Triangle Similarity Criteria

Why This Matters

Triangle similarity is one of the most powerful tools in your geometry toolkit—and it shows up everywhere on exams. When you understand similarity, you unlock the ability to solve problems involving indirect measurement, proportional reasoning, and geometric proofs. You're being tested on your ability to recognize when triangles are similar, why a particular criterion applies, and how to use that relationship to find missing measurements.

The key insight? Similar triangles have the same shape but different sizes, which means 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

These are your primary tools for proving triangles are similar. Each criterion gives you a different "shortcut" so you don't have to verify all angles AND all sides—just enough information to guarantee similarity.

Angle-Angle (AA) Similarity

• Two pairs of congruent angles—that's all you need, since the third angle is automatically determined by the Triangle Angle Sum Theorem
• No side lengths required, making AA the most commonly used criterion when angle measures are given or can be found
• Best for parallel line problems where alternate interior or corresponding angles create the congruent pairs you need

Side-Angle-Side (SAS) Similarity

• One pair of congruent angles with the sides that form that angle in proportion—the angle must be the included angle between the two sides
• Ratio of sides matters, not actual lengths; if $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A \cong \angle D$, you've got SAS similarity
• Use when you have mixed information—some angle measures and some side lengths, but not enough for AA or SSS

Side-Side-Side (SSS) Similarity

• All three pairs of corresponding sides must be proportional—same ratio for every pair
• No angle information needed, making this ideal when you're given only side lengths
• Check all three ratios: if $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$, the triangles are similar

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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

• Same shape, different size—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 order
• Foundation for all similarity work—if you can't identify which parts correspond, you can't set up correct proportions

Scale Factor

• The ratio of corresponding side lengths between similar triangles, often written as $k$
• Determines enlargement or reduction: $k > 1$ means the second triangle is larger; $k < 1$ means it's smaller
• 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 $4 \times 3 = 12$

Corresponding Parts

• Corresponding angles are equal; corresponding sides are proportional—this is the defining property of similar triangles
• Order in the similarity statement tells you everything: in $\triangle ABC \sim \triangle XYZ$, side $AB$ corresponds to $XY$, $BC$ to $YZ$, and $AC$ to $XZ$
• Misidentifying corresponding parts is the most common error on similarity problems—always match vertices carefully

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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 (Parallel Line Theorem)

• A line parallel to one side of a triangle divides the other two sides proportionally
• Creates similar triangles: the smaller triangle formed is similar to the original by AA (parallel lines create congruent corresponding angles)
• Key setup for many proofs: look for parallel lines as your signal to apply this theorem

Similarity Transformations (Dilation)

• Dilation preserves shape but changes size—the definition of a similarity transformation
• Center point and scale factor determine the transformation: every point moves along a ray from the center, multiplied by the scale factor
• Proves similarity visually: if one triangle is a dilation of another, they're similar by definition

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Applications and Proof Strategies

This is where similarity becomes a problem-solving powerhouse.

Real-World Applications

• Indirect measurement uses similar triangles to find heights or distances you can't measure directly—like using shadows to calculate building heights
• Scale models and maps rely on similarity; the scale factor converts between model measurements and real-world dimensions
• Architecture and engineering use proportional reasoning from similar triangles to ensure structural accuracy

Writing Similarity Proofs

• Identify your given information first, then select the appropriate criterion (AA, SAS, or SSS)
• State corresponding parts explicitly—which angles are congruent? Which sides are proportional? Show your work
• End with a clear conclusion: "Therefore, $\triangle ABC \sim \triangle DEF$ by [criterion]"

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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?