If $ PQRS $ is a rhombus such that $\angle PRQ = 40^\circ $, then $\angle PSQ $ is:

The correct answer is Option B: 50^\circ. In a rhombus, diagonals bisect the opposite angles, and the triangle formed by two sides and a diagonal is isosceles; specifically, the alternate angles subtended by the sides at the vertices are equal. Step-by-Step Solution & Concepts: Shortcut Trick: In a rhombus, diagonals bisect the opposite angles, and the triangle formed by two sides and a diagonal is isosceles; specifically, the alternate angles subtended by the sides at the vertices are equal. Formula Used: $$\angle PSQ = \angle PRQ$$ Given: PQRS is a rhombus and \angle PRQ = 40^\circ. Steps: Analyze Rhombus Properties: In a rhombus $PQRS$, all sides are equal ($PQ=QR=RS=SP$). The diagonal $PR$ divides the rhombus into two congruent triangles $\triangle PQR$ and $\triangle PSR$.. Use Symmetry of Rhombus: Because $PQ \parallel SR$ and $PS \parallel QR$, the diagonal $PR$ acts as a transversal. By alternate interior angles, $\angle QRP = \angle SPR$. Due to the property of rhombus diagonals bisecting angles, $\angle PSQ = \angle PRQ$.. Final Verification: Given $\angle PRQ = 40^\circ$, by the property that diagonals of a rhombus create equal angles at the vertices due to congruent triangles $\triangle PQS$ and $\triangle RQS$, we confirm the value. Answer: C is correct because in a rhombus, the diagonals bisect the vertex angles and the triangles formed are symmetric, leading to $\angle PSQ = \angle PRQ = 40^\circ$. Exam Tip: Always remember that a rhombus is a special type of parallelogram where diagonals are perpendicular bisectors; utilize these properties to solve angle geometry questions rapidly.

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MathematicsPrevious Year Paper 2024

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If $PQRS$ is a rhombus such that $\angle PRQ = 40°$ , then $\angle PSQ$ is:

60°

50°

40°

45°

Correct Answer

📐 MA • Math Geometry PropertyMathematicsPrevious Year Paper 2024

Option B

50°

Quick Summary:

In a rhombus, diagonals bisect the opposite angles, and the triangle formed by two sides and a diagonal is isosceles; specifically, the alternate angles subtended by the sides at the vertices are equal.

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📐MAMath SolutionGeometry Property

📋 Given

PQRS is a rhombus and \angle PRQ = 40°.

🔢 Formula Used

∠PSQ=∠PRQ\angle PSQ = \angle PRQ∠PSQ=∠PRQ

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often incorrectly assume \angle PSQ is complementary to \angle PRQ (i.e., 50°) because they mistake the triangle for a right triangle, forgetting that only the intersection of diagonals forms right angles.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Analyze Rhombus Properties

In a rhombus $PQRS$ , all sides are equal ( $PQ=QR=RS=SP$ ). The diagonal $PR$ divides the rhombus into two congruent triangles $\triangle PQR$ and $\triangle PSR$ .

PQ=QR=RS=SPPQ = QR = RS = SPPQ=QR=RS=SP

Use Symmetry of Rhombus

Because $PQ \parallel SR$ and $PS \parallel QR$ , the diagonal $PR$ acts as a transversal. By alternate interior angles, $\angle QRP = \angle SPR$ . Due to the property of rhombus diagonals bisecting angles, $\angle PSQ = \angle PRQ$ .

∠PSQ=∠PRQ=40°\angle PSQ = \angle PRQ = 40°∠PSQ=∠PRQ=40°

Final Verification

Given $\angle PRQ = 40°$ , by the property that diagonals of a rhombus create equal angles at the vertices due to congruent triangles $\triangle PQS$ and $\triangle RQS$ , we confirm the value.

∠PSQ=40°\angle PSQ = 40°∠PSQ=40°

✅

C is correct because in a rhombus, the diagonals bisect the vertex angles and the triangles formed are symmetric, leading to $\angle PSQ = \angle PRQ = 40°$ .

Core Concepts Used

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Rhombus Properties Congruent Triangles Parallel Line Theorems

💡 EXAM TIP

Always remember that a rhombus is a special type of parallelogram where diagonals are perpendicular bisectors; utilize these properties to solve angle geometry questions rapidly.

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