In triangleRST, the measure of angleR is 63°. Which of the following could be the measure, in degrees, of angleS?

1. TRANSLATE the problem information

• Given information:
  • Triangle RST has \(\angle\mathrm{R} = 63°\)
  • Need to find which value could be \(\angle\mathrm{S}\)

• What this tells us: We need to use triangle angle relationships

2. TRANSLATE the angle sum constraint

• Set up the fundamental triangle equation:
  \(\angle\mathrm{R} + \angle\mathrm{S} + \angle\mathrm{T} = 180°\)

• Substitute the known value:
  \(63° + \angle\mathrm{S} + \angle\mathrm{T} = 180°\)

3. SIMPLIFY to find the constraint on S and T

• Rearrange the equation:
  \(\angle\mathrm{S} + \angle\mathrm{T} = 180° - 63°\)
  \(\angle\mathrm{S} + \angle\mathrm{T} = 117°\)

4. INFER the constraint on angle S

• Since \(\angle\mathrm{T}\) must be positive (angles in triangles are always positive):
  \(\angle\mathrm{T} \gt 0°\)

• This means: \(\angle\mathrm{S} \lt 117°\)

5. APPLY CONSTRAINTS to eliminate invalid choices

• Check each answer choice against \(\angle\mathrm{S} \lt 117°\):
  • A. \(116° \lt 117°\) ✓ (valid)
  • B. \(118° \gt 117°\) ✗ (invalid)
  • C. \(126° \gt 117°\) ✗ (invalid)
  • D. \(180° \gt 117°\) ✗ (invalid)

Answer: A. 116

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly set up \(\angle\mathrm{S} + \angle\mathrm{T} = 117°\) but fail to recognize that \(\angle\mathrm{T}\) must be positive, so \(\angle\mathrm{S}\) must be less than 117°.

Without this insight, they might think any of the answer choices could work as long as there's some positive value for \(\angle\mathrm{T}\). This leads to confusion and guessing among the choices.

Second Most Common Error:

Conceptual confusion about triangle constraints: Students might forget that all angles in a triangle must be positive, or they might not connect this constraint to the problem.

This may lead them to select Choice B (118°) or Choice C (126°) without realizing these values would require \(\angle\mathrm{T}\) to be negative.

The Bottom Line:

The key challenge is recognizing that finding one angle in a triangle creates constraints on the others - it's not just about the angle sum equaling 180°, but about ensuring all angles remain positive.