Question

Diagonals of a rectangle PQRS are intersecting in point M. If ∠QMR = 50° find the measure of ∠MPS.

Answer 1

${\displaystyle \square }$PQRS is a rectangle.

∴ PM = ${\displaystyle \frac{1}{2}}$ PR     …(i)

MS = ${\displaystyle \frac{1}{2}}$ QS       …(ii) ...(Diagonals of a rectangle bisect each other)

Also, PR = QS      …(iii) ...(Diagonals of a rectangle are congruent)

∴ PM = MS      …(iv)   ...[From (i), (ii) and (iii)]

In ∆PMS,

PM = MS      ...[From (iv)]

∴ ∠MSP = ∠MPS = x°       ...(v) ...(Isosceles triangle theorem)

∠PMS = ∠QMR = 50°      ...(vi) ...(Vertically opposite angles)

In ∆MPS,

∠PMS + ∠MPS + ∠MSP = 180°      ...(The sum of the measures of the three angles of a triangle is 180°.)

∴ 50° + x + x = 180°      ...[From (v) and (vi)]

∴ 50° + 2x = 180°

∴ 2x = 180° - 50°

∴ 2x = 130°

∴ x = ${\displaystyle \frac{130°}{2}}$

∴ x = 65°

∴ ∠MPS = 65°       ...[From (v)]