In triangle XYZ, angle Y is a right angle, point P lies on XZ, and point Q lies on YZ...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle XYZ, \(\angle \mathrm{Y}\) is a right angle, point P lies on XZ, and point Q lies on YZ such that PQ is parallel to XY. If the measure of \(\angle \mathrm{XZY}\) is \(63°\), what is the measure, in degrees, of \(\angle \mathrm{XPQ}\)?
1. TRANSLATE the problem information
- Given information:
- Triangle XYZ with right angle at Y (\(\mathrm{angle\;Y = 90°}\))
- \(\mathrm{Angle\;XZY = 63°}\)
- Point P lies on XZ, point Q lies on YZ
- PQ is parallel to XY
- Need to find angle XPQ
2. INFER the parallel line relationships
- Since \(\mathrm{PQ \parallel XY}\), corresponding angles are equal
- This means \(\mathrm{angle\;ZQP = angle\;XYZ = 90°}\)
- We now have a right triangle ZPQ with \(\mathrm{angle\;ZQP = 90°}\)
3. INFER the angle calculation in triangle ZPQ
- In triangle ZPQ:
- \(\mathrm{Angle\;ZQP = 90°}\)
- \(\mathrm{Angle\;QZP = angle\;XZY = 63°}\) (same angle)
- \(\mathrm{Angle\;ZPQ = 180° - 90° - 63° = 27°}\)
4. INFER the supplementary relationship
- Angles XPQ and ZPQ form a linear pair on line XZ
- Therefore: \(\mathrm{angle\;XPQ + angle\;ZPQ = 180°}\)
- So: \(\mathrm{angle\;XPQ = 180° - 27° = 153°}\)
Answer: 153°
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students fail to recognize that \(\mathrm{PQ \parallel XY}\) creates corresponding angles, specifically that \(\mathrm{angle\;ZQP = 90°}\).
Without this key insight, students might try to work directly with the given angle of 63° without establishing the right triangle ZPQ. This leads to confusion about which angles to use and how they relate to each other. This causes them to get stuck and guess randomly.
Second Most Common Error:
Poor INFER reasoning: Students correctly find \(\mathrm{angle\;ZPQ = 27°}\) but then confuse this with angle XPQ directly, giving 27° as their final answer.
They miss that XPQ and ZPQ are supplementary angles on a straight line. Instead of recognizing the linear pair relationship, they assume ZPQ is the answer being asked for. This may lead them to select an incorrect choice if 27° were among the options.
The Bottom Line:
This problem requires students to chain together multiple geometric relationships: parallel lines create corresponding angles, which helps establish a right triangle, whose angles can be calculated, leading to a supplementary angle relationship for the final answer. Students who break this chain at any point will struggle to reach the correct solution.