Question

these figures are congruent. what is ( mangle k )?

Explanation:

Step1: Identify congruent polygon angle sum

The sum of interior angles of a quadrilateral is $(4-2)\times180^\circ = 360^\circ$.

Step2: Match congruent corresponding angles

Congruent polygons have matching angles. First, find the unknown angle in the second quadrilateral:
Let $m\angle Q = x$.
$x + 112^\circ + m\angle O + 122^\circ = 360^\circ$.
First, find $m\angle O$: it corresponds to $\angle L = 30^\circ$, so $m\angle O = 30^\circ$.

Step3: Calculate $\angle Q$ (matches $\angle K$)

$x = 360^\circ - 112^\circ - 30^\circ - 122^\circ$
$x = 360^\circ - (112^\circ+30^\circ+122^\circ) = 360^\circ - 264^\circ = 96^\circ$
Alternatively, verify with the first quadrilateral: sum angles to confirm. $\angle K$ corresponds to $\angle Q$, so $m\angle K = m\angle Q$.

Answer:

$96$