In an isosceles $\triangle PQR$, $PQ = PR$. If each of the two equal angles is $15^{\circ}$ more than one-fourth of the third angle, find the measure of the third angle $\angle P$.

In an isosceles $\triangle PQR$, $PQ = PR$. If each of the two... | Mathematics MCQ | Examoogle

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In an isosceles $\triangle PQR$ , $PQ = PR$ . If each of the two equal angles is $15°$ more than one-fourth of the third angle, find the measure of the third angle $\angle P$ .

100°

120°

110°

105°

Correct Answer

📐 MA • Math Linear AlgebraMathematicsMathematics

Option B

120°

Quick Summary:

Use the options to quickly check: If P = 120, then (180 - 120)/2 = 30. Check if 30 is 1/4 of 120 + 15: (120/4) + 15 = 30 + 15 = 45 (incorrect). Wait, let's re-verify: if P=100, (180-100)/2 = 40. 100/4 + 15 = 40 (Correct). Note: The provided answer key says B, but the math proves P = 100 degrees.

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📐MAMath SolutionLinear Algebra

📋 Given

Triangle PQR is isosceles with PQ = PR. The two equal angles are Q = R. Each equal angle Q = R = 1/4 * P + 15 degrees.

🔢 Formula Used

P+Q+R=180°P + Q + R = 180°P+Q+R=180°

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often misinterpret 'one-fourth of the third angle' by adding 15 degrees to the sum instead of the calculated fraction.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define variables

Let the third angle $\angle P = x$ . Since $\triangle PQR$ is isosceles with $PQ = PR$ , the base angles are equal: $\angle Q = \angle R$ . Given $\angle Q = \angle R = \frac{1}{4}x + 15°$ .

∠Q=∠R=x4+15°\angle Q = \angle R = \frac{x}{4} + 15°∠Q=∠R=4x​+15°

Apply angle sum property

The sum of all angles in a triangle is $180°$ . Thus, $P + Q + R = 180°$ . Substitute the expressions in terms of $x$ .

x+(x4+15°)+(x4+15°)=180°x + \left(\frac{x}{4} + 15°\right) + \left(\frac{x}{4} + 15°\right) = 180°x+(4x​+15°)+(4x​+15°)=180°

Solve for x

Combine the terms: $x + \frac{x}{2} + 30° = 180°$ . Subtract $30°$ from both sides to get $\frac{3x}{2} = 150°$ . Multiplying by $\frac{2}{3}$ gives $x = 100°$ .

x=150×23=100°x = \frac{150 \times 2}{3} = 100°x=3150×2​=100°

✅

A is correct because the calculated measure of the third angle $\angle P$ is 100 degrees, which satisfies the condition $\frac{100}{4} + 15 = 40$ and $100 + 40 + 40 = 180$ .

Core Concepts Used

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Angle Sum Property of Triangles Isosceles Triangle Properties Linear Equations

💡 EXAM TIP

This type of algebraic geometry problem frequently appears in CAT/SSC exams. Always prioritize setting up the equation using the triangle angle sum theorem ( $180°$ ) before testing options.

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