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In triangle ABC, side BC is produced to D. A student measures carefully and reports that angle ACD = angle A + angle B. What does this measurement tell us about triangle ABC?

Class 7The Triangle and its PropertiesTriangles
  • AThat triangle ABC must be right-angled.
  • BThat triangle ABC must be isosceles.
  • CThat angle ACD must be 90°.
  • DNothing special — this relation holds for the exterior angle of every triangle.

Step-by-Step Solution

Angle ACD is the exterior angle at C, and the exterior angle property says it always equals the sum of the two interior opposite angles, angle A and angle B. The measurement merely confirms a universal rule, so it singles out no special triangle. Trap: reading a general property as a special condition, or expecting the exterior angle to equal all three interior angles added, which would give 180°.
Answer: D) Nothing special — this relation holds for the exterior angle of every triangle.

Key idea: Exterior angle property

Extend one side of a triangle past a vertex. The angle formed outside the triangle is the **exterior angle** at that vertex.

**Exterior angle = sum of the two opposite interior angles.**

So if the two non-adjacent interior angles are 50° and 60°, the exterior angle = 50° + 60° = 110°.

Why? The exterior angle and the adjacent interior angle form a linear pair (sum 180°), and the three interior angles also sum to 180°. Subtract: exterior = sum of the other two interior angles.

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