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Similar Triangles

4 min readMedium10-question checkpoint

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 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

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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Checkpoint

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Where this lesson fits

Before

Right Triangles→

Similar Triangles

You are here

→Volume →Area and Perimeter