Question

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

Explanation:

Step1: Match congruent polygon angles

Congruent polygons have corresponding equal angles. First, identify the corresponding sides: $HI=TS=47$ mm, $HK=UV=46$ mm, $KJ=VS=22$ mm, $JI=RT=47$ mm. So $\angle S$ corresponds to $\angle I$.

Step2: Calculate $\angle I$ (sum of quadrilateral angles)

Sum of interior angles of a quadrilateral is $(4-2)\times180^\circ=360^\circ$. For quadrilateral $HIJK$:
$\angle I = 360^\circ - \angle H - \angle K - \angle J$
$\angle I = 360^\circ - 54^\circ - 154^\circ - 63^\circ$
$\angle I = 360^\circ - 271^\circ = 89^\circ$

Step3: Corresponding angles are equal

Since $\angle S \cong \angle I$, $m\angle S = \angle I$.

Answer:

$89^\circ$