Triangle Similarity Theorems

To prove that two triangles are similar, we do not always need to measure all sides and angles. It is enough to verify one of three conditions — the so-called similarity theorems.

Table of Contents

• SSS Theorem (Side – Side – Side)
• SAS Theorem (Side – Angle – Side)
• AA Theorem (Angle – Angle)
• Comparison of the theorems
• When to use which theorem
• Related Articles

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SSS Theorem (Side – Side – Side)

Two triangles are similar if the ratios of all three pairs of corresponding sides are equal.

Expressed mathematically: triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ are similar if:

Example: $△ABC$: $a=3$ cm, $b=4$ cm, $c=5$ cm $△A^{\prime }{B}^{\prime }{C}^{\prime }$: $a^{\prime }=6$ cm, $b^{\prime }=8$ cm, $c^{\prime }=10$ cm

Verification:

All ratios are equal ($k=2$), so $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$ by the SSS Theorem.

⚠️ Caution: When applying the SSS Theorem, we must compare corresponding sides — the shortest with the shortest, the longest with the longest.

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SAS Theorem (Side – Angle – Side)

Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the included angles are equal.

Mathematically: triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ are similar if:

where $\gamma$ and ${\gamma }^{\prime }$ are the angles included between sides $a$, $b$ and $a^{\prime }$, $b^{\prime }$.

Example: $△ABC$: $a=4$ cm, $b=6$ cm, $\gamma =60°$ $△A^{\prime }{B}^{\prime }{C}^{\prime }$: $a^{\prime }=8$ cm, $b^{\prime }=12$ cm, ${\gamma }^{\prime }=60°$

Verification:

The ratios are equal ($k=2$) and the included angles are equal ($60°=60°$), so $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$ by the SAS Theorem.

⚠️ Caution: The angle must be included between the two sides whose ratios we are verifying.

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AA Theorem (Angle – Angle)

Two triangles are similar if two angles of one triangle are equal to two angles of the other triangle.

The third angle is automatically equal because the sum of interior angles of a triangle is always $180°$:

If we know two angles, the third is uniquely determined.

Example: $△ABC$: $\alpha =50°$, $\beta =70°$ $△A^{\prime }{B}^{\prime }{C}^{\prime }$: ${\alpha }^{\prime }=50°$, ${\beta }^{\prime }=70°$

Third angles: $\gamma =180°-50°-70°=60°$ and ${\gamma }^{\prime }=180°-50°-70°=60°$

All angles match, so $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$ by the AA Theorem.

💡 Tip: The AA Theorem is the most commonly used similarity theorem because it only requires knowing the angles — we do not need to measure any sides.

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Comparison of the theorems

• SSS Theorem — we need to know all 6 sides (3 sides from each triangle) and verify that the ratios are equal
• SAS Theorem — we need to know 2 sides and the included angle from each triangle
• AA Theorem — we only need to know 2 angles from each triangle (the simplest to use)

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When to use which theorem

• You know all the sides of both triangles — use the SSS Theorem
• You know two sides and the angle between them — use the SAS Theorem
• You know the angles — use the AA Theorem

Practical approach:

• First, look at what information is given
• If you have angles, start with the AA Theorem — it is the simplest
• If you have sides, try the SSS Theorem
• If you have a combination of sides and angles, use the SAS Theorem

👉 Examples using each theorem: Triangle Similarity – Solved Examples

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Related Articles

• Comprehensive guide: Triangle Similarity – Comprehensive Guide
• Similarity of figures: Similarity of Geometric Figures
• Similarity ratio: Similarity Ratio (Scale Factor)
• 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