Isosceles triangles ABC and XYZ are congruent, where A corresponds to X, B corresponds to Y, and C corresponds to...

1. INFER the key triangle properties

• Given information:
  • Triangle ABC is isosceles with \(\mathrm{AB = AC}\)
  • Angle A = 48° (vertex angle)
  • Triangle XYZ is congruent to ABC with B corresponding to Y

• Key insight: In an isosceles triangle, the two base angles (opposite the equal sides) are equal in measure. Since \(\mathrm{AB = AC}\), angles B and C are the base angles and \(\mathrm{\angle B = \angle C}\).

2. INFER the solution approach

• We need to find \(\mathrm{\angle B}\) first, then use the correspondence to find \(\mathrm{\angle Y}\)
• Use the angle sum theorem since we know one angle and the relationship between the other two

3. TRANSLATE the angle relationship into an equation

• Angle sum theorem: \(\mathrm{\angle A + \angle B + \angle C = 180°}\)
• Since \(\mathrm{\angle B = \angle C}\) and \(\mathrm{\angle A = 48°}\):
• \(\mathrm{48° + \angle B + \angle B = 180°}\)
• \(\mathrm{48° + 2\angle B = 180°}\)

4. SIMPLIFY to find the base angle measure

• \(\mathrm{2\angle B = 180° - 48°}\)
• \(\mathrm{2\angle B = 132°}\)
• \(\mathrm{\angle B = 66°}\)

5. INFER the final answer using triangle correspondence

• Since triangles ABC and XYZ are congruent and B corresponds to Y
• \(\mathrm{\angle Y = \angle B = 66°}\)

Answer: B) 66°

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about isosceles triangles: Students might think the vertex angle and base angles are equal, or confuse which angle is the vertex angle.

If they incorrectly assume all angles in an isosceles triangle are equal, they'd calculate: \(\mathrm{180° \div 3 = 60°}\) for each angle. This leads to confusion since 60° doesn't match any answer choice exactly, causing them to guess or select the closest value.

Second Most Common Error:

Weak INFER skill regarding triangle correspondence: Students solve correctly for \(\mathrm{\angle B = 66°}\) but then select \(\mathrm{\angle A = 48°}\) as the answer because they don't properly apply the correspondence between congruent triangles.

They might think "angle Y corresponds to angle A" instead of "angle Y corresponds to angle B," leading them to select Choice A (48°).

The Bottom Line:

This problem requires students to work with multiple geometric concepts simultaneously - isosceles triangle properties, angle sum theorem, and congruent triangle correspondence. Success depends on correctly identifying which angles are equal in an isosceles triangle and then properly applying the correspondence between congruent figures.