The speaker shown is an isosceles trapezoid. If the measure of angle 𝐹𝐽𝐻 equals 82 degrees, find the measure of angle 𝐹𝐺𝐻.
Let’s highlight this trapezoidal shape and include the fact that the angle 𝐹𝐽𝐻 is 82 degrees. We can recall that a trapezoid or a trapezium is a quadrilateral with one pair of parallel sides. Further, we’re told that this speaker is an isosceles trapezoid. This means that the nonparallel sides in this quadrilateral are congruent. So here, the side 𝐹𝐽 will be equal in length to the side 𝐺𝐻.
In order to find this unknown angle of 𝐹𝐺𝐻, we’ll need to use the properties of isosceles trapezoids. Because we have two legs that are congruent, then we can say that the lower base angles are congruent and the upper base angles are congruent. So angle 𝐹𝐽𝐻 will be the same as angle 𝐺𝐻𝐽. And the upper base angles are congruent. So angle 𝐺𝐹𝐽 is congruent to angle 𝐹𝐺𝐻. We could find the measure of angle 𝐹𝐺𝐻 using a number of different methods. However, perhaps the quickest one is to use this final isosceles trapezoid property.
Any lower base angle is supplementary to any upper base angle. That means these will add to 180 degrees. We know this to be true because we have a pair of parallel sides. Therefore, the lower base angle and the upper base angle would add to 180 degrees. So we can set up an equation that this upper base angle that we wish to find out, angle 𝐹𝐺𝐻, plus this lower base angle of 𝐹𝐽𝐻 must be equal to 180 degrees. We can plug in the information that angle 𝐹𝐽𝐻 is 82 degrees. And then subtracting 82 degrees from both sides of the equation would give us that the measure of angle 𝐹𝐺𝐻 is 98 degrees.
We could, of course, also have answered this in a different way, remembering that the angles in a quadrilateral add to 360 degrees and given the lower base angles are congruent and the upper base angles are congruent. Defining our unknown upper base angles as the letter 𝑥, we could set up an alternative equation. Solving this, we would find that each of the upper base angles is 98 degrees, which also confirms our original answer that the measure of angle 𝐹𝐺𝐻 is 98 degrees.