Mathematics
ΔPQR is an isosceles triangle such that PQ = QR. If S is a point on QR produced such that PR = RS and ∠QPS = 63°, find ∠PSQ.
Triangles
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Answer
ΔPQR is an isosceles triangle such that PQ = QR.
S is a point on QR produced such that PR = RS.
As we know that in an isosceles triangle, angles opposite to equal sides are equal.
In ΔPQR,
⇒ ∠QPR = ∠QRP = x° (let)………………..(1)
In ΔPRS,
⇒ ∠PSR = ∠SPR = y° (let)………………..(2)
Given,
⇒ ∠QPS = 63°
From figure,
⇒ ∠QPR + ∠SPR = 63°
⇒ x° + y° = 63° ………………(3)
We know that,
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
The angle ∠PRQ (which is x) is an exterior angle to △PRS at vertex R.
⇒ ∠PRQ = ∠SPR + ∠PSR.
⇒ x° = y° + y°
⇒ x° = 2y°
Substituting the above value of x in equation (1), we get :
⇒ 2y° + y° = 63°
⇒ 3y° = 63°
⇒ y° =
⇒ y° = 21°
⇒ ∠PSR = ∠SPR = 21°.
From figure,
⇒ ∠PSQ = ∠PSR = 21°
Hence, ∠PSQ = 21°.
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