Triangles ABC and DEF are congruent, where A corresponds to D, and B and E are right angles. The measure...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Triangles \(\mathrm{ABC}\) and \(\mathrm{DEF}\) are congruent, where \(\mathrm{A}\) corresponds to \(\mathrm{D}\), and \(\mathrm{B}\) and \(\mathrm{E}\) are right angles. The measure of angle \(\mathrm{A}\) is \(69°\). What is the measure, in degrees, of angle \(\mathrm{F}\)?
1. TRANSLATE the problem information
- Given information:
- Triangles ABC and DEF are congruent
- A corresponds to D (correspondence mapping)
- B and E are right angles
- Angle A = 69°
- Need to find angle F
2. INFER what congruence tells us
- Since the triangles are congruent with A corresponding to D:
- Corresponding angles are equal
- Therefore: \(\mathrm{angle\ D = angle\ A = 69°}\)
3. TRANSLATE the right angle information
- Since B and E are right angles:
- \(\mathrm{Angle\ E = 90°}\)
4. INFER the solution strategy
- We have two angles in triangle DEF (angles D and E)
- We can use the triangle angle sum theorem to find the third angle (angle F)
5. SIMPLIFY using triangle angle sum theorem
- In triangle DEF: \(\mathrm{angle\ D + angle\ E + angle\ F = 180°}\)
- Substitute known values: \(\mathrm{69° + 90° + angle\ F = 180°}\)
- Combine: \(\mathrm{159° + angle\ F = 180°}\)
- Solve: \(\mathrm{angle\ F = 180° - 159° = 21°}\)
Answer: 21
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret the correspondence statement "A corresponds to D" and assume they need to find angle A instead of using it to determine angle D.
They might think: "If A corresponds to D, then I need to find what A equals" rather than "Since A corresponds to D, and A = 69°, then D = 69°." This leads to confusion about which triangle to work in and which angles they actually know.
This leads to confusion and guessing.
Second Most Common Error:
Poor INFER reasoning: Students don't recognize that they should work within one triangle (DEF) using the angle sum theorem after establishing the known angle measures.
Instead, they might try to work across both triangles simultaneously or forget to use the triangle angle sum property. They get stuck trying to figure out how to connect the information from both triangles.
This causes them to get stuck and randomly select an answer.
The Bottom Line:
Success requires clearly translating the correspondence relationship (\(\mathrm{A ↔ D}\) means equal angles) and then strategically focusing on one triangle with known angle measures to apply the fundamental triangle angle sum theorem.