In triangle FGH, the measure of an exterior angle at vertex G is 124°. If the measure of interior angle...

1. TRANSLATE the problem information

• Given information:
  • Triangle FGH with exterior angle at vertex G = \(124°\)
  • Interior angle H = \(61°\)
  • Need to find interior angle F

• What this tells us: We have an exterior angle and one of its opposite interior angles

2. INFER the most efficient approach

• The Exterior Angle Theorem gives us a direct relationship
• Since we know the exterior angle at G and one opposite interior angle (H), we can find the other opposite interior angle (F)
• This avoids the need to find interior angle G first

3. TRANSLATE the Exterior Angle Theorem into an equation

• Exterior angle = sum of opposite interior angles
• \(124° = \mathrm{angle\ F} + \mathrm{angle\ H}\)
• \(124° = \mathrm{angle\ F} + 61°\)

4. SIMPLIFY to solve for angle F

• Subtract \(61°\) from both sides:
• \(\mathrm{angle\ F} = 124° - 61° = 63°\)

Answer: B. \(63°\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize which angles are "opposite" to the exterior angle at G. They might incorrectly think that angle G (interior) is opposite to the exterior angle at G, leading them to set up: \(124° = \mathrm{angle\ F} + \mathrm{angle\ G\ (interior)}\). Since they don't know angle G, this creates confusion and they may guess or attempt incorrect calculations.

This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about exterior angles: Students may confuse the Exterior Angle Theorem with the Triangle Angle Sum Theorem and try to write: \(\mathrm{angle\ F} + 124° + 61° = 180°\). This gives \(\mathrm{angle\ F} = 180° - 185° = -5°\), which is impossible. Recognizing this impossibility, they might select the largest positive answer choice available.

This may lead them to select Choice C (\(117°\)).

The Bottom Line:

Success depends on correctly identifying which interior angles are opposite to a given exterior angle. The exterior angle at vertex G is opposite to the interior angles at vertices F and H—not to the interior angle at G itself.