In convex pentagon ABCDE, segment AB is parallel to segment DE. The measure of angle B is 139 degrees, and...

1. TRANSLATE the problem information

• Given information:
  • Pentagon ABCDE is convex
  • AB is parallel to DE
  • Angle B = \(139°\)
  • Angle D = \(174°\)
  • Need to find angle C

2. VISUALIZE the geometric setup

• VISUALIZE by extending sides BC and DE beyond the pentagon until they meet
• This creates a triangle where we can apply our parallel line knowledge
• The extended BC acts as a transversal cutting through the parallel lines AB and DE

3. INFER the relationship between interior and exterior angles

• At vertex B: exterior angle = \(180° - 139° = 41°\)
• At vertex D: exterior angle = \(180° - 174° = 6°\)

4. INFER the parallel line relationships

• Since AB || DE and extended BC is a transversal:
• The exterior angle at B \(41°\) and angle CXD are alternate interior angles
• Therefore: angle CXD = \(41°\)

5. INFER how to find angle C using the exterior angle theorem

• In triangle CDX, angle C of the pentagon is an exterior angle
• By exterior angle theorem: exterior angle = sum of two non-adjacent interior angles
• Angle C = angle CDX + angle CXD = \(6° + 41° = 47°\)

Answer: 47

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students don't extend the sides to create the necessary triangle configuration. Instead, they try to work directly with the pentagon angles using only the angle sum formula for pentagons \(540°\). While they know three angles \(139°\), \(174°\), and the unknown C, they don't have enough information about angles A and E to use this approach effectively. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about parallel lines: Even if students extend the sides correctly, they struggle to identify which angles are alternate interior angles. They might confuse corresponding angles with alternate interior angles, or incorrectly identify which transversal creates the parallel line relationship. This leads them to set up incorrect angle equations and arrive at wrong values.

The Bottom Line:

This problem requires spatial visualization skills to see beyond the given pentagon and create a workable triangle. Most students get stuck because they don't recognize that extending sides strategically unlocks the parallel line properties they need.