Triangle XYZ is similar to triangle RST such that X, Y, and Z correspond to R, S, and T, respectively....
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Triangle \(\mathrm{XYZ}\) is similar to triangle \(\mathrm{RST}\) such that \(\mathrm{X, Y, and Z}\) correspond to \(\mathrm{R, S, and T}\), respectively. The measure of \(\angle \mathrm{Z}\) is \(20°\) and \(2\mathrm{XY} = \mathrm{RS}\). What is the measure of \(\angle \mathrm{T}\)?
1. TRANSLATE the problem information
- Given information:
- \(\triangle \mathrm{XYZ} \sim \triangle \mathrm{RST}\) (similar triangles)
- \(\mathrm{X} \leftrightarrow \mathrm{R}, \mathrm{Y} \leftrightarrow \mathrm{S}, \mathrm{Z} \leftrightarrow \mathrm{T}\) (correspondence)
- \(\angle\mathrm{Z} = 20°\)
- \(2\mathrm{XY} = \mathrm{RS}\) (side length relationship)
- Find: \(\angle\mathrm{T}\)
2. INFER the key relationship
- Similar triangles have a crucial property: corresponding angles are always congruent
- This means: \(\angle\mathrm{X} \cong \angle\mathrm{R}, \angle\mathrm{Y} \cong \angle\mathrm{S}, \angle\mathrm{Z} \cong \angle\mathrm{T}\)
- The side length information (\(2\mathrm{XY} = \mathrm{RS}\)) tells us about scale factor but doesn't affect angle measures
3. Apply the angle congruence
- Since \(\angle\mathrm{Z}\) corresponds to \(\angle\mathrm{T}\), and \(\angle\mathrm{Z} = 20°\)
- Therefore: \(\angle\mathrm{T} = 20°\)
Answer: C. 20°
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students get distracted by the side length relationship (\(2\mathrm{XY} = \mathrm{RS}\)) and think it somehow affects the angle measure.
They might reason: "Since the sides are in a 2:1 ratio, maybe the angle is also modified by this ratio" and calculate something like \(20° \times 2 = 40°\) or \(20° \div 2 = 10°\).
This may lead them to select Choice D (40°) or Choice B (10°).
Second Most Common Error:
Conceptual confusion: Students don't remember or properly apply the fact that similar triangles have congruent corresponding angles.
They might think angles change proportionally with side lengths, or they might confuse similar triangles with other transformations where angles could change.
This causes them to get stuck and guess among the answer choices.
The Bottom Line:
The key insight is that similarity preserves angle measures—only side lengths scale, never angles. The side length ratio is irrelevant information for finding corresponding angle measures.