# Question Video: Finding the Measure of an Angle in a Triangle Given the Corresponding Angleβs Measure in a Similar Triangle Mathematics • 8th Grade

Given that πβ πΈπ΄π· = 80Β° and line segment π΄π· β line segment π΅πΆ, what is πβ πΆπ΅πΈ?

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### Video Transcript

Given that the measure of angle πΈπ΄π· equals 80 degrees and line segment π΄π· is parallel to line segment π΅πΆ, what is the measure of angle πΆπ΅πΈ?

We can begin this problem by adding the measure of angle πΈπ΄π·, which is 80 degrees, to the diagram and note that we are told that line segments π΄π· and π΅πΆ are parallel. We need to find the measure of angle πΆπ΅πΈ. Since we have two parallel lines and the transversal π΄π΅, then the measure of angle πΆπ΅πΈ is simple to calculate. Because these are alternate angles, then the measure of angle πΆπ΅πΈ is equal to the measure of angle πΈπ΄π·, and they will both be 80 degrees.

Although we have found the answer, itβs worth taking a closer look at whatβs going on in this particular diagram. And it involves similar triangles. We can recall that similar triangles have corresponding angles congruent and corresponding sides in proportion. But it might be easy to think that we donβt have enough information to prove that these two triangles are similar. However, letβs consider the angles. If we can prove that all the corresponding pairs of angles in these triangles are congruent, then the triangles are similar. And we can prove this even without knowing that angle πΈπ΄π· is 80 degrees.

We do already know that because of the parallel lines and the transversal, angles πΆπ΅πΈ and π·π΄πΈ will be congruent because these are alternate angles. Angles π΅πΆπΈ and π΄π·πΈ are also alternate angles, and so these are congruent. And finally, angles πΆπΈπ΅ and π·πΈπ΄ are congruent because they are opposite angles. This is sufficient to prove that triangles πΆπ΅πΈ and π·π΄πΈ are similar. And therefore, if we want to find the measure of angle πΆπ΅πΈ, we know that it will be the same as angle π·π΄πΈ. And we were given that angle πΈπ΄π· is 80 degrees.

Notice that it doesnβt matter if we give this angle as angle π·π΄πΈ or πΈπ΄π·, so long as the letter π΄ is in the middle to represent that the vertex of the angle is at point π΄. And because the corresponding angle πΆπ΅πΈ is congruent, it also has a measure of 80 degrees, which confirms the original answer.