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

In the figure, 𝐴𝐢 = 𝐴𝐡, line segment 𝐴𝐢 βˆ₯ line segment 𝐷𝐹, and line segment 𝐴𝐡 βˆ₯ line segment 𝐷𝐸. If π‘šβˆ π΄π΅πΆ = 30Β°, find π‘šβˆ πΈπ·πΉ.

02:29

Video Transcript

In the following figure, 𝐴𝐢 equals 𝐴𝐡, line segment 𝐴𝐢 is parallel to line segment 𝐷𝐹, and line segment 𝐴𝐡 is parallel to line segment 𝐷𝐸. If the measure of angle 𝐴𝐡𝐢 equals 30 degrees, find the measure of angle 𝐸𝐷𝐹.

Let’s begin this question by noting that we have two congruent line segments, as the line segments 𝐴𝐢 and 𝐴𝐡 are congruent. In fact, this also means that the larger triangle, triangle 𝐴𝐡𝐢, is an isosceles triangle. So, given the information that the measure of angle 𝐴𝐡𝐢 is 30 degrees, then we know that the measure of angle 𝐴𝐢𝐡 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 𝐸𝐷𝐹, which is in the smaller triangle. Although we don’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 𝐴𝐡 and 𝐷𝐸 and the transversal of line segment 𝐢𝐡, we can determine that the measure of angle 𝐷𝐸𝐹 is also 30 degrees, as these angles are corresponding. Then, using the other pair of parallel lines segments, 𝐴𝐢 and 𝐷𝐹, with the same transversal, we can determine that the measure of angle 𝐷𝐹𝐸 is 30 degrees.

Finally, we can observe that the two angles we have calculated along with the angle 𝐸𝐷𝐹, whose measure we need to calculate, are all contained in the triangle 𝐸𝐷𝐹. 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 𝐸𝐷𝐹 is 120 degrees.

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