Question:In triangle ABC, the measure of angleB is 74°. The lengths of side AB and side BC are equal. What...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC with \(\text{angle B} = 74°\)
  • Sides AB and BC have equal lengths
  • Need to find measure of angle A

2. INFER the triangle type and key property

• Since \(\mathrm{AB} = \mathrm{BC}\), this is an isosceles triangle
• Key insight: In any triangle, angles opposite equal sides are also equal
• Side BC is opposite angle A, side AB is opposite angle C
• Therefore: \(\text{angle A} = \text{angle C}\)

3. TRANSLATE this insight into an equation

• Sum of angles in any triangle = 180°
• Set up: \(\text{angle A} + \text{angle B} + \text{angle C} = 180°\)
• Since \(\text{angle A} = \text{angle C}\) and \(\text{angle B} = 74°\):
  \(\text{angle A} + 74° + \text{angle A} = 180°\)

4. SIMPLIFY to solve for angle A

• Combine like terms: \(2(\text{angle A}) + 74° = 180°\)
• Subtract 74° from both sides: \(2(\text{angle A}) = 106°\)
• Divide by 2: \(\text{angle A} = 53°\)

Answer: B (53°)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that equal sides create equal opposite angles. They might know it's an isosceles triangle but don't make the connection to angle relationships.

Without this key insight, they can't set up the fundamental equation that \(\text{angle A} = \text{angle C}\). This leads to confusion about how to use the given information, and they often resort to guessing or trying to use angle B in incorrect ways.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify the isosceles property but confuse which angles are equal. They might think \(\text{angle A} = \text{angle B} = 74°\) because they misidentify which sides are opposite which angles.

This incorrect reasoning leads them to select Choice C (74°) - thinking angle A equals the given angle B.

The Bottom Line:

The key challenge is making the connection between equal sides and equal opposite angles. Once students grasp this relationship, the algebra becomes straightforward, but without this insight, they're stuck with unusable information.