In triangle triangle ABC, angle angle B is a right angle. Point D lies on AC, and point E lies...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle \(\triangle \mathrm{ABC}\), angle \(\angle \mathrm{B}\) is a right angle. Point \(\mathrm{D}\) lies on \(\mathrm{AC}\), and point \(\mathrm{E}\) lies on \(\mathrm{BC}\) such that \(\mathrm{DE}\) is parallel to \(\mathrm{AB}\). If the measure of angle \(\angle \mathrm{ACB} = 62°\), what is the measure, in degrees, of the obtuse angle \(\angle \mathrm{EDA}\)?
1. TRANSLATE the problem information
- Given information:
- Triangle ABC with right angle at B (\(\angle\mathrm{ABC} = 90°\))
- \(\angle\mathrm{ACB} = 62°\)
- \(\mathrm{DE} \parallel \mathrm{AB}\) (DE is parallel to AB)
- Points D on AC, E on BC
- What we need: The obtuse angle EDA
2. INFER what angle information we can find first
- Since we have two angles in triangle ABC, we can find the third angle
- We need \(\angle\mathrm{BAC}\) because it will help us with the parallel line relationships
3. Find angle BAC using angle sum property
- In any triangle, the three angles sum to 180°
- \(\angle\mathrm{BAC} + \angle\mathrm{ABC} + \angle\mathrm{ACB} = 180°\)
- \(\angle\mathrm{BAC} + 90° + 62° = 180°\)
- \(\angle\mathrm{BAC} = 180° - 152° = 28°\)
4. INFER how parallel lines create angle relationships
- Since DE is parallel to AB, and AC cuts through both lines as a transversal
- Corresponding angles are equal when parallel lines are cut by a transversal
- The acute angle that DE makes with AC equals \(\angle\mathrm{BAC} = 28°\)
5. INFER which angle we actually need
- We found the acute angle between DE and DA is 28°
- But the problem asks for the obtuse angle EDA
- These two angles are supplementary (they form a straight line along DA)
6. Calculate the obtuse angle
- Obtuse angle EDA = \(180° - 28° = 152°\)
Answer: 152
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students correctly find \(\angle\mathrm{BAC} = 28°\) but fail to recognize the parallel line relationship that makes this angle equal to the acute angle at D.
Instead, they might try to use the 62° angle directly or get confused about which angles are related. Without recognizing that \(\mathrm{DE} \parallel \mathrm{AB}\) creates corresponding angles, they cannot connect the triangle's angles to the angle at point D. This leads to confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students find the correct 28° acute angle at D but don't realize the problem asks for the "obtuse angle EDA."
They submit 28° instead of recognizing they need the supplementary angle \(180° - 28° = 152°\). This causes them to select an answer that represents the acute angle rather than the obtuse one.
The Bottom Line:
This problem requires students to bridge two different geometric concepts: triangle angle relationships and parallel line properties. The key insight is recognizing that information from the triangle (\(\angle\mathrm{BAC}\)) transfers to the parallel line configuration through corresponding angles, and then distinguishing between acute and obtuse angles at the same vertex.