Page 183 - ICSE Math 8

In quadrilateral PQRS (Fig. 16.16), PQ || SR and PS || QR. P Q
So, PQRS is a parallelogram.

Properties of a parallelogram O
(a) Opposite sides are of equal length (PQ = SR and PS = QR).
(b) Opposite angles are of equal measure (–P = –R and –Q = –S). S R
(c) The sum of adjacent angles is 180º. Fig. 16.16
( –P + –Q = 180º, –Q + –R = 180º, –R + –S = 180º and –S + –P = 180º).
(d) The diagonals bisect each other (Diagonals PR and SQ bisect each other at point O, so OP = OR and
OQ = OS).
P
Rhombus
A rhombus is a quadrilateral having all sides equal. Hence, a parallelogram
in which two adjacent sides are equal is a rhombus.
In parallelogram PQRS (Fig. 16.17), PQ = PS or QR = SR. S O Q
So, PQRS is a rhombus.
Properties of a rhombus
R
(a) All sides are of equal length (PQ = QR = RS = SP). Fig. 16.17
(b) Opposite angles are of equal measure (–P = –R and –Q = –S).
(c) The sum of adjacent angles is 180º (–P + –Q = 180º, –Q + –R = 180º, –R + –S = 180º and –S + –P
= 180º).
(d) The diagonals bisect each other at right angles (Diagonals PR and SQ bisect each other at point O, so
OP = OR and OQ = OS, also PR ^ SQ at point O).

Example 4: Look at the following parallelogram and find the value of the S R
unknown x, y, z. 50° y
Solution: In parallelogram PQRS,

–S = –PQR fi –PQR = 50º (Opposite angles of a
parallelogram are equal)
Ray QR stands on PT, therefore x z
–PQR + –RQT = 180º (Linear pair) P Q T

fi 50º + z = 180º fi z = 180º – 50º = 130º
Also PQ || SR and SP is the transversal intersecting them, therefore,
–P + –S = 180º (Interior angles on the same side of transversal are supplementary)
fi x + 50º = 180º fi x = 180º – 50º = 130º
Again, –R = –P (Opposite angles of a parallelogram are equal)
fi y = x fi y = 130º
Therefore, x = y = z = 130º

Example 5: Can a quadrilateral ABCD be a parallelogram if:
(a) –D + –B = 180º? D 9 cm C
(b) AB = DC = 9 cm and AD = 5 cm and BC = 5.5 cm?
Solution: (a) –B + –D = 180º
In a parallelogram, opposite angles are equal. 5 cm 5.5 cm
fi –B = –D

Since –B + –D = 180º fi –B = –D = 90º A 9 cm B

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