Question

pqrst ~ vuyxw. what is ( mangle y )? ( mangle y = square^circ )

Explanation:

Step1: Find sum of pentagon angles

The sum of interior angles of an n-sided polygon is calculated by $(n-2)\times180^\circ$. For a pentagon ($n=5$):
$$(5-2)\times180^\circ = 540^\circ$$

Step2: Calculate missing angle in PQRST

First, sum the known angles in PQRST: $138^\circ + 62^\circ = 200^\circ$. Let the sum of the remaining three angles be $S_1$:
$$S_1 = 540^\circ - 200^\circ = 340^\circ$$

Step3: Calculate sum of known angles in VUYXW

Sum the given angles: $100^\circ + 113^\circ + 127^\circ + 62^\circ = 402^\circ$

Step4: Use similarity to find $\angle Y$

Since similar polygons have congruent corresponding angles, the total of angles in VUYXW is also $540^\circ$. Solve for $m\angle Y$:
$$m\angle Y = 540^\circ - 402^\circ = 138^\circ$$
(Note: This matches the corresponding $\angle R$ in PQRST, confirming similarity.)

Answer:

$\boldsymbol{138}$