Advertisements
Advertisements
Question
Diagonals of a rectangle PQRS are intersecting in point M. If ∠QMR = 50° find the measure of ∠MPS.
Advertisements
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)]
APPEARS IN
RELATED QUESTIONS
Name the quadrilaterals whose diagonals are equal
Two adjacent angles of a parallelogram are (3x − 4)° and (3x + 10)°. Find the angles of the parallelogram.
The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram.
In the following figure, BDEF and DCEF are each a parallelogram. Is it true that BD = DC? Why or why not?
Which of the following statement is true for a rectangle?
It has all its sides of equal length.
Which of the following statement is true for a rectangle?
Its diagonals are perpendicular and bisect each other.
Which of the following statement are true for a square?
It has all its sides of equal length.
Which of the following statement true for a square?
Its diagonals are equal to its sides.
The sides of a rectangle are in the ratio 4 : 5. Find its sides if the perimeter is 90 cm.
Adjacent sides of a rectangle are 7 cm and 24 cm. Find the length of its diagonal.
Draw a rectangle ABCD such that l(AB) = 6.0 cm and l (BC) = 4.5 cm.
If diagonal of a rectangle is 26 cm and one side is 24 cm, find the other side.
If the diagonals of a parallelogram are of equal lengths, the parallelogram is a rectangle. Prove it.
The following figure is a rectangle in which x: y = 3: 7; find the values of x and y.
If the adjacent angles of a parallelogram are equal, then the parallelogram is a ______.
All squares are rectangles.
Rectangle is a regular quadrilateral.
If diagonals of a quadrilateral are equal, it must be a rectangle.
In rectangle PAIR, find ∠ARI, ∠RMI and ∠PMA.
A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. Is PXQY a rectangle? Given reason.
