In quadrilateral PQRS, the measures of angleP and angleQ are each 76°, and the measure of angleR is 112°. What...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In quadrilateral PQRS, the measures of \(\angle\mathrm{P}\) and \(\angle\mathrm{Q}\) are each \(76°\), and the measure of \(\angle\mathrm{R}\) is \(112°\). What is the measure of \(\angle\mathrm{S}\), in degrees? (Disregard the degree symbol when entering your answer.)
1. TRANSLATE the problem information
- Given information:
- Quadrilateral PQRS with three known angles
- \(\angle\mathrm{P} = 76°\), \(\angle\mathrm{Q} = 76°\), \(\angle\mathrm{R} = 112°\)
- Need to find \(\angle\mathrm{S}\)
2. INFER the approach
- Key insight: Use the quadrilateral angle sum property
- Strategy: Set up an equation where all four angles sum to 360°
- This gives us one equation with one unknown, which we can solve directly
3. TRANSLATE into mathematical equation
- Set up: \(\angle\mathrm{P} + \angle\mathrm{Q} + \angle\mathrm{R} + \angle\mathrm{S} = 360°\)
- Substitute known values: \(76° + 76° + 112° + \angle\mathrm{S} = 360°\)
4. SIMPLIFY to solve for the unknown angle
- Add the known angles: \(76° + 76° + 112° = 264°\)
- Our equation becomes: \(264° + \angle\mathrm{S} = 360°\)
- Solve for \(\angle\mathrm{S}\): \(\angle\mathrm{S} = 360° - 264° = 96°\)
Answer: 96
Why Students Usually Falter on This Problem
Most Common Error Path:
Missing conceptual knowledge: Students might confuse quadrilaterals with triangles and use 180° as the angle sum instead of 360°.
This leads them to set up the equation as: \(76° + 76° + 112° + \angle\mathrm{S} = 180°\), giving them \(264° + \angle\mathrm{S} = 180°\), so \(\angle\mathrm{S} = 180° - 264° = -84°\). This negative result should alert them that something is wrong, but it often leads to confusion and guessing.
Second Most Common Error:
Weak SIMPLIFY execution: Students make arithmetic mistakes when adding the three given angles or when performing the final subtraction.
For example, they might incorrectly calculate \(76° + 76° + 112°\) as \(254°\) instead of \(264°\), leading to \(\angle\mathrm{S} = 360° - 254° = 106°\). This seems reasonable but is incorrect due to the calculation error.
The Bottom Line:
This problem tests whether students can recall and correctly apply the fundamental property that quadrilateral angles sum to 360°, then execute the arithmetic accurately. The conceptual knowledge is more critical than the computational steps.