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In $\triangle ABC$ , if $\angle A = 2\angle B$ and $\angle B = 3\angle C$ , what is the measure of $\angle A$ ?

90 degrees

100 degrees

108 degrees

120 degrees

Correct Answer

📐 MA • Math Ratio MethodMathematicsGeometry

Option C

108 degrees

Quick Summary:

Express all angles in terms of $\angle C$ (let $\angle C = x$ ). Then $\angle B = 3x$ and $\angle A = 2(3x) = 6x$ . Solve $6x + 3x + x = 180°$ to find $x$ .

📐MAMath SolutionRatio Method

📋 Given

In triangle ABC, angle A is twice angle B and angle B is thrice angle C.

🔢 Formula Used

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

⚡ Exam Hall Shortcut / Speed Trick

Express all angles in terms of $\angle C$ (let $\angle C = x$ ). Then $\angle B = 3x$ and $\angle A = 2(3x) = 6x$ . Solve $6x + 3x + x = 180°$ to find $x$ .

⚠️ Common Student Trap / Pitfall

Students often misinterpret $\angle A = 2\angle B$ and $\angle B = 3\angle C$ as $\angle A = 2\angle B = 6\angle C$ directly without performing the transitive multiplication.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define angles in terms of a single variable

Let $\angle C = x$ . Given $\angle B = 3\angle C$ , we have $\angle B = 3x$ . Given $\angle A = 2\angle B$ , we substitute $\angle B = 3x$ to get $\angle A = 2(3x) = 6x$ .

∠C=x,∠B=3x,∠A=6x\angle C = x, \quad \angle B = 3x, \quad \angle A = 6x∠C=x,∠B=3x,∠A=6x

Apply the angle sum property of triangles

The sum of all internal angles in a triangle is $180°$ . Substituting our expressions for $x$ , we get $6x + 3x + x = 180°$ .

6x+3x+x=180°6x + 3x + x = 180°6x+3x+x=180°

Solve for x

Combining the terms gives $10x = 180°$ , so $x = 18°$ .

10x=180°  ⟹  x=18°10x = 180° \implies x = 18°10x=180°⟹x=18°

Calculate the final value of angle A

Since $\angle A = 6x$ , substitute $x = 18°$ to find the value.

∠A=6×18°=108°\angle A = 6 \times 18° = 108°∠A=6×18°=108°

✅

C is correct because the angles calculate to 108 degrees, 54 degrees, and 18 degrees respectively, summing to 180 degrees.

Core Concepts Used

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Triangle Angle Sum Property Algebraic Substitution Ratio Proportion

💡 EXAM TIP

This variable-substitution technique is fundamental for solving ratio-based problems in both Coordinate Geometry and Arithmetic chapters like Mixture and Alligation.

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