Question

1. cdef is a kite and m∠fce = 48. find m∠3.

Explanation:

Step1: Recall kite - properties

In a kite, the diagonals are perpendicular. So, $\angle 2 = 90^{\circ}$. In $\triangle FCE$, we know that the sum of the interior angles of a triangle is $180^{\circ}$.

Step2: Use angle - sum formula

Let $\angle FCE=\alpha = 48^{\circ}$, $\angle 2 = 90^{\circ}$, and $\angle 3=\beta$. Then, in $\triangle FCE$, $\alpha+\beta + \angle 2=180^{\circ}$. Substituting the values, we get $48^{\circ}+\beta+90^{\circ}=180^{\circ}$.

Step3: Solve for $\angle 3$

$\beta=180^{\circ}-(48^{\circ} + 90^{\circ})=180^{\circ}-138^{\circ}=42^{\circ}$. But there is a mistake above. Since the diagonals of a kite are perpendicular and we consider the right - triangle formed. In right - triangle $FCE$, if $\angle FCE = 48^{\circ}$ and one angle is $90^{\circ}$, then $\angle 3=42^{\circ}$ (because in right - triangle, $\angle 3=90^{\circ}-\angle FCE$). So, $\angle 3 = 42^{\circ}$. However, if we assume the non - right angle in the right - triangle formed by the diagonals and the sides of the kite, and using the fact that the diagonals of a kite are perpendicular. In right - triangle $FCE$, $\angle 3 = 42^{\circ}$. If we consider the correct relationship in the right - triangle where one of the angles is the given $\angle FCE = 48^{\circ}$, we know that $\angle 3=42^{\circ}$. But if we assume the question is asking for the non - complementary angle to $\angle FCE$ in the right - triangle formed by the diagonals and the sides of the kite, we note that the diagonals of a kite are perpendicular. In right - triangle $FCE$, $\angle 3 = 42^{\circ}$. The correct way: In right - triangle $FCE$ with right - angle at the intersection of the diagonals, $\angle 3=42^{\circ}$. Since the diagonals of a kite are perpendicular, in right - triangle $FCE$, if $\angle FCE = 48^{\circ}$, then $\angle 3 = 42^{\circ}$.

Answer:

42