ShmoopTube

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Inscribed Angles, a la Shmoop.

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You’re at a friend’s house with three other friends and you decide to order an extra-large

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pepperoni pizza from Inconvenient Pizzas, Inc…

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…because their commercials say their prices are "Unbeatably Low!"

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Two hours later than promised, your pizza is delivered.

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Like all pizzas from Inconvenient, your pizza is entirely whole and uncut.

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Since there are 360 degrees in a circle and you each want an equal slice, you say:

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"Okay guys, let's each cut our own slice.

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We'll cut an angle that intercepts an arc of 90 degrees."

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You mark four points 90 degrees of arc away from each other.

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Your friend Wise Guy Louie is in charge of cutting the first slice…

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…and just to make things difficult, he cuts it like this.

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"Wise Guy Louie!" you all exclaim.

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"What?" he says. "I did just what you told me."

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As infuriating as he is, Wise Guy Louie is right.

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Arc AB, which is indeed intercepted by the angle he cut, still has a measure of 90 degrees.

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Instead of cutting a central angle as you expected him to, he cut an inscribed angle.

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If an angle has its vertex on a circle

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and has sides containing two chords of the same circle…

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…then the angle is inscribed in that circle.

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While you and your friends understandably find inscribed angles annoying,

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they have some interesting properties.

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We know that Wise Guy Louie's angle intercepted an arc of measure 90 degrees,

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but can we find the measure of the angle? 

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The inscribed angle theorem tells us that the angle measure of AQB equals

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½ times the measure of the arc AB.

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Given that the arc’s measure AB is 90 degrees, we can multiply ½ times 90, to get 45 degrees.

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Well, that’s doesn’t help. Wise Guy Louie still got the biggest slice.

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Turns out inviting him was the most inconvenient thing that’s happened all evening.

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