Question

In the given figure, ABC is a triangle in which ∠A : ∠B : ∠C = 3 : 2 : 1 and AC ⊥ CD. Find the measure of ∠ECD.

Answer 1

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