Question Video: Finding the Measure of an Angle in an Isosceles Trapezoid Using Its Properties Mathematics • 11th Grade

The speaker shown is an isosceles trapezoid. If ๐‘šโˆ ๐น๐ฝ๐ป = 82ยฐ, find ๐‘šโˆ ๐น๐บ๐ป.

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Video Transcript

The speaker shown is an isosceles trapezoid. If the measure of angle ๐น๐ฝ๐ป equals 82 degrees, find the measure of angle ๐น๐บ๐ป.

Letโ€™s highlight this trapezoidal shape and include the fact that the angle ๐น๐ฝ๐ป is 82 degrees. We can recall that a trapezoid or a trapezium is a quadrilateral with one pair of parallel sides. Further, weโ€™re told that this speaker is an isosceles trapezoid. This means that the nonparallel sides in this quadrilateral are congruent. So here, the side ๐น๐ฝ will be equal in length to the side ๐บ๐ป.

In order to find this unknown angle of ๐น๐บ๐ป, weโ€™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 ๐น๐ฝ๐ป will be the same as angle ๐บ๐ป๐ฝ. And the upper base angles are congruent. So angle ๐บ๐น๐ฝ is congruent to angle ๐น๐บ๐ป. We could find the measure of angle ๐น๐บ๐ป 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 ๐น๐บ๐ป, plus this lower base angle of ๐น๐ฝ๐ป must be equal to 180 degrees. We can plug in the information that angle ๐น๐ฝ๐ป is 82 degrees. And then subtracting 82 degrees from both sides of the equation would give us that the measure of angle ๐น๐บ๐ป 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 ๐‘ฅ, 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 ๐น๐บ๐ป is 98 degrees.

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