Question Video: Finding the Measure of an Unknown Angle Using the Properties of Parallelograms Mathematics • First Year of Preparatory School

Video Transcript

In the figure, 𝐴𝐵𝐶𝐷 and 𝐶𝐵𝐻𝑂 are parallelograms. Find the measure of angle 𝐴𝐵𝐻.

In this question, we are told that the two quadrilaterals formed here are parallelograms, which we can recall are defined as quadrilaterals with both pairs of opposite sides parallel. That would mean that the line segments 𝐴𝐷 and 𝐵𝐶 are parallel and line segments 𝐵𝐶 and 𝑂𝐻 are parallel. So, we have three parallel line segments here. The other pair of opposite sides in 𝐴𝐵𝐶𝐷, that’s line segments 𝐷𝐶 and 𝐴𝐵, are parallel. And then in parallelogram 𝐶𝐵𝐻𝑂, line segments 𝐶𝑂 and 𝐵𝐻 are parallel.

Now, we are asked to find the measure of angle 𝐴𝐵𝐻. And so, recalling some of the angle properties of parallelograms will be helpful here. Firstly, we have that in a parallelogram, opposite angles are equal in measure. And the sum of any two consecutive interior angles is 180 degrees. So, let’s consider that we are given the measure of angle 𝐷𝐴𝐵. Angle 𝐴𝐵𝐶 would be a consecutive interior angle to angle 𝐷𝐴𝐵. So, the sum of their measures is 180 degrees.

Filling in the information that the measure of angle 𝐷𝐴𝐵 is 72 degrees, we have that 72 degrees plus the measure of angle 𝐴𝐵𝐶 is 180 degrees. And by subtracting 72 degrees from both sides, we have that the measure of angle 𝐴𝐵𝐶 is 108 degrees.

Next, knowing the measure of angle 𝐶𝐵𝐻 would be useful in helping us work out the required angle measure. We can identify that in parallelogram 𝐶𝐵𝐻𝑂 one of the consecutive interior angles to angle 𝐶𝐵𝐻 is angle 𝐵𝐶𝑂, whose measure we are given. Therefore, using the same property as before, we know that these two angle measures must sum to 180 degrees. Filling in the measure of angle 𝐵𝐶𝑂 as 51 degrees, we can determine that the measure of angle 𝐶𝐵𝐻 is 129 degrees.

Now, we can consider the three angle measures about the point 𝐵. We can recall that the sum of the angle measures about a point is 360 degrees. So, the sum of the angle measures of 108 degrees, 129 degrees, and the measure of angle 𝐴𝐵𝐻 is 360 degrees. We can simplify the left-hand side to give 237 degrees plus the measure of angle 𝐴𝐵𝐻 equals 360 degrees. And subtracting 237 degrees from both sides, we have the answer that the measure of angle 𝐴𝐵𝐻 is 123 degrees.