In a triangle ABC, AB = AC. BC is extended to D such that CD = AB and the angle ADC is 30°. What are the angles of triangle ABC?

The correct option is 60°, 60°, 60°.

Step-by-step Explanation:

Step 1: Understand the given conditions

We are given a triangle $ABC$ where:

1. $AB=AC$ (so $\Delta ABC$ is an isosceles triangle with $\angle ABC=\angle ACB$).

2. Side $BC$ is extended to point $D$ such that $CD=AB$.

3. The angle $\angle ADC=30°$.

Step 2: Analyze triangle ACD

Since $AB=AC$ and $CD=AB$, it follows that:

$AC=CD$

Therefore, triangle $ACD$ is an isosceles triangle with sides $AC$ and $CD$ being equal. The angles opposite to these equal sides are also equal:

$\angle CAD=\angle ADC=30°$

Step 3: Calculate angle ACD

In triangle $ACD$, the sum of all interior angles is $180°$:

$\angle ACD+\angle CAD+\angle ADC=180°$

$\angle AC+30°+30°=180°$

$\angle ACD=180°-60°=120°$

Step 4: Find the interior angles of triangle ABC

Since $BCD$ is a straight line segment, the angle $\angle ACB$ and $\angle ACD$ form a linear pair:

$\angle ACB+\angle ACD=180°$

$\angle ACB=180°-120°=60°$

Since $AB=AC$, the base angles of isosceles triangle $ABC$ are equal:

$\angle ABC=\angle ACB=60°$

Finally, using the angle sum property of triangle $ABC$:

$\angle BAC=180°-(\angle ABC+\angle ACB)=180°-120°=60°$

Thus, all three angles of triangle $ABC$ are $60°,60°,60°$.