In a triangle ABC, side BC is extended to a point D. The exterior angle at C is 105°. The interior angles at A and B are in the ratio 2:5. What is the measure of the largest interior angle of the triangle?

Correct — B. Rule: an exterior angle equals the sum of the two opposite interior angles.

Exterior angle theorem: ∠A + ∠B = 105°
Ratio 2:5, so 2k + 5k = 7k = 105° → k = 15°
∠A = 2 × 15° = 30° and ∠B = 5 × 15° = 75°

Linear pair at C: ∠C = 180° − 105° = 75°
Check: 30° + 75° + 75° = 180° ✓
Largest interior angle = 75° → option (b)

Why the others are wrong

• (a)
  30° — 30° is ∠A, the smallest angle. It is the 2-part share of 105°, and the question asks for the largest.
• (c)
  105° — 105° is the exterior angle at C, outside the triangle. The interior angle beside it is 180° − 105° = 75°.
• (d)
  15° — 15° is one ratio part, k, not an angle of the triangle. The angles are 2k = 30°, 5k = 75° and the 75° at C.

When side BC is extended to D, the exterior angle ∠ACD and the interior ∠ACB sit on a straight line, so they add to 180°.

The three interior angles also add to 180°, so the exterior angle equals ∠A + ∠B, the two interior angles not touching it. That turns the ratio 2:5 into a split of 105°, not of 180°.

Key facts

• An exterior angle of a triangle equals the sum of the two opposite interior angles.
• An exterior angle and its adjacent interior angle add to 180°.
• Here the angles are ∠A = 30°, ∠B = 75° and ∠C = 75°.

Common traps

• Splitting 180° in the ratio 2:5 instead of the 105° exterior angle.
• Giving 105° because it is the biggest number in the stem, though it lies outside the triangle.

No directly related past PYQ was found.