Math

Similar Triangles

5 min readMedium5-question drill

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.

Where this lesson fits

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Right Triangles→

Similar Triangles

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Similar triangles have:

All three pairs of corresponding angles equal.
All three pairs of corresponding sides in the same ratio (the scale factor).

If $\triangle ABC \sim \triangle DEF$ with scale factor $k$ :

$\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k$

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Linear, area, volume ratios on similar figures

Side ratio (linear)	Area ratio	Volume ratio
1 : 2	1 : 4	1 : 8
1 : 3	1 : 9	1 : 27
2 : 5	4 : 25	8 : 125
1 : k	1 : $k^2$	1 : $k^3$

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 $\triangle ABC$ , $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$ . Then $\triangle ADE \sim \triangle ABC$ .

The corresponding sides give a proportion: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$ .

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Recognizing similarity in a triangle problem

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)

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).

$\frac{\text{person height}}{\text{person shadow}} = \frac{\text{tree height}}{\text{tree shadow}} \Rightarrow \frac{6}{4} = \frac{h}{30} \Rightarrow h = 45 \text{ ft}$

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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 similar triangles have corresponding sides in the ratio 1:2. If a side of the smaller triangle is 8, what is the corresponding side of the larger triangle?

Worked examples

Example 1

Triangles $\triangle ABC$ and $\triangle DEF$ are similar with $\frac{AB}{DE} = \frac{2}{5}$ . If the area of $\triangle DEF$ is 100, what is the area of $\triangle ABC$ ?

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Example 2

In $\triangle ABC$ , point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $\overline{DE} \parallel \overline{BC}$ . If $AD = 4$ , $DB = 6$ , and $DE = 8$ , what is $BC$ ?

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Common pitfalls

Using the linear ratio for area

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.

Setting up the proportion with mismatched corresponding sides

When writing $\frac{AB}{DE} = \frac{BC}{EF}$ , $AB$ must correspond to $DE$ , $BC$ to $EF$ — based on which vertices match in the similarity statement $\triangle ABC \sim \triangle DEF$ . Mismatched correspondences give wrong proportions.

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Forgetting that AA only requires two angles

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.

Mixing up similar with congruent

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.

Key takeaways

Similar triangles: same angles, proportional sides. Scale factor $k$ = side ratio.
AA, SSS, SAS — any one establishes similarity.
Area ratio = $k^2$ . Volume ratio = $k^3$ .
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A line parallel to one side of a triangle creates a similar smaller triangle.
Match corresponding sides based on the similarity statement order ( $\triangle ABC \sim \triangle DEF$ means $A \leftrightarrow D$ , etc.).
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Tracks your progress across lessons.

Try it yourself

5 practice questions on Similar Triangles, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Right Triangles→

Similar Triangles

You are here

→Volume →Area and Perimeter