Question:In triangle ABC, point B lies on line l, and side BC is extended to form an exterior angle at...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC with \(\angle\mathrm{ABC} = 50°\)
  • Exterior angle at vertex C = 130°
  • Need to find \(\angle\mathrm{BAC}\)

2. INFER the key relationship

• The exterior angle at C and the interior angle \(\angle\mathrm{ACB}\) form a linear pair
• This means they're supplementary: \(\mathrm{Interior}\;\angle\mathrm{ACB} + \mathrm{Exterior\;angle} = 180°\)
• Calculate: \(\mathrm{Interior}\;\angle\mathrm{ACB} = 180° - 130° = 50°\)

3. INFER the solution strategy

• Now we have two interior angles of the triangle: \(\angle\mathrm{ABC} = 50°\) and \(\angle\mathrm{ACB} = 50°\)
• Use triangle angle sum theorem to find the third angle

4. SIMPLIFY using triangle angle sum theorem

• \(\angle\mathrm{BAC} + \angle\mathrm{ABC} + \angle\mathrm{ACB} = 180°\)
• \(\angle\mathrm{BAC} + 50° + 50° = 180°\)
• \(\angle\mathrm{BAC} = 180° - 100° = 80°\)

Answer: D) 80°

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about exterior angles: Students might think the exterior angle IS one of the triangle's interior angles, rather than understanding it's supplementary to the interior angle at C.

If they use 130° as the interior angle \(\angle\mathrm{ACB}\), they would calculate:

\(\angle\mathrm{BAC} + 50° + 130° = 180°\)
\(\angle\mathrm{BAC} = 0°\)

This creates an impossible result, leading to confusion and guessing.

Second Most Common Error:

Weak INFER skill: Students might recognize that exterior angles relate to triangles but incorrectly try to apply the exterior angle theorem (exterior angle equals sum of two non-adjacent interior angles) without first finding the interior angle.

They might think: \(130° = \angle\mathrm{BAC} + \angle\mathrm{ABC} = \angle\mathrm{BAC} + 50°\), so \(\angle\mathrm{BAC} = 80°\). While this gives the right answer, it's based on incorrect reasoning since \(\angle\mathrm{ABC}\) is adjacent to the exterior angle at C, not non-adjacent.

The Bottom Line:

The key challenge is recognizing that you must first convert the exterior angle to its corresponding interior angle before applying triangle properties. Students often jump directly to triangle theorems without this crucial first step.