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PQRS is a rectangle with $∠ QPR = 35 °$ . Determine $∠ SQR, ∠ QSR$ and $∠ SPR$

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∠ S Q R = 65 °, ∠ Q S R = 45 °, ∠ S P R = 65 °

∠ S Q R = 55 °, ∠ Q S R = 55 °, ∠ S P R = 55 °

∠ S Q R = 55 °, ∠ Q S R = 35 °, ∠ S P R = 55 °

∠ S Q R = 75 °, ∠ Q S R = 35 °, ∠ S P R = 75 °

answer is C.

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Detailed Solution

Given, a rectangle PQRS with

∠ QPR = 35 °

∠ S P R = 90 ° − ∠ Q P R ⇒ ∠ S P R = 90 ° − 35 ° ⇒ ∠ S P R = 55 ° ... (1)

Let A be the intersection points of the diagonals PR and SQ

S A = A Q, P A = A R

[Diagonals of a rectangle bisect each other.]
Also, PS=QR [Opposite sides of rectangle]
Therefore, by SSS test,

Δ S A P ≅ Δ R A Q

⇒ ∠ S P A = ∠ A Q R = 55 °

[CPCT, From (1)]

⇒ ∠ S Q R = 55 °

... (2)
And,

∠ QRS=90 ° [Angle of rectangle]

Δ S Q R

,
By angle sum property of triangle,

∠ S Q R + ∠ Q R S + ∠ Q S R = 180 ° ⇒ 55 ° + 90 ° + ∠ Q S R = 180 ° [From (2)] ⇒ ∠ Q S R = 35 °

Therefore,

∠ S Q R = 55 °, ∠ Q S R = 35 °, ∠ S P R = 55 °

.
Hence, the correct option is 3.

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