Triangle;ABC is drawn so that, at vertex;A, angle;A and its adjacent exterior angle form a linear pair (a straight line).The...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
- \(\mathrm{Triangle\;ABC}\) is drawn so that, at \(\mathrm{vertex\;A}\), \(\mathrm{angle\;A}\) and its adjacent exterior angle form a linear pair (a straight line).
- The exterior angle at \(\mathrm{A}\) measures \(\mathrm{128°}\).
- The measure of \(\mathrm{angle\;B}\) is \(\mathrm{43°}\).
- What is the measure of \(\mathrm{angle\;C}\)?
1. TRANSLATE the problem information
- Given information:
- Triangle ABC has angle A and its adjacent exterior angle forming a linear pair
- The exterior angle at A measures 128°
- Angle B measures 43°
- Need to find angle C
- What this tells us: We have a triangle with one known interior angle and one known exterior angle at the same vertex.
2. INFER the most efficient approach
- We can solve this problem using two different theorem combinations:
- Option 1: Use linear pair property to find angle A, then use triangle angle sum
- Option 2: Use exterior angle theorem directly
- Both approaches will work, so let's use the linear pair method first.
3. SIMPLIFY to find angle A using linear pair property
- Since angle A and its exterior angle form a linear pair:
\(\mathrm{angle\,A + 128° = 180°}\) - Therefore: \(\mathrm{angle\,A = 180° - 128° = 52°}\)
4. SIMPLIFY to find angle C using triangle angle sum
- In any triangle: \(\mathrm{A + B + C = 180°}\)
- Substituting known values: \(\mathrm{52° + 43° + C = 180°}\)
- Solving: \(\mathrm{C = 180° - (52° + 43°) = 180° - 95° = 85°}\)
Answer: (D) 85
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "adjacent exterior angle" and confuse which angle measures 128°. They might think angle A itself is 128°, leading them to calculate \(\mathrm{C = 180° - (128° + 43°) = 9°}\). Since 9° isn't among the choices, this leads to confusion and guessing.
Second Most Common Error:
Missing conceptual knowledge about exterior angle theorem: Students know the linear pair concept but don't recognize they could use the exterior angle theorem as a shortcut. While this doesn't lead to a wrong answer, it makes the problem seem more complicated than necessary and increases chances of arithmetic errors.
The Bottom Line:
This problem tests whether students can translate geometric language into mathematical relationships and recognize that exterior angles provide multiple solution pathways. The key insight is seeing that "linear pair" and "exterior angle" give you two different ways to connect the same geometric information.