Question

Diagonals of a rectangle PQRS are intersecting in point M. If $\angle$QMR = ${50}^{°}$ find the measure of $\angle$MPS.

Answer 1

In the given rectangle PQRS,
$\angle$QMR = $\angle$PMS = ${50}^{°}$ (vertically opposite angles are equal)
Also, SQ = PR (Diagonals of a rectangle are equal)
So,
$$\begin{gathered} \frac{1}{2}\mathrm{SQ}=\frac{1}{2}\mathrm{PR} \\ \Rightarrow \mathrm{SM}=\mathrm{PM} \end{gathered}$$
$\Rightarrow \angle \mathrm{MSP}=\angle \mathrm{MPS}$ (Angle opposite to equal sides are equal)
In $△$PMS,
$$\begin{gathered} \angle \mathrm{PMS}+\angle \mathrm{MSP}+\angle \mathrm{MPS}=180° \\ \Rightarrow 50°+2\angle \mathrm{MPS}=180° \\ \Rightarrow 2\angle \mathrm{MPS}=130° \\ \Rightarrow \angle \mathrm{MPS}=65° \end{gathered}$$