Triangle Similarity Criteria

Why This Matters

Triangle similarity is one of the most powerful tools in geometry, and it shows up constantly on exams. When you understand similarity, you can solve problems involving indirect measurement, proportional reasoning, and geometric proofs.

Similar triangles have the same shape but different sizes. Their corresponding angles are equal and their corresponding sides are proportional. This single concept connects to dilations, parallel line theorems, scale factors, and real-world applications like calculating heights you can't measure directly. Don't just memorize the three criteria. Know what each one requires and when to use it.

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The Three Similarity Criteria

Each criterion gives you a shortcut so you don't have to verify all angles AND all sides. You just need enough information to guarantee similarity.

Angle-Angle (AA) Similarity

Two pairs of congruent angles is all you need. The third angle is automatically determined by the Triangle Angle Sum Theorem (all three angles add to $180°$), so checking two is sufficient.

• No side lengths required, making AA the most commonly used criterion when angle measures are given or can be deduced
• Especially useful in parallel line problems, where alternate interior angles or corresponding angles give you the congruent pairs

Side-Angle-Side (SAS) Similarity

You need one pair of congruent angles with the two sides forming that angle in proportion. The angle must be the included angle between the two sides you're comparing.

• The ratio of sides matters, not actual lengths. If $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A \cong \angle D$, you have SAS similarity.
• Reach for this when you have mixed information: some angle measures and some side lengths, but not enough for AA or SSS.

Watch out: SAS Similarity is not the same as SAS Congruence. For congruence, the two sides must be equal. For similarity, they only need to be proportional.

Side-Side-Side (SSS) Similarity

All three pairs of corresponding sides must share the same ratio. No angle information is needed.

• Check all three ratios: if $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$, the triangles are similar
• If even one ratio doesn't match, similarity fails

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Foundational Concepts

Before you can apply the criteria, you need to understand what similarity actually means and how to work with the measurements involved.

Definition of Similar Triangles

Similar triangles have the same shape but different size. Formally, corresponding angles are congruent and corresponding sides are proportional.

• Notation matters: $\triangle ABC \sim \triangle DEF$ means $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$, and the sides follow the same vertex order
• If you can't identify which parts correspond, you can't set up correct proportions

Note that congruent triangles are a special case of similar triangles where the scale factor equals 1. Congruence is similarity with no size change.

Scale Factor

The scale factor is the ratio of corresponding side lengths between similar triangles, often written as $k$.

• $k > 1$ means the second triangle is larger (enlargement); $k < 1$ means it's smaller (reduction); $k = 1$ means the triangles are congruent
• Use it to find unknowns: if $k = 3$ and one side of the smaller triangle is 4, the corresponding side of the larger triangle is $4 \times 3 = 12$
• The scale factor applies to every pair of corresponding sides. If it doesn't, the triangles aren't similar.
• For areas, the ratio scales by $k^2$. So if $k = 3$, the larger triangle's area is $9$ times the smaller triangle's area.

Corresponding Parts

The order in the similarity statement tells you everything. In $\triangle ABC \sim \triangle XYZ$:

• Side $AB$ corresponds to $XY$, side $BC$ corresponds to $YZ$, and side $AC$ corresponds to $XZ$
• $\angle A \cong \angle X$, $\angle B \cong \angle Y$, $\angle C \cong \angle Z$

Misidentifying corresponding parts is the single most common error on similarity problems. Always match vertices carefully using the similarity statement before setting up any proportion.

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Theorems and Transformations

These extend your similarity toolkit beyond the basic criteria, connecting similarity to parallel lines and transformations.

Triangle Proportionality Theorem (Side-Splitter Theorem)

If a line is parallel to one side of a triangle, it divides the other two sides proportionally. For example, if line $DE$ is parallel to side $BC$ in $\triangle ABC$ and intersects $AB$ at $D$ and $AC$ at $E$, then:

$\frac{AD}{DB} = \frac{AE}{EC}$

This setup also creates similar triangles: $\triangle ADE \sim \triangle ABC$ by AA, because the parallel lines produce congruent corresponding angles ($\angle ADE \cong \angle ABC$ and $\angle AED \cong \angle ACB$ as corresponding angles, with $\angle A$ shared). Look for parallel lines as your signal to apply this theorem.

The converse is also true and useful: if a line divides two sides of a triangle proportionally, then that line is parallel to the third side.

Similarity Transformations (Dilation)

A dilation preserves shape but changes size. It's defined by a center point and a scale factor: every point moves along a ray from the center, with its distance from the center multiplied by the scale factor.

• A dilation with scale factor $k > 0$ maps each point $P$ to a point $P'$ such that $P'$ lies on ray $\overrightarrow{CP}$ (from center $C$) and $CP' = k \cdot CP$
• Dilations preserve angle measures and produce proportional side lengths, which is exactly the definition of similarity

If one triangle is a dilation of another, they're similar by definition. This gives you a transformation-based way to justify similarity, which some proof problems specifically ask for.

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Applications and Proof Strategies

Real-World Applications

Indirect measurement uses similar triangles to find heights or distances you can't measure directly. Here's a classic example worked step by step:

A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 20-foot shadow. How tall is the tree?

1. The sun's rays hit both the person and the tree at the same angle, so the two right triangles formed (person + shadow, tree + shadow) share that angle. Both also have a right angle where the object meets the ground.
2. By AA, the triangles are similar.
3. Set up the proportion with corresponding sides: $\frac{\text{height}}{\text{shadow}} = \frac{6}{4} = \frac{h}{20}$
4. Cross-multiply: $6 \times 20 = 4h$, so $120 = 4h$, giving $h = 30$ feet.

Scale models and maps also rely on similarity. The scale factor converts between model measurements and real-world dimensions.

Writing Similarity Proofs

A reliable process for proof problems:

1. List your given information. Mark up the diagram with angle measures, side lengths, and any parallel lines.
2. Identify corresponding parts. Which angles are congruent? Which sides are proportional? Parallel lines, vertical angles, and shared angles are common sources of congruent angles.
3. Choose the right criterion (AA, SAS, or SSS) based on what you've identified.
4. State corresponding parts explicitly in your proof. Show the angle congruences or side ratios.
5. Write a clear conclusion: "Therefore, $\triangle ABC \sim \triangle DEF$ by [criterion]."

A common proof pattern you'll see: two triangles share a vertex angle (vertical angles or the same angle), and a parallel line gives you a second pair of congruent angles. That's AA, and you're done.

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Quick Reference Table

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Self-Check Questions

1. Two triangles have two pairs of congruent angles. Which similarity criterion applies, and why don't you need to check the third angle?

2. You're given that $\frac{AB}{DE} = \frac{BC}{EF} = 2$, but $\frac{AC}{DF} = 3$. Are the triangles similar? Explain your reasoning.

3. Compare and contrast SAS Similarity with SAS Congruence. What's the key difference in what you're checking for the sides?

4. A line is drawn parallel to side $BC$ of $\triangle ABC$, intersecting $AB$ at point $D$ and $AC$ at point $E$. Explain why $\triangle ADE \sim \triangle ABC$ and identify which criterion you'd use in a proof.

5. If the scale factor from $\triangle PQR$ to $\triangle XYZ$ is $\frac{2}{5}$, and $PQ = 10$, what is $XY$? What does this scale factor tell you about the relative sizes of the triangles?