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Maharashtra State BoardSSC (English Medium) 9th Standard

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

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Question

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

Sum
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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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Chapter 5: Quadrilaterals - Problem Set 5 [Page 74]

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Balbharati Mathematics 2 [English] Standard 9 Maharashtra State Board
Chapter 5 Quadrilaterals
Problem Set 5 | Q 6 | Page 74

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