Similarity Rules

Why This Matters

Similarity is one of geometry's most powerful tools because it lets you solve problems involving shapes you can't directly measure. When two figures are similar, you unlock a predictable relationship between their angles, sides, areas, and volumes—and the exam loves testing whether you understand how these relationships scale. You'll see similarity show up in proofs, coordinate geometry, real-world applications, and especially in problems where you need to find missing measurements.

Don't just memorize that similar triangles have proportional sides—know why each similarity theorem works and when to apply it. The key concepts you're being tested on include proving similarity, applying scale factors, and understanding how scaling affects different dimensions. Master the logic behind these rules, and you'll handle everything from basic proportion problems to complex multi-step proofs with confidence.

---

Proving Triangles Similar

Before you can use similarity to solve problems, you need to establish that two triangles actually are similar. These three theorems give you different entry points depending on what information you have—each requires the minimum evidence needed to guarantee the same shape.

AA (Angle-Angle) Similarity Theorem

• Two pairs of congruent angles—if two angles of one triangle equal two angles of another, the triangles are similar (the third angle is automatically equal since angles sum to $180°$)
• Most efficient method for proving similarity because you only need angle measures, no side lengths required
• Look for parallel lines creating congruent alternate interior or corresponding angles—this is the most common setup for AA similarity on exams

SAS (Side-Angle-Side) Similarity Theorem

• Two proportional sides with an equal included angle—the angle must be between the two sides you're comparing
• Useful when angles aren't given but you can calculate or measure side ratios and one angle
• Watch the order carefully—the proportional sides must correspond, and the angle must be the one they share

SSS (Side-Side-Side) Similarity Theorem

• All three pairs of sides proportional—if $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$, the triangles are similar
• Pure ratio comparison with no angle information needed, making it ideal when you only have side lengths
• Verify all three ratios match—if even one ratio differs, the triangles are not similar

---

Properties of Similar Figures

Once similarity is established, these properties tell you what's true about the relationship between the figures. Understanding these lets you set up equations and solve for unknowns.

Congruent Corresponding Angles

• All matching angles are equal—this is what makes similar figures the "same shape"
• Angle measures don't change regardless of how much the figure is enlarged or reduced
• Use this property to find missing angles without any calculation—just identify the corresponding angle in the similar figure

Proportional Corresponding Sides

• Matching sides maintain a constant ratio—if one pair of sides has ratio $3:5$, all pairs have ratio $3:5$
• Set up proportions to find unknown lengths: $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$
• Order matters—always match corresponding vertices when writing proportions to avoid errors

---

The Scale Factor and Its Powers

The scale factor is the multiplier that relates corresponding lengths between similar figures. What makes this concept exam-critical is understanding how the scale factor behaves differently for length, area, and volume.

Scale Factor (Linear)

• Ratio of corresponding lengths—written as $k = \frac{\text{new length}}{\text{original length}}$
• $k > 1$ means enlargement, $k < 1$ means reduction, and $k = 1$ means congruent figures
• Multiply any length by $k$ to find the corresponding length in the similar figure

Area Ratio (Scale Factor Squared)

• Area scales by $k^2$—if the scale factor is $3$, the area ratio is $9$
• Two-dimensional measure requires squaring because area involves two length dimensions multiplied together
• Common exam trap—students often forget to square and use $k$ instead of $k^2$ for area problems

Volume Ratio (Scale Factor Cubed)

• Volume scales by $k^3$—if the scale factor is $2$, the volume ratio is $8$
• Three-dimensional measure requires cubing because volume involves three length dimensions
• Dramatic scaling effects—doubling linear dimensions makes volume $8$ times larger, which is why this appears in real-world application problems

---

Similarity in Special Configurations

Certain geometric setups automatically create similar triangles. Recognizing these patterns saves time and gives you a starting point for proofs.

Parallel Lines Cut by a Transversal

• Creates congruent corresponding and alternate interior angles—which immediately sets up AA similarity
• Triangle-in-triangle pattern—when a line parallel to one side of a triangle intersects the other two sides, it creates a smaller similar triangle
• Proportional segments result—the parallel line divides the sides proportionally, a key relationship for coordinate geometry problems

Right Triangle Similarity

• Altitude to the hypotenuse creates two smaller triangles, each similar to the original and to each other
• Three similar triangles total—the original plus two smaller ones, all sharing the same angle measures
• Geometric mean relationships emerge from these similarities: the altitude is the geometric mean of the two segments of the hypotenuse

---

Quick Reference Table

---

Self-Check Questions

1. Which similarity theorem would you use if you know two triangles have two pairs of congruent angles but no side length information?

2. If the scale factor between two similar triangles is $\frac{2}{3}$, what is the ratio of their areas? What is the ratio of the volumes of two similar prisms with this same scale factor?

3. Compare and contrast AA and SAS similarity: What information does each require, and in what situation would SAS be more useful than AA?

4. A line parallel to the base of a triangle divides the other two sides. Explain why the smaller triangle formed is similar to the original triangle, and identify which theorem justifies this.

5. Two similar cylinders have surface areas in the ratio $16:25$. What is the ratio of their heights, and what is the ratio of their volumes?