In triangle DEF, the measure of angle D is 47° and the measure of angle E is 97°. In triangle...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle DEF, the measure of angle D is \(47°\) and the measure of angle E is \(97°\). In triangle RST, the measure of angle R is \(47°\) and the measure of angle S is \(97°\). Which of the following additional pieces of information is needed to determine whether triangle DEF is similar to triangle RST?
The measure of angle F
The measure of angle T
The measure of angle F and the measure of angle T
No additional information is needed.
1. TRANSLATE the problem information
- Given information:
- Triangle DEF: \(\angle\mathrm{D} = 47°\), \(\angle\mathrm{E} = 97°\)
- Triangle RST: \(\angle\mathrm{R} = 47°\), \(\angle\mathrm{S} = 97°\)
- Question asks: What additional information is needed to determine if triangles are similar?
2. INFER the similarity strategy
- To prove triangles are similar, I can use several criteria
- The most efficient is AA (Angle-Angle): if two pairs of corresponding angles are congruent, triangles are similar
- I need to check if I already have enough angle information
3. INFER the angle relationships
- Compare corresponding angles:
- \(\angle\mathrm{D} = \angle\mathrm{R} = 47°\) ✓ (first pair of congruent angles)
- \(\angle\mathrm{E} = \angle\mathrm{S} = 97°\) ✓ (second pair of congruent angles)
- Since I have two pairs of congruent corresponding angles, AA similarity criterion is satisfied
4. INFER the conclusion
- AA similarity criterion requires only TWO pairs of congruent corresponding angles
- I already have two pairs: \((47°, 47°)\) and \((97°, 97°)\)
- Therefore, no additional information is needed
Answer: D. No additional information is needed.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing that AA similarity criterion requires only two pairs of congruent angles, not all three.
Students often think they need to know all three angles of both triangles to prove similarity. They might reason: "I only know two angles from each triangle, so I need to find the third angles (F and T) before I can determine similarity." This misconception leads them to believe additional information is required.
This may lead them to select Choice C (The measure of angle F and the measure of angle T).
Second Most Common Error:
Conceptual confusion about similarity criteria: Mixing up similarity requirements with congruence requirements.
Some students confuse triangle similarity with triangle congruence. While congruence often requires more specific information (like side lengths), similarity can be established with just angle relationships. They might think more information is always needed for any triangle comparison.
This leads to confusion and may cause them to select Choice A or B thinking partial additional information is needed.
The Bottom Line:
This problem tests whether students truly understand the efficiency of the AA similarity criterion - that having two pairs of congruent corresponding angles is both necessary AND sufficient for proving triangle similarity.
The measure of angle F
The measure of angle T
The measure of angle F and the measure of angle T
No additional information is needed.