Similarity of Geometric Figures

Table of Contents

• What does it mean for two figures to be similar
• Properties of similar figures
• Examples of similar figures
• Triangle similarity
• Related Articles

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What does it mean for two figures to be similar

Two geometric figures are similar if they have the same shape but can differ in size. You can think of similarity as enlarging or reducing a figure — the shape stays the same, only the size changes.

Imagine a photograph that you enlarge or reduce on a copier. All proportions remain the same — that is exactly the principle of similarity.

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Properties of similar figures

If two figures are similar, the following holds:

• All corresponding angles are equal — angles do not change when the size changes
• Corresponding sides are in the same ratio — there exists a number $k$ (the scale factor) such that each side of one figure is $k$ times the corresponding side of the other figure

We write: $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$ (triangle $ABC$ is similar to triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$)

💡 Important: The order of the letters matters! $A$ corresponds to $A^{\prime }$, $B$ corresponds to $B^{\prime }$, $C$ corresponds to $C^{\prime }$.

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Examples of similar figures

Always similar figures:

• Any two circles are similar
• Any two squares are similar
• Any two equilateral triangles are similar

Not always similar figures:

• Two rectangles may not be similar (for example, a 2 by 4 rectangle and a 2 by 6 rectangle)
• Two triangles may not be similar — they must satisfy certain conditions

Visual example — two similar rectangles:

Side ratios: $\frac{4}{2}=\frac{8}{4}=2$ — the rectangles are similar with a scale factor of $k=2$.

Visual example — two non-similar rectangles:

Side ratios: $\frac{2}{2}=1$, but $\frac{6}{4}=1.5$ — the ratios are not equal, so the figures are not similar.

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Triangle similarity

Triangle similarity is especially important in geometry. Two triangles are similar if:

• They have the same three angles (it is enough to verify two — the third follows automatically)
• Their corresponding sides are in the same ratio

To verify triangle similarity, we use three similarity theorems: SSS, SAS, and AA.

👉 More about the similarity theorems: Triangle Similarity Theorems – SSS, SAS, AA

👉 Scale factor: Similarity Ratio (Scale Factor)

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Related Articles

• Comprehensive guide: Triangle Similarity – Comprehensive Guide
• Similarity ratio: Similarity Ratio (Scale Factor)
• Similarity theorems: Triangle Similarity Theorems – SSS, SAS, AA
• Solved examples: Triangle Similarity – Solved Examples

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Practice

• 🔢 Similarity ratio: Calculate the Similarity Ratio
• 📐 Missing sides: Find the Missing Side
• 🔍 Similarity theorems: Identify the Similarity Theorem
• 🧮 Mixed problems: Mixed Similarity Problems