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Question
Diagonals of a rectangle PQRS are intersecting in point M. If ∠QMR = 50° find the measure of ∠MPS.
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Solution
`square`PQRS is a rectangle.
∴ PM = `1/2` PR …(i)
MS = `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 = `(130°)/2`
∴ x = 65°
∴ ∠MPS = 65° ...[From (v)]
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