In a right triangle, the measure of one of the acute angles is 51°. What is the measure, in degrees,...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In a right triangle, the measure of one of the acute angles is \(51°\). What is the measure, in degrees, of the other acute angle?
1. TRANSLATE the problem information
- Given information:
- We have a right triangle
- One of the acute angles measures \(51°\)
- What we need to find: The measure of the other acute angle
2. INFER the key relationship
- Since we have a right triangle, one angle is exactly \(90°\)
- All triangle interior angles sum to \(180°\)
- This means: \(\mathrm{acute\ angle\ 1} + \mathrm{acute\ angle\ 2} + 90° = 180°\)
- Therefore: \(\mathrm{acute\ angle\ 1} + \mathrm{acute\ angle\ 2} = 90°\)
3. SIMPLIFY to find the unknown angle
- We know one acute angle = \(51°\)
- Other acute angle = \(90° - 51° = 39°\)
Answer: B. 39
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about triangle properties: Students may forget that the angles in ANY triangle sum to \(180°\), or they may not recognize that in a right triangle, the two acute angles specifically must sum to \(90°\).
Some students might incorrectly think the two acute angles should be equal, leading them to calculate \(180° ÷ 3 = 60°\), then get confused when this doesn't match any answer choice. This leads to confusion and guessing.
Second Most Common Error:
Weak SIMPLIFY execution: Students understand the correct approach but make an arithmetic error when calculating \(90° - 51°\).
They might miscalculate and get \(90° - 51° = 49°\) instead of \(39°\), leading them to select Choice C (49).
The Bottom Line:
This problem tests whether students can connect multiple geometric concepts: triangle angle sum, properties of right triangles, and basic subtraction. The key insight is recognizing that in a right triangle, the two acute angles are complementary (sum to \(90°\)).