Question:Trapezoids ABCD and EFGH are congruent, where A corresponds to E, B corresponds to F, C corresponds to G, and...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Trapezoids ABCD and EFGH are congruent, where A corresponds to E, B corresponds to F, C corresponds to G, and D corresponds to H. In trapezoid ABCD, bases AB and DC are parallel, and in trapezoid EFGH, bases EF and HG are parallel. The measure of angle A is \(\mathrm{18^\circ}\), and the measure of angle B is \(\mathrm{90^\circ}\). What is the measure of angle H?
\(18^\circ\)
\(72^\circ\)
\(90^\circ\)
\(162^\circ\)
1. TRANSLATE the problem information
- Given information:
- Trapezoids ABCD and EFGH are congruent
- Vertex correspondences: A↔E, B↔F, C↔G, D↔H
- ABCD has parallel bases AB and DC
- Angle A = 18°, Angle B = 90°
- Need to find: Angle H
2. INFER the key geometric relationship
- Since AB || DC in trapezoid ABCD, angles A and D are co-interior angles
- Co-interior angles are formed when a transversal (side AD) cuts two parallel lines (AB and DC)
- Key insight: Co-interior angles are always supplementary (sum to 180°)
3. SIMPLIFY to find angle D
- Angle A + Angle D = 180° (co-interior angles)
\(\mathrm{Angle\,A + Angle\,D = 180°}\)
\(\mathrm{18° + Angle\,D = 180°}\)
\(\mathrm{Angle\,D = 180° - 18° = 162°}\)
4. INFER the final answer using congruence
- Since trapezoids ABCD and EFGH are congruent
- Corresponding angles are equal: Angle D = Angle H
- Therefore: \(\mathrm{Angle\,H = 162°}\)
Answer: D. 162 degrees
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that angles A and D are co-interior angles formed by the parallel bases and transversal AD. Instead, they might think that since the trapezoids are congruent, angle H simply equals angle A (18°), missing the intermediate step of finding angle D first.
This may lead them to select Choice A (18 degrees).
Second Most Common Error:
Conceptual confusion about angle relationships: Students might correctly identify that angles A and D are related but incorrectly apply vertical angles or alternate interior angles instead of co-interior angles. This could lead to setting Angle A = Angle D, giving them 18° for angle D, and subsequently 18° for angle H.
This also leads them to select Choice A (18 degrees).
The Bottom Line:
This problem requires understanding multiple geometric relationships: trapezoid properties, parallel line angle relationships, and congruent figure correspondences. Students must chain these concepts together rather than jumping directly to the congruence relationship.
\(18^\circ\)
\(72^\circ\)
\(90^\circ\)
\(162^\circ\)