In triangle ABC, the exterior angle at vertex A has a measure of 125°. In triangle PQR, the exterior angle...

1. TRANSLATE the exterior angle information

• Given information:
  • Triangle ABC: exterior angle at A = 125°, \(\mathrm{∠B = 35°}\)
  • Triangle PQR: exterior angle at P = 125°, \(\mathrm{∠Q = 35°}\)

• Convert to interior angles: An exterior angle and its corresponding interior angle are supplementary (sum to 180°)
  • \(\mathrm{∠A = 180° - 125° = 55°}\)
  • \(\mathrm{∠P = 180° - 125° = 55°}\)

2. INFER which similarity criterion applies

• We now have:
  • Triangle ABC: \(\mathrm{∠A = 55°}\), \(\mathrm{∠B = 35°}\)
  • Triangle PQR: \(\mathrm{∠P = 55°}\), \(\mathrm{∠Q = 35°}\)

• Two pairs of corresponding angles are congruent: \(\mathrm{∠A = ∠P}\) and \(\mathrm{∠B = ∠Q}\)

• By the AA (Angle-Angle) similarity criterion, this is sufficient to prove triangle similarity

3. INFER that no additional information is needed

• The AA criterion requires only two pairs of congruent corresponding angles
• The third pair of angles must also be congruent because triangle angles sum to 180°
• No additional angle measurements or side lengths are necessary

Answer: D (No additional information is needed)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recall that AA similarity requires only two pairs of congruent angles, not all three pairs.

They think: "I only know two angles in each triangle, so I need to know the third angles too to be sure the triangles are similar."

This leads them to select Choice C (The measures of angle C and angle R) or Choice A/B thinking they need more angle information.

Second Most Common Error:

Poor TRANSLATE reasoning: Students struggle with the exterior angle to interior angle conversion.

They either forget that exterior and interior angles are supplementary, or they make arithmetic errors in the conversion (\(\mathrm{125° - 180°}\) instead of \(\mathrm{180° - 125°}\)).

This leads to confusion about what angles they actually have, causing them to get stuck and guess.

The Bottom Line:

The key insight is recognizing that triangle similarity doesn't require knowing all three angles explicitly - the AA criterion with just two pairs of congruent corresponding angles is mathematically sufficient because the third pair must automatically be congruent.