How To Determine If Two Triangles Are Similar

The concept of similar triangles is a cornerstone of geometry, providing a foundation for understanding proportions, scale, and geometric relationships. Two triangles are considered similar if they have the same shape but potentially different sizes. This article will delve into the comprehensive methods for determining if two triangles are similar, covering the essential theorems, practical examples, and common pitfalls.

Understanding Similarity

Before diving into the methods, it's crucial to grasp the underlying principles of triangle similarity. Unlike congruent triangles, which are identical in both shape and size, similar triangles only need to maintain proportional relationships. This means their corresponding angles are equal, and their corresponding sides are in proportion.

Key Definitions:

• Similar Triangles: Triangles that have the same shape but may differ in size.
• Corresponding Angles: Angles that occupy the same relative position in two different triangles.
• Corresponding Sides: Sides that are opposite corresponding angles in two different triangles.

Criteria for Determining Triangle Similarity

There are three primary criteria for determining if two triangles are similar. Each criterion provides a different set of conditions that, if met, guarantee the triangles are similar. These criteria are:

1. Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
2. Side-Side-Side (SSS) Similarity: If the three sides of one triangle are proportional to the corresponding three sides of another triangle, then the two triangles are similar.
3. Side-Angle-Side (SAS) Similarity: If two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles are congruent, then the two triangles are similar.

1. Angle-Angle (AA) Similarity

The Angle-Angle (AA) similarity postulate is the most straightforward way to determine if two triangles are similar. It states that if two angles of one triangle are congruent (equal in measure) to two angles of another triangle, then the triangles are similar.

Explanation:

• When two angles of a triangle are known, the third angle can be easily determined since the sum of angles in any triangle is always 180 degrees.
• If two triangles have two pairs of congruent angles, their third angles must also be congruent.
• Having all three angles congruent ensures that the triangles have the same shape, regardless of their size.

Example:

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

• ∠A = 50° and ∠B = 70°
• ∠D = 50° and ∠E = 70°

Since ∠A ≅ ∠D and ∠B ≅ ∠E, Triangle ABC ~ Triangle DEF (where ~ denotes similarity) by AA similarity.

Steps to Apply AA Similarity:

1. Identify two angles in each triangle: Look for given angle measures or relationships that allow you to deduce the measures of two angles in each triangle.
2. Compare the angles: Determine if two angles in one triangle are congruent to two angles in the other triangle.
3. Conclude similarity: If two pairs of congruent angles are found, conclude that the triangles are similar by AA similarity.

Practical Application:

AA similarity is particularly useful in scenarios where angle measures are readily available or can be easily calculated, such as in problems involving parallel lines and transversals, where alternate interior angles or corresponding angles are congruent.

2. Side-Side-Side (SSS) Similarity

The Side-Side-Side (SSS) similarity theorem states that if all three sides of one triangle are proportional to the corresponding three sides of another triangle, then the two triangles are similar.

Explanation:

• Proportionality means that the ratios of the lengths of corresponding sides are equal.
• If all three pairs of corresponding sides have the same ratio, the triangles maintain the same shape, regardless of their size.

Example:

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

• AB = 4, BC = 6, CA = 8
• DE = 6, EF = 9, FD = 12

Check the ratios of the corresponding sides:

• AB/DE = 4/6 = 2/3
• BC/EF = 6/9 = 2/3
• CA/FD = 8/12 = 2/3

Since AB/DE = BC/EF = CA/FD, Triangle ABC ~ Triangle DEF by SSS similarity.

Steps to Apply SSS Similarity:

1. Identify the lengths of all three sides in each triangle: Ensure you have the measurements of all sides for both triangles.
2. Determine corresponding sides: Match the sides based on their relative lengths (e.g., shortest side to shortest side, longest side to longest side).
3. Calculate the ratios of corresponding sides: Divide the length of each side in one triangle by the length of its corresponding side in the other triangle.
4. Compare the ratios: Check if all three ratios are equal.
5. Conclude similarity: If all three ratios are equal, conclude that the triangles are similar by SSS similarity.

Practical Application:

SSS similarity is useful when side lengths are known and angle measures are not readily available. It is commonly applied in problems involving scale drawings, models, and geometric proofs.

3. Side-Angle-Side (SAS) Similarity

The Side-Angle-Side (SAS) similarity theorem states that if two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles (the angles between those sides) are congruent, then the two triangles are similar.

Explanation:

• SAS similarity combines the concepts of proportionality and congruence.
• The proportionality of two pairs of sides ensures that the triangles have a similar "structure" in those regions.
• The congruence of the included angles anchors the triangles, ensuring that they maintain the same shape around that angle.

Example:

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

• AB = 3, AC = 5, ∠A = 60°
• DE = 6, DF = 10, ∠D = 60°

Check the proportionality of the sides and the congruence of the included angles:

• AB/DE = 3/6 = 1/2
• AC/DF = 5/10 = 1/2
• ∠A ≅ ∠D (both are 60°)

Since AB/DE = AC/DF and ∠A ≅ ∠D, Triangle ABC ~ Triangle DEF by SAS similarity.

Steps to Apply SAS Similarity:

1. Identify two sides and the included angle in each triangle: Ensure you have the lengths of two sides and the measure of the angle between them for both triangles.
2. Determine corresponding sides and included angles: Match the sides based on their relative lengths and ensure the included angles are correctly identified.
3. Calculate the ratios of corresponding sides: Divide the length of each side in one triangle by the length of its corresponding side in the other triangle.
4. Compare the ratios and angles: Check if the two ratios are equal and if the included angles are congruent.
5. Conclude similarity: If the two ratios are equal and the included angles are congruent, conclude that the triangles are similar by SAS similarity.

Practical Application:

SAS similarity is particularly useful when only partial information is available about the triangles, such as when two sides and the included angle are known. It is commonly used in problems involving geometric constructions and proofs.

Advanced Tips and Considerations

Reflexive Property

The reflexive property states that any geometric figure is congruent to itself. In the context of triangle similarity, if two triangles share a common side or angle, this property can be useful in establishing similarity.

Example:

Consider two triangles, Triangle ABC and Triangle ADE, sharing a common angle ∠A. If AB/AD = AC/AE, then Triangle ABC ~ Triangle ADE by SAS similarity, with ∠A being the common, congruent angle.

Vertical Angles

Vertical angles are pairs of opposite angles made by intersecting lines. Vertical angles are always congruent. When dealing with triangles that share a vertex formed by intersecting lines, recognizing vertical angles can help establish angle congruence for AA similarity.

Example:

If two lines intersect at point E, forming triangles ABE and CDE, then ∠AEB ≅ ∠CED (vertical angles). If, in addition, ∠A ≅ ∠C, then Triangle ABE ~ Triangle CDE by AA similarity.

Parallel Lines and Transversals

Parallel lines cut by a transversal create congruent corresponding angles and alternate interior angles. Recognizing these angle relationships can be helpful in establishing angle congruence for AA similarity.

Example:

If line AB is parallel to line CD, and a transversal intersects both lines, creating triangles ABE and CDE, then ∠BAE ≅ ∠DCE (alternate interior angles) and ∠ABE ≅ ∠CDE (alternate interior angles). Therefore, Triangle ABE ~ Triangle CDE by AA similarity.

Using Algebra to Find Missing Sides or Angles

In many problems, you may need to use algebraic equations to find missing side lengths or angle measures before you can apply the similarity criteria.

Example:

Suppose you have two triangles, and you know that the sides of one triangle are twice as long as the corresponding sides of the other triangle. If one side of the smaller triangle is x and the corresponding side of the larger triangle is 2x, you can use this information to set up an equation and solve for x.

Similarly, if you know that the sum of the angles in a triangle is 180°, you can use this to find a missing angle measure.

Common Pitfalls

• Assuming Similarity Without Sufficient Evidence: Always verify that the criteria for similarity are fully met before concluding that two triangles are similar.
• Incorrectly Identifying Corresponding Sides or Angles: Ensure that you are comparing the correct pairs of sides and angles based on their relative positions in the triangles.
• Misinterpreting Proportionality: Proportionality means that the ratios of corresponding sides are equal, not that the sides themselves are equal.
• Ignoring the Included Angle in SAS Similarity: The angle must be between the two sides for SAS similarity to apply.
• Confusing Similarity with Congruence: Remember that similar triangles have the same shape but may differ in size, while congruent triangles are identical in both shape and size.

Examples

Example 1: Applying AA Similarity

Given: Triangle ABC and Triangle DEF, where ∠A = 40°, ∠B = 80°, ∠D = 40°, and ∠E = 80°.

Determine if the triangles are similar.

• Solution:
  • ∠A ≅ ∠D (both are 40°)
  • ∠B ≅ ∠E (both are 80°)
  • Therefore, Triangle ABC ~ Triangle DEF by AA similarity.

Example 2: Applying SSS Similarity

Given: Triangle PQR and Triangle STU, where PQ = 3, QR = 4, RP = 5, ST = 6, TU = 8, and US = 10.

Determine if the triangles are similar.

• Solution:
  • PQ/ST = 3/6 = 1/2
  • QR/TU = 4/8 = 1/2
  • RP/US = 5/10 = 1/2
  • Since PQ/ST = QR/TU = RP/US, Triangle PQR ~ Triangle STU by SSS similarity.

Example 3: Applying SAS Similarity

Given: Triangle XYZ and Triangle LMN, where XY = 4, XZ = 6, ∠X = 50°, LM = 6, LN = 9, and ∠L = 50°.

Determine if the triangles are similar.

• Solution:
  • XY/LM = 4/6 = 2/3
  • XZ/LN = 6/9 = 2/3
  • ∠X ≅ ∠L (both are 50°)
  • Since XY/LM = XZ/LN and ∠X ≅ ∠L, Triangle XYZ ~ Triangle LMN by SAS similarity.

Conclusion

Determining whether two triangles are similar involves applying one of three fundamental criteria: AA, SSS, or SAS. Each criterion provides a different set of conditions that, if met, guarantee the triangles are similar. Understanding these criteria, along with the associated theorems and properties, is essential for solving geometric problems and developing a strong foundation in mathematics. By carefully analyzing the given information, identifying corresponding sides and angles, and applying the appropriate similarity criterion, you can confidently determine if two triangles are similar and unlock a deeper understanding of geometric relationships.