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Is the measure of angle π΄π·π less than, equal to, or greater than the measure of angle π΄πΆπ·?
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Letβs begin by identifying the two angles that are referred to in the question.
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Angle π΄π·π is the angle formed by travelling from π΄ to π· to π.
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So itβs the obtuse angle marked in orange.
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Angle π΄πΆπ· is the angle formed by travelling from π΄ to πΆ to π·.
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So itβs the obtuse angle marked in green.
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Now, letβs consider how to answer this question.
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We havenβt been given the length of any sides in the diagram or any of the angles.
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And therefore, we canβt answer this question by calculating the two angles.
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Instead, we need to think about the relationship between the measures of these two angles based on their positions.
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Letβs consider part of the diagram: triangle π΄πΆπ·.
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With respect to this triangle, we see that angle π΄πΆπ· is an interior angle and angle π΄π·π is an exterior angle.
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We can therefore answer this question using the exterior angle inequality, which tells us about the relationship between the measure of an exterior angle of a triangle and the measures of the two nonadjacent interior angles.
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The exterior angle inequality tells us that in a triangle the measure of an exterior angle is greater than each of the two nonadjacent interior angles.
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By nonadjacent interior angles, we mean the two interior angles of the triangle that donβt lie on the straight line with the given exterior angle β the two angles that Iβve marked with stars.
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One of these is of course the angle weβre interested in β angle π΄πΆπ·.
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Therefore, we can conclude that by the exterior angle inequality, the measure of angle π΄π·π is greater than the measure of angle π΄πΆπ·.