In triangle ABC, side AB is extended past B to point D, forming exterior angle CBD. If the measure of...

1. TRANSLATE the problem setup

• Given information:
  • Triangle ABC with AB extended past B to point D
  • Exterior angle \(\mathrm{CBD = 117°}\)
  • Interior angle \(\mathrm{A = 46°}\)
  • Need to find angle C

• VISUALIZE this: Draw or picture triangle ABC with the extension creating exterior angle CBD at vertex B.

2. INFER which theorem applies

• The key insight: When you see an exterior angle problem, think exterior angle theorem
• This theorem states: An exterior angle equals the sum of the two non-adjacent interior angles
• Non-adjacent to exterior angle CBD means angles A and C (not angle B, which is adjacent)

3. TRANSLATE the theorem into an equation

• Set up: \(\angle\mathrm{CBD} = \angle\mathrm{A} + \angle\mathrm{C}\)
• Substitute known values: \(\mathrm{117° = 46° + }\angle\mathrm{C}\)

4. SIMPLIFY to find the answer

• Solve for \(\angle\mathrm{C}\): \(\angle\mathrm{C} = \mathrm{117° - 46° = 71°}\)

Answer: B) 71°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which angles are non-adjacent to the exterior angle. They might think exterior angle CBD equals angle A plus angle B (the adjacent angle), setting up \(\mathrm{117° = 46° + }\angle\mathrm{B}\) instead of \(\mathrm{117° = 46° + }\angle\mathrm{C}\).

This leads them to find \(\angle\mathrm{B} = \mathrm{71°}\), then try to use triangle angle sum to find \(\angle\mathrm{C}\), getting \(\mathrm{180° - 46° - 71° = 63°}\). Since \(\mathrm{63°}\) isn't an answer choice, this causes confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize the exterior angle theorem and instead try to find angle B first using linear pairs (\(\angle\mathrm{B} = \mathrm{180° - 117° = 63°}\)), then use triangle angle sum. While this method works, it's more complex and creates more opportunities for arithmetic errors.

Students might calculate:

\(\mathrm{46° + 63° + }\angle\mathrm{C} = \mathrm{180°}\)

\(\mathrm{109° + }\angle\mathrm{C} = \mathrm{180°}\)

\(\angle\mathrm{C} = \mathrm{71°}\)

However, calculation errors in this longer path might lead them to Choice A (46°) if they mistakenly think \(\angle\mathrm{C} = \angle\mathrm{A}\).

The Bottom Line:

Success depends on recognizing the exterior angle theorem and correctly identifying which interior angles are non-adjacent to the given exterior angle.