PQRS is a trapezium in which $ QR \parallel PS $ and $ PR = RS $. If $\angle PQR = 96^{\circ}$ and $\angle QPR = 34^{\circ}$, then what is the measure of $\angle PRS $?

The correct answer is Option C: 80°. In a trapezium with parallel sides, alternate interior angles and the sum of angles in a triangle are the fastest tools. Find the missing angle in $\triangle PQR$ first to relate it to side $RS$. Step-by-Step Solution & Concepts: Shortcut Trick: In a trapezium with parallel sides, alternate interior angles and the sum of angles in a triangle are the fastest tools. Find the missing angle in $\triangle PQR$ first to relate it to side $RS$. Formula Used: $$\angle PQR + \angle QPS = 180^{\circ}$$, $$\sum \text{angles in } \triangle PQR = 180^{\circ}$$, $$\angle PRS = \angle QRP + \angle QPS - \angle PSR$$ Given: Trapezium PQRS where QR || PS, PR = RS, angle PQR = 96°, and angle QPR = 34°. Steps: Calculate interior angle of triangle PQR: In $\triangle PQR$, the sum of interior angles is $180^{\circ}$. Given $\angle PQR = 96^{\circ}$ and $\angle QPR = 34^{\circ}$, we find $\angle PRQ$.. Use parallel lines property for angle QPS: Since $QR \parallel PS$, the consecutive interior angles $\angle PQR$ and $\angle QPS$ sum to $180^{\circ}$. Thus, $\angle QPS = 180^{\circ} - 96^{\circ} = 84^{\circ}$.. Apply triangle properties to PRS: Given $PR = RS$, $\triangle PRS$ is isosceles with $\angle SPR = \angle QPS - \angle QPR = 84^{\circ} - 34^{\circ} = 50^{\circ}$. Since $PR = RS$, base angles are equal: $\angle RSP = \angle SPR = 50^{\circ}$.. Final calculation for angle PRS: In $\triangle PRS$, the sum of angles is $180^{\circ}$. $\angle PRS = 180^{\circ} - (\angle RSP + \angle SPR) = 180^{\circ} - (50^{\circ} + 50^{\circ}) = 80^{\circ}$. Answer: C is correct because the interior angle sum in $\triangle PRS$ leads to $180^{\circ} - 100^{\circ} = 80^{\circ}$. Exam Tip: Always verify if the parallel sides result in an isosceles property for the non-parallel sides; identifying congruent triangles or segments is key in coordinate-free geometry.

PQRS is a trapezium in which $ QR \parallel PS $ and $ PR = RS... | Mathematics MCQ

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PQRS is a trapezium in which $QR \parallel PS$ and $PR = RS$ . If $\angle PQR = 96°$ and $\angle QPR = 34°$ , then what is the measure of $\angle PRS$ ?

75°

70°

80°

65°

Correct Answer

📐 MA • Math Geometric Property AnalysisMathematicsPrevious Year Paper 2024

Option C

80°

Quick Summary:

In a trapezium with parallel sides, alternate interior angles and the sum of angles in a triangle are the fastest tools. Find the missing angle in $\triangle PQR$ first to relate it to side $RS$ .

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📐MAMath SolutionGeometric Property Analysis

📋 Given

Trapezium PQRS where QR || PS, PR = RS, angle PQR = 96°, and angle QPR = 34°.

🔢 Formula Used

∠PQR+∠QPS=180°\angle PQR + \angle QPS = 180°∠PQR+∠QPS=180°, ∑angles in △PQR=180°\sum \text{angles in } \triangle PQR = 180°∑angles in △PQR=180°, ∠PRS=∠QRP+∠QPS−∠PSR\angle PRS = \angle QRP + \angle QPS - \angle PSR∠PRS=∠QRP+∠QPS−∠PSR

⚡ Exam Hall Shortcut / Speed Trick

In a trapezium with parallel sides, alternate interior angles and the sum of angles in a triangle are the fastest tools. Find the missing angle in $\triangle PQR$ first to relate it to side $RS$ .

⚠️ Common Student Trap / Pitfall

Confusing the parallel lines condition and assuming PQRS is an isosceles trapezium; here only $PR = RS$ is specified, not necessarily $PQ = RS$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate interior angle of triangle PQR

In $\triangle PQR$ , the sum of interior angles is $180°$ . Given $\angle PQR = 96°$ and $\angle QPR = 34°$ , we find $\angle PRQ$ .

∠PRQ=180°−(96°+34°)=180°−130°=50°\angle PRQ = 180° - (96° + 34°) = 180° - 130° = 50°∠PRQ=180°−(96°+34°)=180°−130°=50°

Use parallel lines property for angle QPS

Since $QR \parallel PS$ , the consecutive interior angles $\angle PQR$ and $\angle QPS$ sum to $180°$ . Thus, $\angle QPS = 180° - 96° = 84°$ .

∠QPS=84°\angle QPS = 84°∠QPS=84°

Apply triangle properties to PRS

Given $PR = RS$ , $\triangle PRS$ is isosceles with $\angle SPR = \angle QPS - \angle QPR = 84° - 34° = 50°$ . Since $PR = RS$ , base angles are equal: $\angle RSP = \angle SPR = 50°$ .

∠RSP=50°\angle RSP = 50°∠RSP=50°

Final calculation for angle PRS

In $\triangle PRS$ , the sum of angles is $180°$ . $\angle PRS = 180° - (\angle RSP + \angle SPR) = 180° - (50° + 50°) = 80°$ .

∠PRS=80°\angle PRS = 80°∠PRS=80°

✅

C is correct because the interior angle sum in $\triangle PRS$ leads to $180° - 100° = 80°$ .

Core Concepts Used

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Trapezium properties Isosceles triangle base angles Consecutive interior angles

💡 EXAM TIP

Always verify if the parallel sides result in an isosceles property for the non-parallel sides; identifying congruent triangles or segments is key in coordinate-free geometry.

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