Criteria for Similarity of Triangles: Definitions, Properties, and Thales’ Theorem

Criteria for the similarity of triangles exist in different angle-angle, side-angle-side, and side-side-side forms. For two similar triangles, their corresponding angles are identical, and their corresponding sides are proportional. This fundamental idea forms the basis for various criteria that determine similarity. 

• According to the Angle-Angle (AA) criteria, two triangles are comparable if their corresponding angles are equal.
• The Side-Angle-Side (SAS) criterion asserts that if one pair of corresponding sides is proportional and the included angles are congruent, the triangles are similar.
• If all three pairs of corresponding sides are proportional, and the triangles are similar, then this criterion is expressed as Side-Side-Side (SSS) criterion.
• Thales' Theorem and the Converse of the Pythagorean Theorem also play a role in establishing similarity.
• Understanding these criteria for the similarity of triangles is crucial for solving geometric problems, analyzing proportional relationships, and making accurate measurements.

Read More: Geometry

Key Terms: Triangle, Similarity, Congruence, Scale factor, AA, ASA, SAS, Thales’ Theorem, Pythagorean Theorem.

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Definition of Similar Triangles

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A pair of similar triangles is a pair of triangles with the same shape but not necessarily the same size. When two triangles are similar, their congruent (equal) angles and proportionate with the same ratio sides. In other words, if it was to scale one of the triangles up or down, the resulting triangles would still have the same shape and angles as the original triangles. 

• The fundamental idea behind the similarity of triangles is the existence of proportional relationships between corresponding sides. 
• When two triangles are similar, the ratio of the lengths of their corresponding sides remains constant. 
• This proportionality can be expressed using the concept of scale factor. 
• The scale factor, denoted as "k," is the ratio of the length of any corresponding sides between two similar triangles. 
• For example, if the scale factor is 2, it means that one triangle is twice as large as the other in terms of side lengths. 
• The proportional relationships between corresponding sides allow to establish a predictable pattern of ratios. 
• For instance, if the lengths of two corresponding sides in one triangle are in a 1:2 ratio, then the lengths of the corresponding sides in the other triangle will also be in a 1:2 ratio. 
• This consistency in proportions holds true for all pairs of corresponding sides in similar triangles. 
• By recognizing and utilizing these proportional relationships, unknown side lengths or comparing the dimensions of different triangles can be determined.

Read More: Coordinate Geometry

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Properties of Similar Triangles

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Few important properties of similar triangles are mentioned below– 

• Corresponding angles are congruent: Similar triangles have the same shape, meaning that their corresponding angles are congruent (equal), known as the angle-angle (AA) similarity criterion.
• Corresponding sides are proportional: The sides of similar triangles are proportional to each other, known as the side-side-side (SSS) similarity criterion.
• Ratios of corresponding sides are equal: The ratio of the lengths of corresponding sides of two similar triangles is constant, regardless of the size of the triangles, known as the side-angle-side (SAS) similarity criterion.
• Areas are proportional to the square of the scale factor: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
• Perimeters are proportional to scale factor: the ratio of two identical triangles' perimeters equals the ratio of their corresponding sides.
• Heights and medians are proportional to the corresponding sides: The heights and medians of similar triangles are proportional to the corresponding sides of the triangles.

Read More: Three-Dimensional Geometry

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Angle-Angle (AA) Criterion

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The Angle-Angle (AA) criterion is a condition for establishing the similarity of two triangles. According to this criterion, if the corresponding angles of two triangles are congruent, then the triangles are similar. In other words, if the angles of one triangle match the angles of another triangle, they are considered similar.

• The AA criterion is based on the idea that the measure of angles determines the shape and configuration of a triangle. 
• When two triangles have congruent angles, it implies that their shapes are the same, but they may differ in size or scale. 
• The congruence of angles ensures that the triangles have the same internal structure and that the corresponding angles are in the same position relative to each other.

For example, considering Triangle ABC and Triangle DEF:

∠A ≅∠D

∠B ≅∠E

∠C ≅∠F

In this case, the corresponding angles of Triangle ABC and Triangle DEF are congruent. Angle A corresponds to Angle D, Angle B corresponds to Angle E, and Angle C corresponds to Angle F can be observed. As a result, it can be concluded that Triangle ABC and Triangle DEF are similar according to the AA criterion.

Read More: Area Perimeter Formula

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Side-Angle-Side (SAS) Criterion

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The Side-Angle-Side (SAS) criterion is a condition for establishing the similarity of two triangles. According to this criterion, if one pair of corresponding sides between two triangles is proportional, and the included angles are congruent, then the triangles are considered similar.

• The SAS criterion combines both side proportions and angle congruence to establish similarity between triangles. 
• It ensures that not only are the corresponding sides of the triangles in proportion, but the included angles between those sides are also congruent. 
• This criterion provides a stronger condition for similarity by considering both side lengths and angles.

For example, considering Triangle ABC and Triangle DEF:

AB/DE = BC/EF

∠B ≅ ∠E

In this case, the ratio of corresponding sides AB/DE is equal to the ratio of BC/EF can be observed. Additionally, the included angles ∠B and ∠E are congruent. Finally, the ratio of sides AC/DF is also equal to the ratio of BC/EF. These conditions fulfill the SAS criterion, indicating that Triangle ABC and Triangle DEF are similar.

Read More: Analytical Geometry

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Side-Side-Side (SSS) Criterion

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The Side-Side-Side (SSS) criterion is a condition for establishing the similarity of two triangles. According to this criterion, if all three pairs of corresponding sides between two triangles are proportional, then the triangles are considered similar.

• The SSS criterion is based on the idea that the ratio of corresponding side lengths determines the similarity between triangles. 
• When all three pairs of corresponding sides are proportional, it implies that the triangles have the same shape and configuration, regardless of their size or scale.

For example, let's consider Triangle ABC and Triangle DEF:

AB/DE = BC/EF

AC/DF = AB/DE

BC/EF = AC/DF

In this case, the ratios of the corresponding sides (AB/DE, BC/EF, and AC/DF) are all equal can be observed. These proportional side lengths fulfill the SSS criterion, indicating that Triangle ABC and Triangle DEF are similar.

Read More: Euclid's Geometry

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Thales' Theorem

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Thales' Theorem, also referred to as the Basic Proportionality Theorem, is a fundamental geometric principle that relates the lengths of the sides of a triangle intersected by a line parallel to one side. This theorem states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.

To prove Thales' Theorem, consider a triangle ABC, with a line DE parallel to side BC, intersecting sides AB and AC at points D and E, respectively.

To prove AD/DB = AE/EC

Proof: Step 1: Draw line DF parallel to AB, where F lies on line AC.

Step 2: Using the Alternate Interior Angles theorem, it can be established that angles ADE and DFB are congruent, as they are corresponding angles formed by the transversal line DE and parallel lines AB and DF.

Step 3: Similarly, the angles AED and FEC are congruent using the same reasoning.

Step 4: Now, observe triangles ADE and DFB. Both triangles share an angle (ADE = DFB) and have parallel sides (DE || AB). Hence, by the AA similarity criterion, triangles ADE and DFB are similar.

Step 5: By the similarity of triangles ADE and DFB, the proportionality of their corresponding sides can be established:

AD/DB = AE/FB ...(1)

Step 6: In a similar manner, consider triangles AED and FEC. These triangles share an angle (AED = FEC) and have parallel sides (DE || AB). Therefore, triangles AED and FEC are also similar.

Step 7: By the similarity of triangles AED and FEC, the proportionality of their corresponding sides can be established:

AE/EC = FD/FC ...(2)

Step 8: Since triangles DFB and FEC share the side DF, the ratios can be equated from equations (1) and (2):

AD/DB = AE/EC = FD/FC

Thus, Thales' Theorem has been proven, also known as the Basic Proportionality Theorem.

Read More: Two Dimensional Coordinate Geometry

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Converse of Thales' Theorem

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The Converse of Thales' Theorem states that if a line intersects two sides of a triangle proportionally, then it is parallel to the third side of the triangle.

Aim: to prove that line DE is parallel to side BC.

Proof: Step 1: Assume that line DE is not parallel to side BC.

Step 2: Extend line DE to meet side BC at point F.

Step 3: By drawing line AF, two triangles ADE and ADF is created.

Step 4: From the assumption that DE is not parallel to BC, it can be concluded that triangle ADE and triangle ADF are not similar.

Step 5: Since triangles ADE and ADF are not similar, their corresponding sides are not proportional.

Step 6: However, this contradicts the given condition that AD/DB = AE/EC.

Step 7: Therefore, our initial assumption that DE is not parallel to BC is false.

Step 8: Consequently, line DE must be parallel to side BC.

Thus, it has been proven that the Converse of Thales' Theorem, states that if a line intersects two sides of a triangle proportionally, it is parallel to the third side of the triangle.

Read More: Collinear Points

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Converse of the Pythagorean Theorem

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The Converse of the Pythagorean Theorem is a statement that establishes a relationship between the lengths of the sides of a triangle and its similarity to a right triangle. It states that if the squares of the lengths of two sides of a triangle are equal to the square of the third side, then the triangle is similar to a right triangle.

Explanation: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The converse of this theorem takes the opposite direction and deals with similarity rather than right triangles.

• According to the converse, if the squares of the lengths of two sides of a triangle are equal to the square of the third side, it implies that the triangle is similar to a right triangle. 
• This means that the given triangle has the same shape as a right triangle, although it may not necessarily be a right triangle itself.
• The Converse of the Pythagorean Theorem plays a significant role in determining similarity because it provides a condition based on the lengths of the sides of a triangle. 
• By identifying equality between the squares of the side lengths, it can be concluded that the triangle is similar to a right triangle, even if the angles themselves are not right angles.

Read More: Geometric Progression

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Things to Remember

• The scale factor determines the ratio of the lengths of the corresponding sides of the new triangle to those of the original triangle.
• If the given scale factor is greater than 1, the new triangle will be larger than the original triangle, and if the scale factor is less than 1, the new triangle will be smaller than the original triangle.
• Criteria for similarity of triangles: AA Criterion, SAS Criterion, SSS Criterion.
• AA Criterion: Corresponding angles must be congruent for similarity.
• SAS Criterion: One pair of corresponding sides must be proportional, with included angles congruent.
• SSS Criterion: All three pairs of corresponding sides must be proportional.