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Is the given statement true or false?

In the given figure, $P Q R S$ is a quadrilateral in which $P Q = P S$ and $Q R = S R .$ Then $∠ P Q R = 2 ∠ P S R$ .

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True

False

answer is B.

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Detailed Solution

We are given that PQ = PS and QR = SR in the quadrilateral PQRS.
Question Image

From the given quadrilateral PQRS, in

Δ P Q R

and

Δ P S R

,
PQ = PS            … [Given]
QR = SR           … [Given]
PR = PR           … [Common]
Thus,

Δ P Q R ≅ Δ P S R

. [By SSS criterion] … (i)

∠ Q P R = ∠ S P R

… [By C.P.C.T]
Then,

∠ Q P E = ∠ S P E

, it means that PR bisects

∠ Q P S

,i.e.,

∠ P

∠ Q R P = ∠ S R P

[By C.P.C.T]
Then,

∠ Q R E = ∠ S R E

which means PR bisects

∠ Q P S

,i.e.,

∠ R

.
In

Δ P Q E

and

Δ P S E

,
PQ = PS … [Given]

∠ Q P E = ∠ S P E

… [From (i)]
PE = PE … [Common]
Thus,

Δ P Q E ≅ Δ P S E

. [By SAS criterion]
Then, QE = SE. … [By C.P.C.T]
Consider the equation (i),

Δ P Q R ≅ Δ P S R

.
Thus,

∠ P Q R = ∠ P S R

. … [By C.P.C.T]
So, given, statement that

∠ P Q R = 2 ∠ P S R

is false.
If

P Q R S

is a quadrilateral in which

P Q = P S

and

Q R = S R .

Then

∠ P Q R = ∠ P S R

.
So, option 2 is the correct choice.

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