In triangle PQR, angle P = 30°. The triangle is either isosceles or right-angled, but not both. Which of the...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR with \(\mathrm{angle\ P = 30°}\)
  • Triangle is either isosceles OR right-angled (but not both)
  • Need to find which angle Q is NOT possible

2. CONSIDER ALL CASES systematically

• Since the triangle must be either isosceles OR right-angled, we need to analyze both scenarios
• For each scenario, we must check all possible configurations

3. INFER the isosceles case possibilities

• An isosceles triangle has two equal angles, so there are three ways this can happen:
  • Case 1a: \(\mathrm{P = Q}\) (both \(\mathrm{30°}\))
  • Case 1b: \(\mathrm{P = R}\) (both \(\mathrm{30°}\))
  • Case 1c: \(\mathrm{Q = R}\) (unknown value)

4. Calculate each isosceles sub-case

• Case 1a: If \(\mathrm{P = Q = 30°}\), then
  \(\mathrm{R = 180° - 30° - 30° = 120°}\)
  So \(\mathrm{Q = 30°}\) is possible
• Case 1b: If \(\mathrm{P = R = 30°}\), then
  \(\mathrm{Q = 180° - 30° - 30° = 120°}\)
  So \(\mathrm{Q = 120°}\) is possible
• Case 1c: If \(\mathrm{Q = R}\), then
  \(\mathrm{30° + Q + Q = 180°}\)
  This gives \(\mathrm{2Q = 150°}\), so \(\mathrm{Q = 75°}\)

5. INFER the right triangle case possibilities

• A right triangle has one 90° angle, so there are three possibilities:
  • Case 2a: \(\mathrm{P = 90°}\) (impossible since \(\mathrm{P = 30°}\))
  • Case 2b: \(\mathrm{Q = 90°}\)
  • Case 2c: \(\mathrm{R = 90°}\)

6. Calculate each right triangle sub-case and APPLY CONSTRAINTS

• Case 2b: If \(\mathrm{Q = 90°}\), then
  \(\mathrm{R = 180° - 30° - 90° = 60°}\)
  Check constraint: angles are \(\mathrm{30°, 90°, 60°}\) (all different, so not isosceles) ✓
• Case 2c: If \(\mathrm{R = 90°}\), then
  \(\mathrm{Q = 180° - 30° - 90° = 60°}\)
  Check constraint: angles are \(\mathrm{30°, 60°, 90°}\) (all different, so not isosceles) ✓

7. INFER the final answer

• Possible values for Q: \(\mathrm{\{30°, 60°, 75°, 90°, 120°\}}\)
• Looking at answer choices: \(\mathrm{45°}\) is not in this set

Answer: B (\(\mathrm{45°}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often analyze only one or two configurations instead of systematically checking all possibilities for both isosceles and right triangle cases.

For example, they might only consider the case where \(\mathrm{Q = R}\) for isosceles triangles, getting \(\mathrm{Q = 75°}\), and only consider \(\mathrm{Q = 90°}\) for right triangles. This incomplete analysis might lead them to think other values like \(\mathrm{45°}\) could be possible, causing them to guess or select the wrong answer.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly find possible angle measures but forget to verify that each configuration satisfies 'either isosceles OR right-angled, but not both.'

They might incorrectly include cases where a triangle is both isosceles and right-angled (like a \(\mathrm{45-45-90}\) triangle), or miss that certain angle combinations violate the constraint. This leads to confusion about which values are actually valid.

The Bottom Line:

This problem requires systematic case analysis combined with careful constraint checking. Students who jump to conclusions without methodically examining all possibilities will miss valid angle measures or incorrectly include invalid ones.