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Find the value of 𝑥.
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Let’s have a look at the diagram we’ve been given.
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It consists of a triangle 𝐴𝐵𝐶 inside a circle.
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We’ve been given one angle in the triangle.
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It’s 50 degrees.
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And 𝑥 is the measure of the other angle.
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We’ve also been given the measure of the arc 𝐵𝐶.
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It’s 168 degrees.
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This arc 𝐵𝐶 is the intercepted arc of the inscribed angle 𝐵𝐴𝐶.
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Let’s recall what we know about the measures of intercepted arcs and their inscribed angles.
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We recall that if an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
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So the measure of the angle 𝐵𝐴𝐶 is equal to a half of the measure of the arc 𝐵𝐶.
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We might not be able to work out angle 𝑥 straight away.
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But using this result, we can work out the measure of angle 𝐵𝐴𝐶.
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It’s equal to a half of 168 degrees or a half multiplied by 168 degrees, which is 84 degrees.
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Let’s look at our diagram again.
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We’ve now filled in one more angle in this triangle, which means in triangle 𝐴𝐵𝐶, we know two of the angles.
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And we wish to calculate the third.
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We know that the angle sum in any triangle is 180 degrees.
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So if we know the other two angles, we can subtract them from 180 to find the value of 𝑥.
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𝑥 is equal to 180 minus 84 minus 50, which is 46.
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The value of 𝑥, found by recalling first of all that the measure of an inscribed angle in a circle is one-half the measure of its intercepted arc and then using the fact that the angle sum in a triangle is 180 degrees, is 46.