Question

In ∆ABC, ∠A : ∠B : ∠C = 3 : 2 : 1 and CD ⊥ AC. Then, ∠ECD = ?
(a) 60°
(b) 45°
(c) 75°
(d) 30°

Answer 1

Let $\angle A=\left(3x\right)°,\angle B=\left(2x\right)°\text{and}\angle C=x°$
Then,
$$\begin{gathered} 3x+2x+x=180°\left[\text{Sum of the angles of a triangle}\right] \\ \Rightarrow 6x=180° \\ \Rightarrow x=30° \end{gathered}$$
Hence, the angles are
$\angle A=3\times 30°=90°,\angle B=2\times 30°=60°\text{and}\angle C=30°$
Side BC of triangle ABC is produced to E.
$$\begin{gathered} \therefore \angle ACE=\angle A+\angle B \\ \Rightarrow \angle ACD+\angle ECD=90°+60° \\ \Rightarrow 90°+\angle ECD=150° \\ \Rightarrow \angle ECD=60° \end{gathered}$$