# Question Video: Proving the Angle Sum of a Triangle Mathematics

In the given figure, if π΄πΆβ₯π·πΈ, πβ π΄π΅πΈ = 55Β°, and πβ πΆ = 75Β°, find the measure of β π΄π΅πΆ.

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### Video Transcript

In the given figure, if π΄πΆ is parallel to π·πΈ, the measure of angle π΄π΅πΈ equals 55 degrees, and the measure of angle πΆ equals 75 degrees, find the measure of angle π΄π΅πΆ.

The first piece of information that we are given here is that π΄πΆ is parallel to π·πΈ. We can also fill in the two angle measures that we are given that the measure of angle π΄π΅πΈ is 55 degrees and the measure of angle πΆ is equal to 75 degrees. We can then establish that the angle that we wish to calculate is here, the measure of angle π΄π΅πΆ.

So there are a couple of ways in which we could find the measure of this unknown angle. But both methods will use the fact that we have these parallel line segments. By using the parallel lines π΄πΆ and πΈπ· and the transversal π΅πΆ, we can identify a pair of alternate interior angles. And since alternate interior angles are congruent, we can say that the measure of angle π·π΅πΆ is equal to the measure of angle πΆ. These will both be 75 degrees.

We can notice then that these three angles made at the vertex π΅ all lie on a straight line. We recall that the angle measures on a straight line sum to 180 degrees. Therefore, we can write that the measure of angle π΄π΅πΈ plus the measure of angle π΄π΅πΆ plus the measure of angle π·π΅πΆ must be equal to 180 degrees. We can then simply fill in the angle information that we know. Then, by adding 55 degrees and 75 degrees, we get 130 degrees. We can then simplify this equation by subtracting 130 degrees from both sides, leaving us with the measure of angle π΄π΅πΆ is equal to 50 degrees. And so we have found the value of this unknown angle.