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Given the π΄π΅πΆπ·πΈ is a regular pentagon, find the measure of angle π΄π΅π.
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Firstly, letβs mark the angle that weβre looking to find on the diagram.
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π΄π΅π is the angle formed when you travel from π΄ to π΅ to π.
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So itβs this angle here, marked in orange.
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We can see that the diagram consists of a triangle, triangle π΅ππ, and then the pentagon π΄π΅πΆπ·πΈ which weβre told is regular.
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Letβs think about how weβre going to approach this problem.
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The angle that weβre looking for, angle π΄π΅π, sits on a straight line with two other angles: angle ππ΅π inside the triangle and angle π΄π΅πΆ inside the pentagon.
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If we can work out these two other angles, then we can calculate angle π΄π΅π using the fact that angles on a straight line sum to 180 degrees.
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Letβs think about the angle in the triangle first of all.
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Remember, the angle sum in a triangle is always 180 degrees.
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And as weβve been given the measures of the other two angles, we can calculate the third.
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So angle ππ΅π is 180 degrees minus 79 degrees minus 64 degrees which is 37 degrees.
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Next, letβs think about the angle in the pentagon, angle π΄π΅πΆ.
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A key fact about polygons is that the sum of their interior angles can be calculated by multiplying 180 by π minus two, where π represents the number of sides in the polygon.
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Our polygon is a pentagon which has five sides.
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Therefore, the sum of its interior angles is found by multiplying 180 by three which is 540.
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Now this is the sum of all of the interior angles in the pentagon, not the size of each individual angle.
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The key piece of information given in the question is that π΄π΅πΆπ·πΈ is a regular pentagon.
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Which means that all of its interior angles are the same size.
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Therefore, each interior angle can be found by dividing the sum by five.
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540 divided by five which is 108 degrees.
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So now we know the size of angle π΄π΅πΆ and the size of angle ππ΅π.
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Remember, these two angles sit on a straight line with angle π΄π΅π, which weβre looking to calculate.
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So angle π΄π΅π is equal to 180 degrees minus 108 degrees minus 37 degrees.
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The measure of angle π΄π΅π is 35 degrees.