Parallelograms ABCD and EFGH are graphed in the xy-plane. Parallelogram ABCD has vertices A, B, C, and D at \((1,1)\),...

1. TRANSLATE the problem information

• Given information:
  • Parallelogram ABCD: \(\mathrm{A(1,1), B(5,1), C(7,4), D(3,4)}\)
  • Parallelogram EFGH: \(\mathrm{E(1,1), F(5+m,1), G(7,4+m), H(3-m,4+m)}\)
  • Angle A measures \(\mathrm{p°}\)
  • Need to find angle F

2. INFER the geometric relationship

• Key insight: When \(\mathrm{m = 0}\), parallelogram EFGH becomes identical to ABCD
• This means \(\mathrm{E = A, F = B, G = C, H = D}\) when \(\mathrm{m = 0}\)
• Therefore, angle F corresponds to angle B in the original parallelogram

3. INFER the angle relationship in parallelograms

• In any parallelogram, consecutive angles are supplementary
• In parallelogram ABCD: \(\mathrm{∠A + ∠B = 180°}\)
• Since \(\mathrm{∠A = p°}\), we get: \(\mathrm{∠B = 180° - p°}\)

4. INFER the final answer

• Since angle F corresponds to angle B, and this relationship is preserved by the transformation
• Therefore: \(\mathrm{∠F = ∠B = 180° - p°}\)

Answer: C. \(\mathrm{(180-p)°}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize the correspondence between angles in the two parallelograms. They might try to directly calculate angle F using complex vector dot product formulas with the transformed coordinates, getting lost in algebraic manipulation without recognizing the simpler geometric relationship.

This leads to confusion and abandoning systematic solution, often resulting in guessing.

Second Most Common Error:

Inadequate INFER reasoning: Students recognize that consecutive angles are supplementary but incorrectly identify which angle F corresponds to in the original parallelogram. They might think F corresponds to A rather than B, leading them to conclude \(\mathrm{∠F = p°}\).

This may lead them to select Choice E (\(\mathrm{p°}\)).

The Bottom Line:

The key insight is recognizing that geometric transformations preserve certain angle relationships, and that the special case when \(\mathrm{m = 0}\) reveals the underlying correspondence between angles in the two parallelograms.