Question

1. find the values of a and b.
2. find m∠1 and m∠3 in the kite.

Explanation:

Step1: Recall sum of angles in a quadrilateral

The sum of interior angles of a quadrilateral is $360^{\circ}$. For the first - figure (a quadrilateral), we have the equation $a + 113+36 + b=360$.

Step2: Simplify the angle - sum equation

$a + b+149 = 360$, so $a + b=360 - 149=211$. But if the quadrilateral is a trapezoid with one pair of parallel sides, and assuming the angles $a$ and $113^{\circ}$ are supplementary (if the non - parallel sides are transversals to the parallel sides), then $a=180 - 113 = 67^{\circ}$.

Step3: Find the value of $b$

Substitute $a = 67^{\circ}$ into $a + b=211$. Then $b=211 - 67=144^{\circ}$.

Step4: Recall properties of a kite

In a kite, the diagonals are perpendicular, so $\angle 2 = 90^{\circ}$. In the right - triangle formed by the diagonals of the kite, if one of the non - right angles is $39^{\circ}$, for $\angle 1$:
Since the diagonals of a kite are perpendicular and the angles of a triangle sum to $180^{\circ}$, in the upper - triangle of the kite, $\angle 1=90^{\circ}-39^{\circ}=51^{\circ}$.

Step5: Find $\angle 3$

The diagonals of a kite bisect the non - vertex angles. So $\angle 3 = 39^{\circ}$.

Answer:

$a = 67^{\circ}$, $b = 144^{\circ}$, $m\angle1 = 51^{\circ}$, $m\angle3 = 39^{\circ}$