Criteria for Similarity of Triangle: Theorem, Formulas, Examples

Criteria for Similarity of Triangle have comparable sides that are proportional to each other and corresponding angles equal to each other. Congruent figures are always similar, whereas similar figures are not necessarily congruent. Two triangles are similar unless they have an identical shape but may differ in size. The same shape means the angles of one triangle are equal to the corresponding angles of another triangle. 

The same size means the sides of one triangle are equal to the other triangle’s corresponding sides. There are several methods for determining if two triangles are similar or not. Let’s study more about similar triangles and their attributes and a few examples that have been solved.

Criteria for Similar Triangles

We shall state some criteria (or rules or axioms) for triangles’ similarity, involving fewer triangle elements. Those are the angle-angle \(\left({{\text{AA}}} \right)\) rule of similarity, side-angle-side \(\left({{\text{SAS}}} \right)\) rule of similarity, side-side-side rule of similarity \(\left({{\text{SSS}}} \right)\), right angle-hypotenuse-side \(\left({{\text{RHS}}} \right)\) similarity criterion.

Two triangles are similar if

1. their corresponding angles are the same and
2. their corresponding sides are in the same proportion (or ratio).

That is, in \(\Delta XYZ\) and \(\Delta PQR\), if

(I) \(\angle X = \angle P,\,\angle Y = \angle Q,\,\angle Z = \angle R\) and
(ii) \(\frac{{XY}}{{PQ}} = \frac{{YZ}}{{QR}} = \frac{{ZX}}{{RP}}\) then the two triangles are similar

Symbolically, we write \(\Delta XYZ \sim \Delta PQR\); where symbol \(\sim \) is read as “is similar to”.

Note: If the two triangles’ corresponding angles are equal, they are called equiangular triangles. A famous Greek mathematician Thales gave a significant fact regarding two equiangular triangles, which will be as follows:

The ratio between any two corresponding sides in two equiangular triangles is always the same. It is thought that had used a result called the Basic Proportionality Theorem (currently known as the Thales Theorem) for the same.

Corresponding Sides of Similar Triangles

If the two triangles are similar, their corresponding angles shall be equal, and corresponding sides must be proportional, i.e., the ratios between the lengths of corresponding sides are the same.

Example: If triangles \(XYZ\) and \(DEF\) are two similar triangles

then, their corresponding angles are equal, i.e., \(\angle X = \angle D,\,\angle Y = \angle E\) and \(\angle Z = \angle F\) and their corresponding sides are in proportion, i.e., \(\frac{{XY}}{{DE}} = \frac{{YZ}}{{EF}} = \frac{{XZ}}{{DF}}\).

Note: The ratio of every two corresponding sides in two equiangular triangles is always the same.

AAA or AA Similarity Theorem

If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar \({\text{AAA}}\) or \({\text{AA}}\) similarity criterion.

Example:

If in \(\Delta ABC\) and \(\Delta DEF,\,\angle A = \angle D\) and, \(\angle B = \angle E\), then by \(AA\) criterion, \(\Delta ABC\sim \Delta DEF\).

Note: Since the sum of the angles of a triangle is \({\rm{18}}{{\rm{0}}^{\rm{o}}}\) when two angles of a triangle are equal to two angles of a different triangle, their third angles are also identical.

SAS Criterion for Similarity of Triangles

If anyone angle of a triangle is equal to one angle of the other triangle and the sides involving these angles are proportional, then the two triangles are similar. This criterion is referred to as the \({\text{SAS}}\) (Side–Angle–Side) similarity criterion for two triangles.

Example:

If in \(\Delta ABC\) and \(\Delta DEF,\,\angle A = \angle D\) and \(\frac{{AB}}{{DE}} = \frac{{AC}}{{DF}}\) then by \({\text{SAS}}\) criterion, \(\Delta ABC \sim \Delta DEF\).

Similarly, if \(\angle B = \angle E\) and \(\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}}\) then also \(\Delta ABC \sim \Delta DEF\) and so on.

SSS Criterion for Similarity of Triangles

Suppose in two triangles, sides of one triangle are proportionate to the sides of the other triangle. In that case, their corresponding angles are equal, and hence the two triangles are similar. This criterion refers to the \({\text{SSS}}\) (Side–Side–Side) similarity criterion for two triangles.

Example:

If in \(\Delta ABC\) and \(\Delta PQR,\,\frac{{AB}}{{PQ}} = \frac{{BC}}{{QR}} = \frac{{AC}}{{PR}}\) then by \(“SSS”\) criterion, \(\Delta ABC \sim \Delta PQR\).

Criteria for Similarity of Triangle Important Points

The given figure shows two similar triangles \(ABC\) and \(DEF\) such that vertex \(A\) corresponds to vertex \(D\) (as, \(\angle A = \angle D\)); vertex \(B\) corresponds to vertex \(E\) (as, \(\angle B = \angle E\)) and, vertex \(C\) corresponds to vertex \(F\) (as, \(\angle C = \angle F\)). We write: \(\Delta ABC \sim \Delta DEF\) and not \(\Delta ABC \sim \Delta DFE\) or \(\Delta BAC \sim \Delta DEF\) etc.

The vertices of two similar triangles must be written so that the corresponding vertices occupy the same position. Thus, \(\Delta ABC\) is similar to triangle \(\Delta DEF\). 

\( \Rightarrow \Delta ABC \sim \Delta DEF\) [\(A \leftrightarrow D,\,B \leftrightarrow E\) and \(C \leftrightarrow F\)]

The figure given below shows two similar triangles such that the corresponding vertices are: \(A \leftrightarrow E,\,B \leftrightarrow D\,{\rm{and}}\,C \leftrightarrow F\).

Therefore, \(\Delta ABC \sim\Delta EDF\).

The figure given below shows two similar triangles such that, the corresponding vertices are: \(A\) and \(F,\,B\) and \(D\), and \(C\) and \(E\)

Therefore, \(\Delta ABC\sim\Delta FDE\).

Similar Triangles Formula

If two triangles \(ABC\) and \(DEF\) are similar.

Then, their corresponding angles are equal, i.e., \(\angle A = \angle D,\,\angle B = \angle E\) and \(\angle C = \angle F\) and their corresponding sides are in proportion, i.e., \(\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}\).

Relation between the areas of two triangles: The areas of two similar triangles are proportional to the squares on their corresponding sides.

That is, \(\frac{{{\text{Area}}\,{\text{of}}\,\Delta XYZ}}{{{\text{Area}}\,{\text{of}}\,\Delta DEF}} = \frac{{A{B^2}}}{{D{E^2}}} = \frac{{B{C^2}}}{{E{F^2}}} = \frac{{A{C^2}}}{{D{F^2}}}\).

Solved Examples

Q.1. In the given figure, \(\Delta ABC\) is similar to \(\Delta DEF,\,AB = (x – 0.5){\rm{cm}},\,AC = 1.5x\;{\rm{cm}},\,DE = 9\;{\rm{cm}}\), and \(DF = 3x\;{\rm{cm}}\). Find the lengths of \(AB\) and \(DF\).

Ans: Given, \(\Delta ABC \sim \Delta DEF\)
We know that,in similar triangles, the corresponding sides will be in proportion.
Therefore, \(\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}\)
\( \Rightarrow \frac{{x – 0.5}}{9} = \frac{{BC}}{{EF}} = \frac{{1.5x}}{{3x}}\)
\( \Rightarrow \frac{{x – 0.5}}{9} = \frac{1}{2}\)
\( \Rightarrow 2x – 1 = 9 \Rightarrow x = 5\)
Therefore, length of \(AB = (x – 0.5){\rm{cm}}\)
\( = (5 – 0.5){\rm{cm}} = 4.5\;{\rm{cm}}\) and
Length of \(DF = 3\,x = 3 \times 5\;{\rm{cm}} = 15\;{\rm{cm}}\).

Q.2. In the given figure, \(AB\) and \(DE\) are perpendicular to \(BC\). If \(AB = 9\;{\rm{cm,}}\,DE = 3\;{\rm{cm}}\) and \(AC = 24\;{\rm{cm}}\), calculate \(AD\).

Ans:
In \(\Delta ABC\) and \(\Delta DEC\):
\(\angle ABC = \angle DEC\) (Each \({{{90}^{\rm{o}}}}\) )
\(\angle C\) is common.
Therefore, \(\Delta ABC\sim\Delta DEC\) (By \(AA\))
\( \Rightarrow \frac{{AC}}{{DC}} = \frac{{AB}}{{DE}}\)
\( \Rightarrow \frac{{24\;{\rm{cm}}}}{{DC}} = \frac{{9\;{\rm{cm}}}}{{3\;{\rm{cm}}}}\)
\( \Rightarrow DC = \frac{{24 \times 3}}{9}\;{\rm{cm}} = 8\;{\rm{cm}}\)
Therefore, \(AD = AC – DC = 24\;{\rm{cm}} – 8\;{\rm{cm}} = 16\;{\rm{cm}}\).

Q.3. It is given that \(\Delta FED\sim\Delta STU\). Is it true to say that \(\frac{{DE}}{{ST}} = \frac{{EF}}{{TU}}\)? Why?
Ans: No because the correspondence is \(F \leftrightarrow S,\,E \leftrightarrow T\) and \(D \leftrightarrow U\).
With this correspondence, we have \(\frac{{DE}}{{UT}} = \frac{{EF}}{{TS}}\).

Q.4. In triangles \(PQR\) and \(MST,\,\angle P = {55^{\rm{o}}},\,\angle Q = {25^{\rm{o}}},\,\angle M = {100^{\rm{o}}}\) and \(S = {25^{\rm{o}}}\). Is \(\Delta QPR\sim\Delta TSM\)? Why?
Ans: In \(\Delta MST,\,\angle M = {100^{\rm{o}}}\) and \(\angle S = {25^{\rm{o}}}\).
Therefore, \(\angle T = {180^{\rm{o}}} – \angle M – \angle S\) (angle sum property of a triangle)
\( \Rightarrow \angle T = {180^{\rm{o}}} – {100^{\rm{o}}} – {25^{\rm{o}}} = {55^{\rm{o}}}\)
In \(\Delta PQR\) and \(\Delta MST\), we have
\(\angle P = \angle T = {55^{\rm{o}}}\) and \(\angle Q = \angle S = {25^{\rm{o}}}\)
Therefore, by \(AA\) criterion of similarity, the two triangles are similar.
The correspondence is \(P \leftrightarrow T,\,Q \leftrightarrow S\) and \(R \leftrightarrow M\),
Therefore, \(\Delta QPR\sim\Delta STM\).
Hence, it is wrong to say that \(\Delta QPR\sim\Delta TSM\).

Q.5. In the figure give below \(CA\) is parallel to \(BD,\,AC = 6\;{\rm{cm}},\,AE = 3\;{\rm{cm}},\,EB = 4\;{\rm{cm}},\,ED = 8\;{\rm{cm}}\). Calculate \(CE\) and \(BD\)

.

Ans: In \(\Delta ACE\) and \(\Delta BDE\)
\(\angle A = \angle B\) (Alternate angles, since \(CA\parallel BD\))
\(\angle ACE = \angle BDE\) (Vertically opposite angles)
Therefore, \(\Delta ACE\sim\Delta BDE\) (\(AA\) rule of similarity)
Therefore, \(\frac{{CE}}{{ED}} = \frac{{AE}}{{EB}} = \frac{{AC}}{{BD}}\)
\( \Rightarrow \frac{{CE}}{8} = \frac{3}{4} = \frac{6}{{BD}}\)
\( \Rightarrow CE = \frac{3}{4} \times 8 = 6\) and \(BD = 6 \times \frac{4}{3} = 8\).
Hence, \(CE = 6\;{\rm{cm}}\) and \(BD = 8\;{\rm{cm}}\).

Summary

In this article, we learnt about,criteria for similar triangles, corresponding sides of similar triangles, \({\text{AAA}}\) (or \({\text{AA}}\) similarity theorem, \({\text{SAS}}\) criterion for similarity of triangles, \({\text{SSS}}\) criterion for similarity of triangles, \({\text{RHS}}\) similarity criterion, some important points on criteria for similarity of triangles, solved examples on criteria for similarity of the triangle and Frequently asked questions on criteria for similarity of the triangle.

The learning outcome of this article is how to prove similarity between two triangles, using fewer elements of the triangle.

FAQs

Q.1. What are similar triangles and their properties?
Ans: Two triangles are similar when they have the same shape but not necessarily the same size.
Properties: Two triangles are similar if
1. their corresponding angles are equal and
2. their corresponding sides are in the same ratio (or proportion).

Q.2. What are the criteria of similarity of triangles?
Ans: Criteria of similar triangles: \({\text{SAS,}}\,{\text{AA}}\) or \({\text{AAA,}}\,{\text{SSS}}\) and \({\text{RHS}}\)
1. If any one angle of a triangle is equal to any angle of the other triangle and in both the triangles, the sides including the equal angles are in proportion, then the triangles are similar. (\({\text{SAS}}\) postulate)
2. If two triangles have at least two pairs of corresponding angles equal, the triangles are similar. (\({\text{AA}}\) or \({\text{AAA}}\) postulate)
3. If two triangles have their three pairs of corresponding sides proportional, then the triangles are similar. (\({\text{SSS}}\) postulate)
4. Suppose the ratio of the hypotenuse and one side of a right-angled triangle is equal to the ratio of the hypotenuse and one side of another right-angled triangle. In that case, the two triangles are similar. (\({\text{RHS}}\) postulate)

Q.3. How do we represent two similar triangles?
Ans: We represent two similar triangles using symbol \({\text{” }} \sim {\text{” }}\), where symbol \( \sim \) is read as “is similar to”.
Example: If \(\Delta XYZ\) and \(\Delta PQR\) are two similar triangles, symbolically, we write \(\Delta XYZ \sim \Delta PQR\).

Q.4. Explain SSS criterion for similarity of triangles.
Ans: If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal, and hence the two triangles are similar. This criterion refers to the \({\text{SSS}}\) (Side–Side–Side) similarity criterion for two triangles.

Q.5. What will affect the similarity of two triangles?
Ans: The scale factor affects the similarity of two triangles.
The ratio of two corresponding sides in similar figures is called the scale factor.

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