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Given that π΄π΅πΆπ· is a square and the measure of angle πΈπ·πΆ equals 62 degrees, find the measure of angle π·πΈπ΄.
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Letβs start this question by reminding ourselves of some of the properties of a square.
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We know that a square has four equal length sides and four equal angles that are 90 degrees each.
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So in our diagram, the square π΄π΅πΆπ· is split into three different triangles.
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Weβre told that the measure of angle πΈπ·πΆ is 62 degrees.
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And we need to work out the angle marked pink, that is, the measure of angle π·πΈπ΄.
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If we look at the triangle π΄π·πΈ, we could, in theory, find our missing angle if we knew the two other angles in this triangle.
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So letβs have a look and see if we can calculate either of these two angles.
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Letβs start by looking at the angle π΄π·πΈ.
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We can see that itβs part of the larger angle π΄π·πΆ, which is one of the angles on our square.
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So to find π΄π·πΈ then, we could find our right angle π΄π·πΆ and subtract 62 degrees, which is the angle πΈπ·πΆ.
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So then we can say that the measure of angle π΄π·πΈ must be 28 degrees.
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Next, letβs see if we can calculate the measure of angle π·π΄πΈ.
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We noticed that the line πΆπ΄ is a diagonal of our square.
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We can recall that the diagonals of a square bisect the angles.
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The word bisect means to split exactly in two.
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This means that the 90-degree angles at the vertices of our square will be split into two equal 45-degree angles by the diagonal.
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This means that the measure of angle π·π΄πΈ must be equal to 45 degrees.
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So letβs now return to our triangle π΄π·πΈ and see if we can find the measure of our third missing angle π·πΈπ΄.
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We can use the fact that the angles in a triangle add up to 180 degrees.
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So this means that the measure of angle π΄π·πΈ plus the measure of angle π·π΄πΈ plus the measure of angle π·πΈπ΄ must be equal to 180 degrees.
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We can fill in the angles that we know to give us 28 degrees plus 45 degrees plus the measure of angle π·πΈπ΄ equals 180 degrees.
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So 73 degrees plus the measure of angle π·πΈπ΄ would be a 180 degrees.
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So to find the measure of angle π·πΈπ΄, we would subtract 73 from both sides, giving us a 180 degrees minus 73 degrees, giving us 107 degrees.
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So our final answer for the measure of angle π·πΈπ΄ is 107 degrees.