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Is π΄π΅πΆπ· a cyclic quadrilateral?
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We can remember that a cyclic quadrilateral is a quadrilateral which has all four vertices inscribed on a circle.
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We can prove if a quadrilateral is cyclic or not by checking some angle properties.
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Given that we have the diagonals marked on this quadrilateral, then that might give us a clue.
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We can check that if an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic.
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In the figure, we are given the measure of this angle π΄π·π΅.
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This is an angle, which is made from a diagonal and side.
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The angle which is created by the other diagonal and the opposite side would be here, angle π΄πΆπ΅.
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If we could demonstrate that the measure of this angle was the same as the measurement of angle π΄π·π΅, then we would have a cyclic quadrilateral.
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Letβs see if we can calculate this angle measure.
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Letβs use the fact that itβs part of this triangle πΈπΆπ΅ to help.
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We should notice that we have a right angle here at angle π΄πΈπ΅.
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And because the angles on a straight line sum to 180 degrees, then we know that the angle measure of π΅πΈπΆ is also 90 degrees.
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So now within this triangle π΅πΈπΆ, we can use the fact that the interior angles of triangle add up to 180 degrees.
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And so weβll have 63 degrees plus 90 degrees plus the measure of angle π΅πΆπΈ must be equal to 180 degrees.
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Adding 63 degrees and 90 degrees gives us 153 degrees, and subtracting 153 degrees from both sides gives us that the measure of angle π΅πΆπ΄ is 27 degrees.
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So now if we compare the angles made at the diagonals, we have this angle measure of 27 degrees and this angle of 38 degrees.
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Of course, 27 degrees is not equal to 38 degrees.
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Therefore, an angle created by a diagonal and side is not equal to an angle created by the other diagonal and opposite side.
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And so π΄π΅πΆπ· is not a cyclic quadrilateral.
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And so we can give the answer no.
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There is an alternative angle pair that we could also have checked.
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The angle πΆπ΅π· is an angle made by a diagonal and side.
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The angle created by the other diagonal and opposite side would be this angle at πΆπ΄π·.
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We could have established that this angle at π΄πΈπ· is a right angle and then use the fact that the angles in a triangle add up to 180 degrees to work out the unknown angle.
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We would have then been able to calculate that the measure of angle πΆπ΄π· is 52 degrees.
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This time, we would be able to show that 63 degrees is not equal to 52 degrees.
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And so this would show that π΄π΅πΆπ· is not a cyclic quadrilateral.
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But we donβt need to show that there are two angle pairs which are not equal.
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Itβs sufficient just to have one pair of angles at the diagonals which are not equal in order to prove that the quadrilateral is not cyclic.