Question

quadrilateral cdef is a kite. what is m∠c? m∠c = °

Explanation:

Step1: Recall the angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. In kite $CDEF$, let $m\angle C = x$.

Step2: Set up the equation

We know that $m\angle D=89^{\circ}$, $m\angle F = 111^{\circ}$, and since a kite has two pairs of adjacent - congruent sides, and the non - vertex angles between the pairs of congruent sides are equal. So, $x + 89^{\circ}+111^{\circ}+m\angle E=360^{\circ}$. Also, in a kite, the vertex angles are bisected by the diagonal, and the non - vertex angles are related to the sum of angles in the quadrilateral. Since the sum of the interior angles of a quadrilateral is $360^{\circ}$, we have $x+89^{\circ}+111^{\circ}+m\angle E = 360^{\circ}$. And because of the properties of a kite, we can also note that the sum of the two non - vertex angles and the two vertex angles gives the total sum of interior angles. So, $x+89^{\circ}+111^{\circ}+(360^{\circ}-x - 89^{\circ}-111^{\circ})=360^{\circ}$. In a simpler way, we know that $x+89 + 111+m\angle E=360$. Since the sum of the known non - $x$ angles is $89 + 111=200^{\circ}$, then $x+m\angle E=360 - 200=160^{\circ}$. Also, in a kite, the non - vertex angles are equal in pairs. So, $x = 160^{\circ}\div2 = 80^{\circ}$.

Answer:

$80$