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$P Q R S$ is a quadrilateral. $P R$ is a quadrilateral. and $Q S$ and intersect each other at $O$ intersect each other at . In which of the following cases, $P Q R S$ . In which of the following cases, is a parallelogram?

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∠P=100°, ∠Q=80°, ∠R=100°

∠P=85°, ∠Q=85°, ∠R=95°

PQ=7 cm, QR=7 cm, RS=8 cm, SP=8 cm

OP=6.5 cm, OQ=6.5 cm, OR=5.2 cm, OS=5.2 cm

answer is A.

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Detailed Solution

Given that,

P Q R S

is a quadrilateral.

P R

and

Q S

intersect each other at

O

.
For option (1),
Use the angle sum property of a quadrilateral.

∠ P + ∠ Q + ∠ R + ∠ S = 360 ° ⇒ 100 ° + 80 ° + 100 ° + ∠ S − 360 ° ⇒ 280 ° + ∠ S = 360 ° ⇒ ∠ S = 80 °

Establish the relation between the angles.

∠ P = ∠ R ∠ Q = ∠ S

Thus, we have

P Q R S

, a quadrilateral, with opposite angles are equal.
Therefore,

P Q R S

is a parallelogram.
For option (2),
Use angle sum property of a quadrilateral.

∠ P + ∠ Q + ∠ R + ∠ S = 360 ° ⇒ 85 ° + 85 ° + 95 ° + ∠ S = 360 ° ⇒ 265 ° + ∠ S = 360 ° ⇒ ∠ S = 95 °

Establish the relation between the angles.

∠ P ≠ ∠ R ∠ Q ≠ ∠ S

Thus,

P Q R S

is a quadrilateral with opposite angles not equal.
Therefore,

P Q R S

is not a parallelogram.
For option (3),
Establish the relation between the side

P Q

and

R S

PQ ≠ RS

Establish the relation between the side

Q R

and

S P

QR ≠ SP

Thus,

P Q R S

is a quadrilateral with opposite sides not equal.
Therefore,

P Q R S

is not a parallelogram.
For option (4),
Establish the relation between the side

O P

and

O R

OP ≠ OR

Establish the relation between the side

O Q

and

O S

The diagonals of a parallelogram bisect each other. Therefore,

P Q R S

is not a parallelogram.
Hence, correct option is 1.

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