Question Video: Finding the Measure of an Inscribed Angle given the Measure of Another Circle’s Inscribed Angle Involving Concentric Circles Mathematics

In the figure, line segment 𝐴𝐸 and line segment 𝐡𝐢 pass through the midpoint of the circles. Given that π‘šβˆ πΉπΈπ· = 50Β° and π‘šβˆ πΆπ΅π΄ = (2π‘₯ βˆ’ 10)Β°, find π‘₯.

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Video Transcript

In the figure, line segment 𝐴𝐸 and line segment 𝐡𝐢 pass through the midpoint of the circles. Given that the measure of angle 𝐹𝐸𝐷 is 50 degrees and the measure of angle 𝐢𝐡𝐴 is equal to two π‘₯ minus 10 degrees, find π‘₯.

We begin by adding the relevant measurements to our diagram. 𝐹𝐸𝐷 is 50 degrees, and 𝐢𝐡𝐴 is two π‘₯ minus 10 degrees. Now, we do know that angles subtended from the same arc are equal. But we also know that angles subtended from arcs with equal measure are equal. And this is really useful when we’re working with a pair of concentric circles, because we’re able to say that the measure of arc 𝐹𝐷 is equal to the measure of arc 𝐢𝐴. They’re both equal to this angle here. Since the measure of those two arcs are equal, then the measure of any angles subtended from the arcs must also be equal. In other words, the measure of angle 𝐹𝐸𝐷 must be equal to the measure of angle 𝐢𝐡𝐴.

And so, we can say that 50 must be equal to two π‘₯ minus 10. Then, we simply have an equation that we can solve for π‘₯. We’ll begin by adding 10 to both sides, giving us 60 equals two π‘₯. And then we divide through by two, giving us 30 is equal to π‘₯, or π‘₯ is equal to 30.

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