In the following figure, 𝐴𝐶
equals 𝐴𝐵, line segment 𝐴𝐶 is parallel to line segment 𝐷𝐹, and line segment
𝐴𝐵 is parallel to line segment 𝐷𝐸. If the measure of angle 𝐴𝐵𝐶
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 𝐴𝐶 and 𝐴𝐵 are
congruent. In fact, this also means that the
larger triangle, triangle 𝐴𝐵𝐶, is an isosceles triangle. So, given the information that the
measure of angle 𝐴𝐵𝐶 is 30 degrees, then we know that the measure of angle 𝐴𝐶𝐵
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 𝐴𝐵 and 𝐷𝐸 and the transversal of line segment 𝐶𝐵, 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, 𝐴𝐶 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.