Question

find ( mangle p ) in rhombus ( pqrs ).
( mangle p = square^circ )

Explanation:

Step1: Use rhombus consecutive angles property

In a rhombus, consecutive angles are supplementary, so $m\angle Q + m\angle S = 180^\circ$.
Substitute the given expressions:
$$3v - 15 + v + 39 = 180$$

Step2: Simplify and solve for $v$

Combine like terms and isolate $v$:
$$4v + 24 = 180$$
$$4v = 180 - 24$$
$$4v = 156$$
$$v = \frac{156}{4} = 39$$

Step3: Find $m\angle Q$

Substitute $v=39$ into $\angle Q$:
$$m\angle Q = 3(39) - 15 = 117 - 15 = 102^\circ$$

Step4: Use rhombus opposite angles property

In a rhombus, opposite angles are equal, so $m\angle P = m\angle R$, and $m\angle P$ is supplementary to $m\angle Q$:
$$m\angle P = 180^\circ - m\angle Q$$
$$m\angle P = 180 - 102 = 78$$

Answer:

$78^\circ$