### Video Transcript

π΄π΅πΆπ· is a parallelogram, and the measure of angle π·π΄πΆ equals five π₯ degrees, the measure of angle π΅π΄πΆ equals three π₯ degrees, and the measure of angle π΄π΅πΆ equals four π₯ degrees. Determine the measure of angle π΅πΆπ· and the measure of angle π΄π·πΆ.

When weβre given a question like this, the best way to begin is by filling in the angle information that weβre given. Angle π·π΄πΆ is five π₯ degrees, angle π΅π΄πΆ is three π₯ degrees, and angle π΄π΅πΆ is four π₯ degrees. The two angles that weβre asked to find out are the angle π΅πΆπ· and the angle π΄π·πΆ. Weβre told that π΄π΅πΆπ· 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, weβll 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 π΅ must be equal to the angle at π·, which is four π₯ degrees. In the same way, this whole angle at π·π΄π΅, which is made up of five π₯ degrees and three π₯ degrees, will be equal to this angle at π·πΆπ΅, meaning that it is also eight π₯ degrees. Looking at the diagram, we can see that weβve found the two unknown angles in terms of π₯. However, we might wonder if we could find the value of π₯ 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 π₯. 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 letβs take the pair of angles of four π₯ degrees and eight π₯ degrees, and we would set this equal to 180 degrees. Simplifying the left-hand side, we have 12π₯ degrees is equal to 180 degrees. We will then divide both sides by 12 to find that π₯ 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 π₯, eight π₯, four π₯, and eight π₯ degrees, and set this equal to 360 degrees. Simplifying this would give us 24π₯ is equal to 360 degrees. Dividing by 24 then would give us once again that π₯ is equal to 15 degrees. But of course, we werenβt asked to simply find π₯. We were asked for two angle measurements. For the angle π΅πΆπ·, we established that this was eight π₯ degrees, and we worked out that π₯ was 15. So, we calculate eight multiplied by 15. And that is 120 degrees. Angle π΄π·πΆ we worked out as four π₯ degrees, so this time weβre multiplying four by 15, which is 60 degrees.

And therefore, we have the answers for the two angle measurements: 120 degrees and 60 degrees.