Question

PQR is a triangle in which /PQ/ = /PR/. S is a point on /PR/ such that /QS/ = /QR/. If <PQS = 30°. Calculate <QPR. l​

Answer 1

• m∠QPR = 40°

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Refer to attachment

Given ΔPQR with two equal sides, PQ and PR. It is therefore isosceles. Hence the angles opposite to equal sides are equal.

We are looking for ∠QPR, let it measure be x and the measure of the other two angles be y.

Also we have a point S on PR making another isosceles ΔQSR as two sides are equal:

• QS = QR

It makes ∠QSR and ∠QRS equal. Since we said angle on vertex R is y, we also get ∠QSR of same value y.

Now, using triangle angle sum theorem, with regards to ΔPQR we can determine that:

• x + 2y = 180 ⇒ x = 180 - 2y

Same approach to ΔQSR:

• m∠SQR + 2y = 180

We are given that m∠PQS = 30°, therefore:

• m∠SQR = m∠PQR - m∠PQS

Substitute values of angles:

• m∠SQR = y - 30

Now substitute this into angle sum equation for ΔQSR:

• y - 30 + 2y = 180
• 3y = 210
• y = 70

Find x by substituting 70 for y into first equation:

• x = 180 - 2y
• x = 180 - 2(70)
• x = 180 - 140
• x = 40

Hence the angle QPR is 40°.