Quadrilaterals PQRS and WXYZ are similar, where P, Q, and R correspond to W, X, and Y, respectively. The measure...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Quadrilaterals PQRS and WXYZ are similar, where P, Q, and R correspond to W, X, and Y, respectively. The measure of \(\angle\mathrm{S}\) is \(135°\), \(\mathrm{PS} = 45\), and \(\mathrm{WZ} = 9\). What is the measure of \(\angle\mathrm{Z}\)?
1. TRANSLATE the correspondence information
- Given information:
- Quadrilaterals PQRS and WXYZ are similar
- P, Q, and R correspond to W, X, and Y, respectively
- \(\angle\mathrm{S} = 135°\)
- What this correspondence pattern tells us:
- \(\mathrm{P} \leftrightarrow \mathrm{W}\)
- \(\mathrm{Q} \leftrightarrow \mathrm{X}\)
- \(\mathrm{R} \leftrightarrow \mathrm{Y}\)
- Therefore: \(\mathrm{S} \leftrightarrow \mathrm{Z}\) (following the same order)
2. INFER the relationship between corresponding angles
- Key insight: In similar figures, corresponding angles are always equal in measure
- Since S corresponds to Z, we know \(\angle\mathrm{S} = \angle\mathrm{Z}\)
3. Apply the angle relationship
- Given: \(\angle\mathrm{S} = 135°\)
- Since \(\angle\mathrm{S} = \angle\mathrm{Z}\), we have \(\angle\mathrm{Z} = 135°\)
Answer: D. 135°
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand the correspondence pattern and think they need to use the given side lengths (\(\mathrm{PS} = 45\), \(\mathrm{WZ} = 9\)) to find the angle measure.
They might try to set up a proportion or think the angle measure is related to the side length ratio, not realizing that angle measures in similar figures are simply equal. This leads to confusion and guessing among the numerical choices.
Second Most Common Error:
Missing conceptual knowledge about similar figures: Students don't remember that corresponding angles in similar figures are equal.
They might think \(\angle\mathrm{Z}\) is the supplement of \(\angle\mathrm{S}\) (\(180° - 135° = 45°\)), leading them to select Choice C (45°).
The Bottom Line:
This problem tests whether students understand the fundamental property of similar figures - that corresponding angles are equal - and whether they can correctly interpret correspondence statements to identify which angles correspond to each other.