Question:Convex quadrilaterals JKLM and PQRS are congruent, with vertex J corresponding to P, K to Q, L to R, and...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Convex quadrilaterals JKLM and PQRS are congruent, with vertex J corresponding to P, K to Q, L to R, and M to S. Angles \(\angle \mathrm{J}\) and \(\angle \mathrm{K}\) are right angles. If the measure of angle S is \(73^\circ\), what is the measure, in degrees, of angle L?
1. TRANSLATE the congruence information
- Given information:
- Quadrilaterals JKLM and PQRS are congruent
- Correspondence: J↔P, K↔Q, L↔R, M↔S
- Angles J and K are right angles (\(90°\) each)
- Angle S = \(73°\)
- What this tells us: Since the quadrilaterals are congruent, corresponding angles must be equal in measure.
2. INFER which angles we can determine immediately
- Since M corresponds to S, and we know angle S = \(73°\), then angle M = \(73°\)
- We already know angles J and K are both \(90°\)
- This leaves only angle L unknown
3. INFER the solution strategy
- We can use the fact that interior angles of any quadrilateral sum to \(360°\)
- Set up equation: Angle J + Angle K + Angle L + Angle M = \(360°\)
4. SIMPLIFY to solve for angle L
- Substitute known values: \(90° + 90° + \mathrm{L} + 73° = 360°\)
- Combine known angles: \(253° + \mathrm{L} = 360°\)
- Solve for L: \(\mathrm{L} = 360° - 253° = 107°\)
Answer: 107
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret the correspondence notation and think that angle L corresponds to angle S directly, leading them to conclude that angle L = \(73°\). They miss that L corresponds to R, while M corresponds to S, so it's angle M that equals \(73°\).
This may lead them to select an answer of 73 if it were an option, or causes confusion about which angle relationships to use.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly set up the equation \(90° + 90° + \mathrm{L} + 73° = 360°\) but make arithmetic errors. Common mistakes include incorrectly adding \(90° + 90° + 73°\) (getting \(243°\) instead of \(253°\)) or making subtraction errors with \(360° - 253°\).
This leads to wrong numerical answers and forces them to guess among the available choices.
The Bottom Line:
This problem tests whether students truly understand what "corresponding parts" means in congruent figures. The correspondence notation can be confusing, and students must carefully track which vertex corresponds to which to identify the correct angle relationships.