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In △ABC, if ∠A=2∠B and ∠B=3∠C, what is the measure of ∠A?
90 degrees
100 degrees
108 degrees
120 degrees
108 degrees
Express all angles in terms of ∠C (let ∠C=x). Then ∠B=3x and ∠A=2(3x)=6x. Solve 6x+3x+x=180° to find x.
In triangle ABC, angle A is twice angle B and angle B is thrice angle C.
∠A+∠B+∠C=180°
Express all angles in terms of ∠C (let ∠C=x). Then ∠B=3x and ∠A=2(3x)=6x. Solve 6x+3x+x=180° to find x.
Students often misinterpret ∠A=2∠B and ∠B=3∠C as ∠A=2∠B=6∠C directly without performing the transitive multiplication.
Define angles in terms of a single variable
Let ∠C=x. Given ∠B=3∠C, we have ∠B=3x. Given ∠A=2∠B, we substitute ∠B=3x to get ∠A=2(3x)=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°
Solve for x
Combining the terms gives 10x=180°, so x=18°.
10x=180°⟹x=18°
Calculate the final value of angle A
Since ∠A=6x, substitute x=18° to find the value.
∠A=6×18°=108°
C is correct because the angles calculate to 108 degrees, 54 degrees, and 18 degrees respectively, summing to 180 degrees.
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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