A hiker travels along a path made of straight segments, visiting five points A, B, C, D, and E in...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A hiker travels along a path made of straight segments, visiting five points A, B, C, D, and E in order. The hiker turns left at points B, C, and D. The measure of the left turn at B is \(97°\), and the measure of the left turn at D is \(56°\). The final segment DE is parallel to the initial segment AB. If each left turn is greater than \(0°\) and less than \(180°\), what is the measure, in degrees, of the left turn at C?
1. TRANSLATE the problem information
- Given information:
- Hiker travels path A → B → C → D → E
- Left turns at B (\(97°\)), C (unknown), and D (\(56°\))
- Each turn is between \(0°\) and \(180°\)
- Final segment DE is parallel to initial segment AB
- What this tells us: We need to find the turn angle at C
2. INFER the key geometric relationship
- When DE is parallel to AB, the hiker's final direction relates to the initial direction in a specific way
- Each left turn changes the hiker's heading by that many degrees
- The sum of all turns determines the total change in direction from start to finish
3. INFER the constraint from parallel segments
- Since DE ∥ AB, the final direction must be either:
- Exactly the same as initial direction (\(0°\) total change), OR
- Exactly opposite to initial direction (\(180°\) total change)
- With three positive left turns, total change can't be \(0°\)
- With each turn \(\lt 180°\), maximum possible is \(3 \times 180° = 540°\), but we need a multiple of \(180°\)
- Therefore: Total change = \(180°\)
4. TRANSLATE into an equation
- Sum of left turns = Total directional change
- \(97° + \mathrm{C} + 56° = 180°\)
5. SIMPLIFY to find the answer
- \(153° + \mathrm{C} = 180°\)
- \(\mathrm{C} = 180° - 153° = 27°\)
Answer: 27
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't connect "parallel segments" to "specific directional relationships"
Many students see that DE is parallel to AB but don't realize this creates a constraint on the total turning. They might try to use properties of parallel lines like "corresponding angles are equal" or get confused about interior vs exterior angles. Without understanding that parallel segments mean the net directional change must be \(0°\) or \(180°\), they can't set up the key equation.
This leads to confusion and guessing.
Second Most Common Error:
Inadequate APPLY CONSTRAINTS reasoning: Students set up \(97° + \mathrm{C} + 56° = 0°\) instead of \(180°\)
Some students correctly understand that parallel segments create a constraint on total turning, but they think "parallel means same direction means no net change." They don't consider that with all positive left turns, the total can't be \(0°\). Setting \(97° + \mathrm{C} + 56° = 0°\) gives \(\mathrm{C} = -153°\), which violates the constraint that turns are between \(0°\) and \(180°\).
This causes them to get stuck and realize something's wrong, leading to guessing.
The Bottom Line:
This problem requires connecting geometric intuition (what "parallel" means for direction) with algebraic constraint reasoning (what sums are actually possible given the restrictions).