐶 will also be 72 degrees. Using the fact that the angles in a quadrilateral add up to 360 degrees means that we can also find the measurements of 𝐵 and 𝐷. We subtract two lots of 72 degrees from 360 degrees, which gives us 216 degrees. Since we know that opposite angles 𝐵 and 𝐷 are equal in size, then we have 216 degrees, giving us 108 degrees for each of these angles.
Let’s take our second parallelogram 𝐶𝐵𝐻𝑂. Once again, we know that opposite angles are equal. And the angle opposite to 𝐶 is 𝐻, meaning that this must also be 51 degrees. Subtracting these two angles from 360 degrees tells us that the sum of angle 𝐵 and angle 𝑂 is 258 degrees. And so the angle at 𝐵, or more specifically angle 𝐶𝐵𝐻, is 129 degrees. Finally, we can find this obtuse angle of 𝐴𝐵𝐻 by using the fact that the angles about a point sum to 360 degrees. Therefore, the obtuse angle 𝐴𝐵𝐻 is equal to 360 degrees subtract 108 degrees subtract 129 degrees. Evaluating this gives us the answer that the obtuse angle 𝐴𝐵𝐻 is 123 degrees.