How To Tell If Triangles Are Similar

Determining whether triangles are similar involves understanding their properties and applying specific theorems. Similarity, in geometry, means that two shapes have the same shape but can be different sizes. This article explores the methods to determine if triangles are similar, providing detailed explanations, examples, and practical applications.

Understanding Similar Triangles

Similar triangles have corresponding angles that are congruent (equal) and corresponding sides that are in proportion. This means that if you were to enlarge or reduce one triangle, it would perfectly overlap with the other. The symbol for similarity is ~. So, if triangle ABC is similar to triangle DEF, it's written as ∆ABC ~ ∆DEF.

Key Properties of Similar Triangles

• Corresponding Angles are Congruent: If two triangles are similar, each pair of corresponding angles are equal in measure. For example, if ∆ABC ~ ∆DEF, then ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.
• Corresponding Sides are Proportional: The ratios of the lengths of corresponding sides are equal. If ∆ABC ~ ∆DEF, then AB/DE = BC/EF = AC/DF.

Theorems and Methods to Prove Triangle Similarity

Several theorems and methods can be used to determine if two triangles are similar. These include Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).

1. Angle-Angle (AA) Similarity

The Angle-Angle (AA) similarity postulate is the most straightforward method to prove that two triangles are similar. It states:

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Explanation:

When two angles of one triangle are equal to two angles of another triangle, the third angles are also equal because the sum of angles in any triangle is always 180 degrees. If all three angles are equal, the triangles are similar because they have the same shape.

Steps to Apply AA Similarity:

1. Identify Two Angles in Each Triangle: Look for given angle measures or angle relationships (such as vertical angles or alternate interior angles) that can help you determine the measures of two angles in each triangle.
2. Compare the Angle Measures: Check if two angles in one triangle are congruent (equal in measure) to two angles in the other triangle.
3. Conclude Similarity: If the condition in step 2 is met, you can conclude that the triangles are similar based on the AA similarity postulate.

Example:

Consider two triangles, ∆ABC and ∆DEF, where:

• ∠A = 50° and ∠B = 70° in ∆ABC
• ∠D = 50° and ∠E = 70° in ∆DEF

Since ∠A = ∠D and ∠B = ∠E, by the AA similarity postulate, ∆ABC ~ ∆DEF.

2. Side-Angle-Side (SAS) Similarity

The Side-Angle-Side (SAS) similarity theorem states:

If two sides of one triangle are proportional to two corresponding sides of another triangle, and the included angles are congruent, then the two triangles are similar.

Explanation:

This theorem combines the concepts of proportional sides and congruent angles. The included angle is the angle formed by the two sides being compared.

Steps to Apply SAS Similarity:

1. Identify Two Sides in Each Triangle: Determine which two sides in each triangle you want to compare.
2. Check for Proportionality: Calculate the ratios of the corresponding sides. If these ratios are equal, the sides are proportional. For example, check if AB/DE = BC/EF.
3. Identify the Included Angle: Find the angle that is formed by the two sides you compared in each triangle.
4. Check for Congruence: Verify that the included angles are congruent (equal in measure).
5. Conclude Similarity: If the sides are proportional and the included angles are congruent, then the triangles are similar by the SAS similarity theorem.

Example:

Consider two triangles, ∆ABC and ∆DEF, where:

• AB = 4, BC = 6, and ∠B = 45° in ∆ABC
• DE = 6, EF = 9, and ∠E = 45° in ∆DEF

1. Check Proportionality:

   • AB/DE = 4/6 = 2/3
   • BC/EF = 6/9 = 2/3
   • The sides are proportional since AB/DE = BC/EF.

2. Check Congruence:

   • ∠B = ∠E = 45°
   • The included angles are congruent.

Since the sides are proportional and the included angles are congruent, ∆ABC ~ ∆DEF by the SAS similarity theorem.

3. Side-Side-Side (SSS) Similarity

The Side-Side-Side (SSS) similarity theorem states:

If all three sides of one triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar.

Explanation:

This theorem relies solely on the proportionality of the sides. If all three pairs of corresponding sides have the same ratio, the triangles are similar.

Steps to Apply SSS Similarity:

1. Identify All Three Sides in Each Triangle: Measure or find the lengths of all three sides in both triangles.
2. Check for Proportionality: Calculate the ratios of the corresponding sides. If all three ratios are equal, the sides are proportional. For example, check if AB/DE = BC/EF = AC/DF.
3. Conclude Similarity: If all three ratios are equal, then the triangles are similar by the SSS similarity theorem.

Example:

Consider two triangles, ∆ABC and ∆DEF, where:

• AB = 3, BC = 4, AC = 5 in ∆ABC
• DE = 6, EF = 8, DF = 10 in ∆DEF

1. Check Proportionality:

   • AB/DE = 3/6 = 1/2
   • BC/EF = 4/8 = 1/2
   • AC/DF = 5/10 = 1/2
   • All sides are proportional since AB/DE = BC/EF = AC/DF.

Since all three sides are proportional, ∆ABC ~ ∆DEF by the SSS similarity theorem.

Practical Examples and Applications

Understanding these similarity theorems is crucial for various applications in geometry, engineering, architecture, and everyday problem-solving.

Example 1: Using AA Similarity in Real Life

Suppose you want to measure the height of a tree using shadows. You know that a meter stick casts a shadow of 1.5 meters, and at the same time, the tree casts a shadow of 12 meters. You can use similar triangles to find the height of the tree.

1. Set up the Triangles:
   • Triangle 1: Formed by the meter stick and its shadow.
   • Triangle 2: Formed by the tree and its shadow.
2. Identify the Angles:
   • Both triangles have a right angle (where the stick and tree meet the ground).
   • The angle of elevation of the sun is the same for both triangles (since the measurements are taken at the same time).
3. Apply AA Similarity:
   • Since both triangles have two congruent angles, they are similar by the AA similarity postulate.
4. Set up Proportions:
   • (Height of meter stick) / (Shadow of meter stick) = (Height of tree) / (Shadow of tree)
   • 1 / 1.5 = H / 12
5. Solve for H (Height of the Tree):
   • H = (1 * 12) / 1.5
   • H = 8 meters

Therefore, the height of the tree is 8 meters.

Example 2: Using SAS Similarity in Construction

An architect is designing a roof with two triangular sections. The dimensions of one section are 8 feet and 12 feet with an included angle of 60 degrees. The second section has dimensions of 12 feet and 18 feet with an included angle of 60 degrees. Are the two sections similar?

1. Set up the Triangles:

   • Triangle 1: Dimensions 8 ft and 12 ft, angle 60°.
   • Triangle 2: Dimensions 12 ft and 18 ft, angle 60°.

2. Check Proportionality:

   • 8/12 = 2/3
   • 12/18 = 2/3
   • The sides are proportional.

3. Check Congruence:

   • The included angles are both 60°, so they are congruent.

4. Apply SAS Similarity:

   • Since the sides are proportional and the included angles are congruent, the two triangular sections are similar by the SAS similarity theorem.

Example 3: Using SSS Similarity in Map Scaling

A cartographer is creating a map where a triangular region needs to be scaled down. The original region has sides of 5 km, 7 km, and 9 km. The corresponding sides on the map are 2 cm, 2.8 cm, and 3.6 cm. Is the map representation similar to the original region?

1. Set up the Triangles:

   • Triangle 1: Sides 5 km, 7 km, 9 km.
   • Triangle 2: Sides 2 cm, 2.8 cm, 3.6 cm.

2. Check Proportionality:

   • 5/2 = 2.5
   • 7/2.8 = 2.5
   • 9/3.6 = 2.5
   • All sides are proportional with the same ratio of 2.5.

3. Apply SSS Similarity:

   • Since all three sides are proportional, the map representation is similar to the original region by the SSS similarity theorem.

Common Mistakes to Avoid

• Assuming Similarity Without Proof: Always use the theorems to prove similarity, rather than assuming it based on appearance.
• Incorrectly Matching Corresponding Sides or Angles: Ensure you are comparing corresponding sides and angles correctly. Use the order of vertices in the similarity statement (e.g., ∆ABC ~ ∆DEF) to help match corresponding parts.
• Misinterpreting Proportionality: Understand that proportionality means the ratios of corresponding sides are equal, not that the sides themselves are equal.
• Ignoring the Included Angle in SAS: The included angle must be between the two sides you are comparing.
• Confusing Similarity with Congruence: Similarity means the shapes are the same but can be different sizes. Congruence means the shapes are identical in both shape and size.

Advanced Tips and Techniques

Using Auxiliary Lines

Sometimes, drawing auxiliary lines (additional lines in the diagram) can help create similar triangles. For instance, drawing a line parallel to one side of a triangle can create smaller triangles that are similar to the original.

Applying Similarity in Coordinate Geometry

In coordinate geometry, you can use the distance formula and slope to determine the lengths of sides and measures of angles. If the corresponding sides of two triangles satisfy the conditions of SSS or SAS similarity, or if the angles satisfy AA similarity, you can prove that the triangles are similar.

Utilizing Similarity in Trigonometry

Similar triangles have the same trigonometric ratios (sine, cosine, tangent) for corresponding angles. This principle is fundamental in trigonometry and can be used to solve problems involving unknown side lengths or angle measures.

Conclusion

Determining whether triangles are similar involves understanding and applying the AA, SAS, and SSS similarity theorems. By carefully checking for congruent angles and proportional sides, you can accurately determine if two triangles share the same shape. These concepts are not only essential in geometry but also have practical applications in various fields such as architecture, engineering, and map-making. By mastering these theorems and avoiding common mistakes, you can confidently solve problems involving similar triangles.