Question

In a △ABC, if 2∠A = 3∠B = 6∠C, then th s measure of the smallest angle is
(a) 90°
(b) 60°
(c) 40°
(d) 30°

Answer 1

We have
2∠A = 3∠B = 6∠C
$$\begin{gathered} \therefore 3\angle \mathrm{B}=2\angle \mathrm{A}\mathrm{and}6\angle \mathrm{C}=2\angle \mathrm{A} \\ \Rightarrow \angle \mathrm{B}=\frac{2}{3}\angle \mathrm{A}\mathrm{and}\angle \mathrm{C}=\frac{2}{6}\angle \mathrm{A}=\frac{1}{3}\angle \mathrm{A} \\ \mathrm{Now},\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180°[\mathrm{Angle}\mathrm{sum}\mathrm{property}\mathrm{of}\mathrm{triangle}] \\ \Rightarrow \angle \mathrm{A}+\frac{2}{3}\angle \mathrm{A}+\frac{1}{3}\angle \mathrm{A}=180° \\ \Rightarrow 3\angle \mathrm{A}+2\angle \mathrm{A}+\angle \mathrm{A}=180°\times 3 \\ \Rightarrow 6\angle \mathrm{A}=540° \\ \Rightarrow \angle \mathrm{A}=90° \end{gathered}$$

∴ Smallest angle = $\angle \mathrm{C}=\frac{1}{3}\angle \mathrm{A}=\frac{1}{3}\times 90°=30°$
Hence, the correct answer is option (d).