Question Video: Determining Whether a Given Quadrilateral Is Cyclic Mathematics

Is the quadrilateral ๐ด๐ต๐ถ๐ท cyclic?

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Video Transcript

Is the quadrilateral ๐ด๐ต๐ถ๐ท cyclic?

We can begin by reminding ourselves that a cyclic quadrilateral is a quadrilateral with all four vertices inscribed on a circle. One property of cyclic quadrilaterals is that opposite angles are supplementary. We can check if a quadrilateral is cyclic by checking if opposite angles are supplementary. So letโ€™s have a closer look at the figure that weโ€™re given. The angle which is opposite to this given angle of ๐ด๐ต๐ถ would be the angle at ๐ท. If the angle at ๐ท and this angle at ๐ต add to 180 degrees, then ๐ด๐ต๐ถ๐ท would be cyclic.

So letโ€™s see if we can indeed work out the measure of this angle. We are given that the measure of angle ๐น๐ถ๐ท is 49 degrees. And we can observe that the angle measure at ๐ธ๐ถ๐น is marked as congruent. Itโ€™s also 49 degrees. The total angle measure then of angle ๐ธ๐ถ๐ท will be 49 degrees plus 49 degrees, which is 98 degrees. We can then use the fact that we have a pair of parallel lines. And so angle ๐ด๐ท๐ถ is alternate to angle ๐ธ๐ถ๐ท. Itโ€™s also 98 degrees

Letโ€™s remember that weโ€™re checking if opposite angles are supplementary. Well, when we add together 98 degrees and 82 degrees, we do indeed get 180 degrees. So that means that opposite angles are supplementary. And so we can give the answer yes, since ๐ด๐ต๐ถ๐ท is cyclic.

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