𝐴𝐵𝐶𝐷 is a parallelogram, and the measure of angle 𝐷𝐴𝐶 equals five 𝑥 degrees, the measure of angle 𝐵𝐴𝐶 equals three 𝑥 degrees, and the measure of angle 𝐴𝐵𝐶 equals four 𝑥 degrees. Determine the measure of angle 𝐵𝐶𝐷 and the measure of angle 𝐴𝐷𝐶.
When we’re given a question like this, the best way to begin is by filling in the angle information that we’re given. Angle 𝐷𝐴𝐶 is five 𝑥 degrees, angle 𝐵𝐴𝐶 is three 𝑥 degrees, and angle 𝐴𝐵𝐶 is four 𝑥 degrees. The two angles that we’re asked to find out are the angle 𝐵𝐶𝐷 and the angle 𝐴𝐷𝐶. We’re told that 𝐴𝐵𝐶𝐷 is a parallelogram, which means that the opposite sides are parallel and congruent. In order to answer this question with the angles in the parallelogram, we’ll need to recall an important fact about the angles in a parallelogram, which is that opposite angles in a parallelogram are equal or congruent.
We could therefore immediately look at our diagram and see that the angle at 𝐵 must be equal to the angle at 𝐷, which is four 𝑥 degrees. In the same way, this whole angle at 𝐷𝐴𝐵, which is made up of five 𝑥 degrees and three 𝑥 degrees, will be equal to this angle at 𝐷𝐶𝐵, meaning that it is also eight 𝑥 degrees. Looking at the diagram, we can see that we’ve found the two unknown angles in terms of 𝑥. However, we might wonder if we could find the value of 𝑥 and give these answers in terms of a numerical value.
There are in fact two different ways in which we could find the value of 𝑥. The first method involves using a property of parallelograms, which is that any two consecutive angles at a parallelogram are supplementary, which means they add up to 180 degrees. So let’s take the pair of angles of four 𝑥 degrees and eight 𝑥 degrees, and we would set this equal to 180 degrees. Simplifying the left-hand side, we have 12𝑥 degrees is equal to 180 degrees. We will then divide both sides by 12 to find that 𝑥 is equal to 15 degrees.
The alternative method we could use is to recall that the angles in any quadrilateral add up to 360 degrees. We could then add our four angles, four 𝑥, eight 𝑥, four 𝑥, and eight 𝑥 degrees, and set this equal to 360 degrees. Simplifying this would give us 24𝑥 is equal to 360 degrees. Dividing by 24 then would give us once again that 𝑥 is equal to 15 degrees. But of course, we weren’t asked to simply find 𝑥. We were asked for two angle measurements. For the angle 𝐵𝐶𝐷, we established that this was eight 𝑥 degrees, and we worked out that 𝑥 was 15. So, we calculate eight multiplied by 15. And that is 120 degrees. Angle 𝐴𝐷𝐶 we worked out as four 𝑥 degrees, so this time we’re multiplying four by 15, which is 60 degrees.
And therefore, we have the answers for the two angle measurements: 120 degrees and 60 degrees.