In an isosceles $\triangle ABC $, $ AC = BC $, if each of the ... | Mathematics MCQ | Examoogle

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In an isosceles $\triangle ABC$ , $AC = BC$ , if each of the two equal angles of $\triangle ABC$ is 15 degrees more than one-third of the third angle. Find the value of $\left( \frac{2\angle C - 3\angle A + \angle B}{2} \right)$ .

Correct Answer

📐 MA • Math Linear Algebra SystemMathematicsMathematics

Option A

Quick Summary:

Since AC = BC, $\angle A = \angle B$ . Substituting this into the triangle sum formula $2\angle A + \angle C = 180°$ and the given condition $\angle A = \frac{1}{3}\angle C + 15$ allows for instant substitution.

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📐MAMath SolutionLinear Algebra System

📋 Given

Triangle ABC is isosceles with AC = BC, which implies angle A = angle B. Each equal angle (A or B) is 15 degrees more than one-third of the third angle (C).

🔢 Formula Used

∠A+∠B+∠C=180°\angle A + \angle B + \angle C = 180°∠A+∠B+∠C=180°

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often misidentify which angles are equal; here, since $AC=BC$ , the base angles are $\angle A$ and $\angle B$ , while $\angle C$ is the vertex angle.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Set up equations

Since $AC = BC$ , base angles $\angle A = \angle B$ . We are given $\angle A = \frac{1}{3}\angle C + 15$ .

∠A=∠C3+15\angle A = \frac{\angle C}{3} + 15∠A=3∠C​+15

Use Angle Sum Property

The sum of angles in a triangle is $180°$ . Replacing $\angle B$ with $\angle A$ gives $2\angle A + \angle C = 180$ .

2(∠C3+15)+∠C=1802\left(\frac{\angle C}{3} + 15\right) + \angle C = 1802(3∠C​+15)+∠C=180

Solve for angles

Multiplying out, $\frac{2}{3}\angle C + 30 + \angle C = 180$ , so $\frac{5}{3}\angle C = 150$ , giving $\angle C = 90°$ . Then $\angle A = \angle B = \frac{90}{3} + 15 = 45°$ .

∠C=90°,∠A=45°,∠B=45°\angle C = 90°, \angle A = 45°, \angle B = 45°∠C=90°,∠A=45°,∠B=45°

Evaluate target expression

Substitute the angles into the expression $\frac{2\angle C - 3\angle A + \angle B}{2}$ .

2(90)−3(45)+452=180−135+452=902=45\frac{2(90) - 3(45) + 45}{2} = \frac{180 - 135 + 45}{2} = \frac{90}{2} = 4522(90)−3(45)+45​=2180−135+45​=290​=45

✅

A is correct because substituting $\angle A = 45°, \angle B = 45°$ , and $\angle C = 90°$ into the expression yields $45$ .

Core Concepts Used

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Triangle angle sum property Isosceles triangle properties Linear equation solving

💡 EXAM TIP

This type of algebraic substitution is common in coordinate geometry problems where you define points or slopes using unknown variables.

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