Video: Using the Properties of Cyclic Quadrilaterals to Verify Whether a given Quadrilateral Is Cyclic

Is 𝐴𝐡𝐢𝐷 a cyclic quadrilateral?

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Video Transcript

Is 𝐴𝐡𝐢𝐷 a cyclic quadrilateral?

We know that the opposite angles in a cyclic quadrilateral sum to 180 degrees. In this question, we will consider the opposite angles 𝐡 and 𝐷. It follows that if these two angles sum to 180 degrees, then angles 𝐴 and 𝐢 must also sum to 180 degrees, as the four angles inside a quadrilateral sum to 360 degrees.

We can begin this question by recalling that the three angles inside any triangle sum to 180 degrees. This means that angle 𝐡 plus 48 degrees plus 29 degrees must equal 180 degrees. 48 plus 29 is equal to 77. Subtracting 77 degrees from both sides of this equation gives us angle 𝐡 is equal to 103 degrees. We can now go back to our statement about a cyclic quadrilateral. We know that angle 𝐡 is equal to 103 degrees and angle 𝐷 is equal to 77 degrees. These two indeed sum to 180 degrees.

As previously mentioned, if angle 𝐡 plus angle 𝐷 sum to 180 degrees, then angle 𝐴 plus angle 𝐢 must also sum to 180 degrees. This is because the four angles inside the quadrilateral 𝐴𝐡𝐢𝐷 must sum to 360 degrees. We can therefore conclude that yes, 𝐴𝐡𝐢𝐷 is a cyclic quadrilateral.

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