# Question Video: Finding Unknown Angle Measures by Using the Properties of Isosceles Triangles Mathematics

In the figure, π΄πΆ = π΄π΅, line segment π΄πΆ β₯ line segment π·πΉ, and line segment π΄π΅ β₯ line segment π·πΈ. If πβ π΄π΅πΆ = 30Β°, find πβ πΈπ·πΉ.

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### Video Transcript

In the following figure, π΄πΆ equals π΄π΅, line segment π΄πΆ is parallel to line segment π·πΉ, and line segment π΄π΅ is parallel to line segment π·πΈ. If the measure of angle π΄π΅πΆ equals 30 degrees, find the measure of angle πΈπ·πΉ.

Letβs begin this question by noting that we have two congruent line segments, as the line segments π΄πΆ and π΄π΅ are congruent. In fact, this also means that the larger triangle, triangle π΄π΅πΆ, is an isosceles triangle. So, given the information that the measure of angle π΄π΅πΆ is 30 degrees, then we know that the measure of angle π΄πΆπ΅ is 30 degrees, since, by the isosceles triangle theorem, we know that an isosceles triangle has two congruent angles.

Now, we need to find the measure of angle πΈπ·πΉ, which is in the smaller triangle. Although we donβt yet have any angle measures for this triangle, we can calculate some of them by using the properties of parallel lines. Using the two parallel line segments π΄π΅ and π·πΈ and the transversal of line segment πΆπ΅, we can determine that the measure of angle π·πΈπΉ is also 30 degrees, as these angles are corresponding. Then, using the other pair of parallel lines segments, π΄πΆ and π·πΉ, with the same transversal, we can determine that the measure of angle π·πΉπΈ is 30 degrees.

Finally, we can observe that the two angles we have calculated along with the angle πΈπ·πΉ, whose measure we need to calculate, are all contained in the triangle πΈπ·πΉ. And since the interior angle measures in a triangle sum to 180 degrees, we know that these three angle measures sum to 180 degrees. We can simplify this equation and then subtract 60 degrees from both sides to give us the answer that the measure of angle πΈπ·πΉ is 120 degrees.