Criteria for Similarity of Triangles: AAA, SSS, and SAS Explained

The similarity of triangles is a fundamental concept in geometry, allowing us to compare triangles based on their shapes, regardless of their size. Two triangles are similar if their corresponding angles are equal, and their corresponding sides are proportional. This principle is crucial in various applications, from scaling objects in real life to solving complex mathematical problems. To determine similarity, three primary criteria are used:

1. AAA (Angle-Angle-Angle) Criterion

If all three angles of one triangle are equal to all three angles of another triangle, the two triangles are similar.

Explanation:

Since the sum of the angles of a triangle is always $\boldsymbol{180^\circ}$ , if all three angles of one triangle match those of another, the two triangles will have identical shapes, though their sizes may differ. This is the strongest criterion for determining similarity and is widely used in geometry.

Mathematical Representation:

For $\boldsymbol{\triangle ABC}$ and $\boldsymbol{\triangle DEF}$ , $\boldsymbol{\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F \implies \triangle ABC \sim \triangle DEF}$

Example:

In $\boldsymbol{\triangle ABC}$ and $\boldsymbol{\triangle PQR}$ , let:

$\boldsymbol{\angle A = 50^\circ, \angle B = 60^\circ, \angle C = 70^\circ \quad \text{and}\quad \angle P = 50^\circ, \angle Q = 60^\circ, \angle R = 70^\circ.}$

Since all three angles of $\boldsymbol{\triangle ABC}$ are equal to all three angles of $\boldsymbol{\triangle PQR}$ , the triangles are similar by the AAA Criterion. This simple yet powerful criterion is commonly employed in theoretical problems and practical applications.

Key Observation:

Unlike other criteria, the AAA criterion does not involve the comparison of sides, making it a pure angle-based approach. It is particularly useful in problems where angles are easier to measure or calculate than sides.

2. SSS (Side-Side-Side) Criterion

If the corresponding sides of two triangles are in proportion, then the triangles are similar.

Explanation:

When the sides of two triangles are proportional, their shapes are identical, making them similar. This criterion establishes similarity by focusing solely on the dimensions of the triangles without involving angles.

Mathematical Representation:

For $\boldsymbol{\triangle ABC}$ and $\boldsymbol{\triangle DEF}$ ,

$\boldsymbol{\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \implies \triangle ABC \sim \triangle DEF.}$

Example:

Let the sides of $\boldsymbol{\triangle ABC}$ be $6$ , $8$ , and $10$ , and the sides of $\boldsymbol{\triangle DEF}$ be $3$ , $4$ , and $5$ . Calculate:

$\boldsymbol{\frac{AB}{DE} = \frac{6}{3} = 2, \quad \frac{BC}{EF} = \frac{8}{4} = 2, \quad \frac{AC}{DF} = \frac{10}{5} = 2.}$

Since the sides are proportional, $\boldsymbol{\triangle ABC \sim \triangle DEF}$ by the SSS Criterion.

Additional Insight:

The SSS criterion is especially useful in practical scenarios where only side lengths are available, such as scaling models or verifying geometric designs.

3. SAS (Side-Angle-Side) Criterion

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, then the triangles are similar.

Explanation:

The SAS Criterion focuses on the proportionality of two sides and the equality of the included angle. This ensures that the triangles share the same shape, as the included angle serves as a bridge to establish similarity.

Mathematical Representation:

For $\boldsymbol{\triangle ABC}$ and $\boldsymbol{\triangle DEF}$ ,

$\boldsymbol{\angle A = \angle D \quad \text{and} \quad \frac{AB}{DE} = \frac{AC}{DF} \implies \triangle ABC \sim \triangle DEF.}$

Example:

In $\boldsymbol{\triangle ABC}$ , let:

$\boldsymbol{AB = 6, AC = 9, \angle A = 60^\circ.}$

In $\boldsymbol{\triangle DEF}$ , let:

$\boldsymbol{DE = 3, DF = 4.5, \angle D = 60^\circ.}$

Calculate:

$\boldsymbol{\frac{AB}{DE} = \frac{6}{3} = 2 \quad \text{and} \quad \frac{AC}{DF} = \frac{9}{4.5} = 2.}$

Since the sides are proportional and the included angles are equal, $\boldsymbol{\triangle ABC \sim \triangle DEF}$ by the SAS Criterion.

Key Application:

The SAS criterion is frequently applied in problems where angle measurement is straightforward, and only two sides are required for comparison, reducing complexity in calculations.

Applications of the Similarity Criteria

Map Scaling: Similar triangles are used to create accurate scale models and maps for navigation and construction purposes. For instance, cartographers employ triangle similarity to ensure proportional scaling between real-world distances and their representations.
Architectural Design: Architects use similar triangles to design and scale buildings proportionally. This ensures that the structural dimensions are accurate, while maintaining aesthetic symmetry.
Indirect Measurements: Heights of buildings, mountains, and trees can be calculated using similar triangles and shadow measurements. This application is critical in fields like civil engineering, astronomy, and land surveying.
Optical Instruments: In devices like periscopes and telescopes, similar triangles help maintain proportionality in light paths for accurate image formation.

Key Observations

The AAA Criterion establishes similarity based solely on angle equality, making it unique among the criteria.
The SSS Criterion compares the proportionality of all three sides of two triangles, ensuring geometric accuracy when angles are not known.
The SAS Criterion combines proportionality of two sides with the equality of the included angle, providing a balance between the AAA and SSS methods.
All three criteria are interdependent, ensuring a comprehensive approach to proving similarity in any scenario.

Practice Problems

Problem 1: Prove that $\boldsymbol{\triangle ABC \sim \triangle DEF}$ if: $\boldsymbol{\angle A = \angle D, \quad \angle B = \angle E, \quad \text{and}\quad \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}.}$
Problem 2: In $\boldsymbol{\triangle PQR}$ , $\boldsymbol{ST \parallel QR}$ . Prove that $\boldsymbol{\triangle PST \sim \triangle PQR}$ .
Problem 3: The sides of two triangles are proportional as follows: $\boldsymbol{\frac{3}{6} = \frac{4}{8} = \frac{5}{10}}$ . Prove that the two triangles are similar.
Problem 4: In $\boldsymbol{\triangle XYZ}$ , $\boldsymbol{PQ \parallel YZ}$ . If $\boldsymbol{XP = 5}$ cm, $\boldsymbol{PY = 10}$ cm, and $\boldsymbol{XZ = 15}$ cm, find $\boldsymbol{QZ}$ .
Problem 5: Verify whether $\boldsymbol{\triangle ABC \sim \triangle DEF}$ if $\boldsymbol{\angle A = \angle D = 90^\circ}$ , $\boldsymbol{AB = 12, BC = 16, AC = 20}$ , and $\boldsymbol{DE = 6, EF = 8, DF = 10}$ .

Criteria for Similarity of Triangles

1. AAA (Angle-Angle-Angle) Criterion

Explanation:

Mathematical Representation:

Example:

Key Observation:

2. SSS (Side-Side-Side) Criterion

Explanation:

Mathematical Representation:

Example:

Additional Insight:

3. SAS (Side-Angle-Side) Criterion

Explanation:

Mathematical Representation:

Example:

Key Application:

Applications of the Similarity Criteria

Key Observations

Practice Problems

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