Page 185 - ICSE Math 8

9. Find x and y in the parallelograms given below.
D C S R W V
y 10 y
15

2x – 5
x 110° y + 7 75°
A B P Q T U x S
(a) (b) (c)

10. Two adjacent angles of a parallelogram are (2y + 10°) and (3y – 40°), find the measure of all the angles
of the parallelogram.
11. Two adjacent sides of a parallelogram are such that one is twice the other. If the perimeter of the
parallelogram is 120 m, find the length of its adjacent sides.
12. The point of intersection of the diagonals of a quadrilateral divides the diagonals in the ratio 1 : 2. Is it
a parallelogram? Give reasons to support of your answer.
Rectangle P Q
A rectangle is a parallelogram in which all the interior angles are right angles.
In parallelogram PQRS (Fig. 16.18),
–P = –Q = –R = –S = 90º. O

Thus, PQRS is a rectangle.
Properties of a rectangle S Fig. 16.18 R
(a) Opposite sides are of equal length (PQ = SR and PS = QR).

(b) All interior angles measure 90º (–P = –Q = –R = –S = 90º).
(c) Diagonals are of equal length (PR = SQ).
(d) Diagonals bisect each other. Hence, OP = OR = OQ = OS.

Square P Q
A rectangle in which two adjacent sides are equal is a square. Or, a rhombus
in which all the interior angles are right angles is a square.
In rectangle PQRS (Fig. 16.19), PQ = QR or PS = SR.
So, PQRS is a square. O

Properties of a square
S R
(a) All sides are of equal length (PQ = QR = SR = SP).
Fig. 16.19
(b) All interior angles measure 90º (–P = –Q = –R = –S = 90º).
(c) Diagonals are of equal length (PR = SQ).
(d) Diagonals bisect each other at right angles (OP = OR and OQ = OS, also PR ^ SQ at point O). Hence,
OP = OR = OQ = OS.

Example 7: Explain how a square is a: (a) quadrilateral (b) parallelogram (c) rhombus (d) rectangle.
Solution: (a) Square is a quadrilateral because it is a closed figure with four sides and four angles.
(b) Square is a parallelogram because its opposite sides are parallel.

(c) Square is a rhombus because all its sides are equal and the diagonals bisect each other at
right angles.
(d) Square is a rectangle because all its angles are of 90º and the diagonals are equal in length.

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