Two triangles are *similar* when they have the same shape but different size — corresponding angles match and corresponding sides are in the same ratio. The SAT uses similarity to test proportional reasoning in geometry.
Are two angles of one triangle equal to two angles of the other?
Yes ↓
→ Triangles are similar by AA — set up the ratio of corresponding sides
No ↓
Is there a line drawn parallel to one side of a triangle?
Yes ↓
→ The smaller triangle is similar to the larger by AA (same angle at the shared vertex + parallel lines = same angles)
No ↓
Are all three pairs of sides in the same ratio?
Yes ↓
→ Triangles are similar by SSS
No ↓
→ Need more info — check for SAS (two sides proportional + included angle)
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Quick check
Set up the proportion using corresponding sides (read the similarity statement carefully). Remember to SQUARE the linear ratio if the question is about area.
Two sides of a triangle have lengths 5 and 11. Which of the following could be the length of the third side?
Worked examples
Example 1
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Example 2
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Common pitfalls
Using the linear ratio for area
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Setting up the proportion with mismatched corresponding sides
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Forgetting that AA only requires two angles
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Mixing up similar with congruent
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Key takeaways
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Tracks your progress across lessons.
Skill check
10 questions on Lines, Angles, and Triangles, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.
Similar Triangles — SAT, PSAT & ACT Lesson | UnlimitedTests
Similar triangles have:
All three pairs of corresponding angles equal.
All three pairs of corresponding sides in the same ratio (the scale factor).
If △ABC∼△DEF with scale factor k:
ABDE=BCEF=ACDF=k
Three ways to prove triangles similar (any one is enough):
AA (angle-angle): two pairs of corresponding angles equal. (The third pair must match because angles sum to 180°.)
SSS (side-side-side): all three pairs of corresponding sides are in the same ratio.
SAS (side-angle-side): two pairs of sides in proportion AND the included angle equal.
The SAT favorite — AA. Most SAT problems set up similar triangles by showing two pairs of angles. Often it's a triangle inside another triangle sharing one angle with parallel-line splits.
Common SAT setup: a line drawn parallel to one side of a triangle creates a smaller triangle similar to the original.
Example: In △ABC, DE∥BC with D on AB and E on AC. Then △ADE∼△ABC.
The corresponding sides give a proportion: ABAD=ACAE=BCDE.
Critical SAT facts about similarity:
Perimeter ratio = scale factor. If sides scale by k, perimeter scales by k.
Area ratio = scale factor SQUARED. If sides scale by 3, area scales by 9.
Volume ratio = scale factor CUBED. If sides scale by 2, volume scales by 8.
This is the most common SAT trap — students forget to square the ratio for area, and apply linear scaling instead.
Similar vs congruent.Congruent triangles are similar with scale factor 1 (same size and shape). Similar allows different sizes.
Quick test in word problems:if a 6-foot tall person casts a 4-foot shadow, how tall is a tree casting a 30-foot shadow at the same time? The two right triangles (person + shadow, tree + shadow) are similar (same sun angle).
person shadowperson height=tree shadowtree height⇒46=30h⇒h=45 ft
Triangles △ABC and △DEF are similar with DEAB=52. If the area of △DEF is 100, what is the area of △ABC?
In △ABC, point D is on AB and point E is on AC such that DE∥BC. If AD=4, DB=6, and DE=8, what is BC?
If sides scale by 3, area scales by 9 (not 3). The most common SAT trap on similar-triangle area problems. Square the side ratio for area. Cube it for volume.
When writing DEAB=EFBC, AB must correspond to DE, BC to EF — based on which vertices match in the similarity statement △ABC∼△DEF. Mismatched correspondences give wrong proportions.
If two pairs of angles are equal, the third pair MUST also be equal (angles sum to 180° in every triangle). So AA is enough to prove similarity — you don't need a third angle or any side.
Similar = same shape, different size (scale factor can be anything). Congruent = same shape, same size (scale factor = 1). SAT problems usually want similar — corresponding sides PROPORTIONAL, not equal.
Similar triangles: same angles, proportional sides. Scale factor k = side ratio.
AA, SSS, SAS — any one establishes similarity.
Area ratio = k2. Volume ratio = k3.
A line parallel to one side of a triangle creates a similar smaller triangle.
Match corresponding sides based on the similarity statement order (△ABC∼△DEF means A↔D, etc.).