In the figure above, triangle FGH is similar to triangle JKL. The area of triangle JKL is 9 times the...

1. TRANSLATE the similarity statement

The problem states "triangle FGH is similar to triangle JKL," written as \(\triangle\mathrm{FGH} \sim \triangle\mathrm{JKL}\).

What this notation tells us:

• The ORDER of the vertices matters
• F corresponds to J
• G corresponds to K
• H corresponds to L

This vertex correspondence is the key to identifying which angles are equal.

2. INFER which angles we're comparing

We need angle JLK (the angle at vertex L in triangle JKL).

From our vertex correspondence, L corresponds to H.

Therefore, angle JLK corresponds to angle FHG.

3. Apply the similarity property

INFER what similarity means for angles: Similar triangles have congruent corresponding angles. This is a fundamental property—angles don't change even when triangles are different sizes.

Since angle JLK corresponds to angle FHG:

• \(\angle\mathrm{JLK} = \angle\mathrm{FHG} = 32°\)

4. Check the area information

The problem mentions that the area of △JKL is 9 times the area of △FGH.

INFER whether this matters: This tells us the triangles are different sizes (scale factor is 3:1), but angle measures are UNCHANGED in similar figures. This information is designed to distract you—don't use it!

Answer: 32°, which is Choice C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill—Getting distracted by the area ratio: Students see "9 times the area" and think this must be used in the solution. They might multiply the angle by 3 (since \(\sqrt{9} = 3\)) or by 9, getting:

• \(32° \times 3 = 96°\) → This may lead them to select Choice B (96)
• \(32° \times 9 = 288°\) → This may lead them to select Choice D (288)

The misconception is thinking that ALL measurements scale proportionally, when in fact angles remain constant in similar figures.

Second Most Common Error:

Poor TRANSLATE reasoning—Misunderstanding vertex correspondence: Students don't realize the order of vertices matters in similarity statements. They might:

• Not establish the correspondence \(\mathrm{H} \leftrightarrow \mathrm{L}\)
• Randomly guess which angles correspond
• Try to compare angle FHG with angle JKL instead of angle JLK

This leads to confusion and guessing among all answer choices.

The Bottom Line:

This problem tests whether you understand that similarity affects LINEAR measurements and AREAS (which scale by k²), but ANGLES remain congruent. The area information is deliberately placed to see if you'll get sidetracked. The key is staying focused on the fundamental property: corresponding angles in similar triangles are always equal.