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In general two objects are similar if they look alike. In layman’s words, objects with the same shape, whether they have the same size or not are usually called similar. In mathematics, it is quite different. In mathematics, two objects are similar when their shapes are the same but their sizes are different.
Let’s understand what is the similarity of triangles and the criteria for similarity of triangles with examples.
What are Similar Triangles?
Two triangles will be similar if the corresponding angles are equal and the corresponding sides are in the same ratio or proportion. Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides must be the same. If two triangles are similar that means,
- All corresponding angle pairs of triangles are equal.
- All corresponding sides of triangles are proportional.
In the above figure, two triangles $\triangle \text{ABC}$ and $\triangle \text{XYZ}$ are similar. We use the ‘$\sim$’ symbol to represent the similarity. So, if two triangles are similar, we show it as $\triangle \text{ABC} \sim \triangle \text{XYZ}$.
Criteria for Similarity of Triangles – Similar Triangles Formula
There are two conditions using which we can verify if the given set of triangles is similar or not. According to these conditions, two triangles can be said similar if either their corresponding angles are equal or congruent or if their corresponding sides are in proportion. Therefore, two triangles $\triangle \text{ABC}$ and $\triangle \text{XYZ}$ can be proved similar $\left( \triangle \text{ABC} \sim \triangle \text{XYZ} \right)$ using either condition among the following set of similar triangles formulas
- AA (Angle Angle): If any two of the angles of the triangles are equal, then the triangles are said to be similar.
- SAS (Side Angle Side): If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
- SSS (Side Side Side): If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
AA (or AAA) or Angle-Angle Similarity Criterion
AA similarity criterion states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles. AA similarity rule is applied when we only know the measure of the angles and have no idea about the length of the sides of the triangle.
In the above figure, if it is known that $\angle \text{A} = \angle \text{D}$, and $\angle \text{B} = \angle \text{E}$.
So we can say that by the AA similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{ABC}$.
This means, $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}}$ and $\angle \text{C} = \angle \text{F}$.
SAS or Side-Angle-Side Similarity Criterion
SAS similarity criterion states that if any two sides of the first triangle are in exact proportion to the two sides of the second triangle along with the angle formed by these two sides of the individual triangles are equal, then they must be similar triangles. This rule is generally applied when we only know the measure of two sides and the angle formed between those two sides in both triangles respectively.
In the above figure if it is known that $\frac{\text{AB}}{\text{DE}} = \frac{\text{CA}}{\text{FD}}$, and $\angle \text{A} = \angle \text{D}$.
And we can say that by the SAS similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{DEF}$.
SSS or Side-Side-Side Similarity Criterion
SSS similarity criterion states that two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles is equal. This criterion is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.
In the above figure if it is known that $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}}$, then we can say that by the SSS similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{DEF}$.
Difference Between Similar Triangles and Congruent Triangles
Similarity and congruency are two different properties of triangles. The following are the differences between similar triangles and congruent triangles.
| Similar Triangles | Congruent Triangles |
| Similar triangles have the same shape but may be different in size. They superimpose each other when magnified or demagnified. | Congruent triangles are the same in shape and size. They superimpose each other in their original shape. |
| They are represented using the symbol $’\sim’$. For example, similar triangles $\text{ABC}$ and $\text{XYZ}$ will be represented as $\triangle \text{ABC} \sim \triangle \text{XYZ}$. | They are represented using the symbol $’\cong’$. For example, congruent triangles $\text{ABC}$ and $\text{XYZ}$ will be represented as $\triangle \text{ABC} \cong \triangle \text{XYZ}$. |
| The ratio of all the corresponding sides is equal in similar triangles. This common ratio is also called as ‘scale factor’ in similar triangles. | The ratio of corresponding sides is equal to 1 for congruent triangles. |
Key Takeaways
If two triangles are similar or proved similar by any of the above-stated criteria, then they possess few properties of similar triangles. The following are the properties of similar triangles
- Similar triangles have the same shape but different sizes.
- In similar triangles, corresponding angles are equal.
- Corresponding sides of similar triangles are in the same ratio.
- The ratio of the area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides.
Practice Problems
- What is the meaning of the similarity of figures in math?
- What does the similarity of triangles mean?
- Explain the AA similarity criterion.
- Explain the SAS similarity criterion.
- Explain the SSS similarity criterion.
FAQs
What is meant by similar triangles in Geometry?
In geometry, similar triangles are triangles that are the same in shape, but may not be equal in size. For example, all equilateral triangles are similar triangles as each triangle has three angles of $60^{\circ}$ each but the length of the sides are different.
What are the 3 similar triangle Theorems?
The three theorems that are used to prove the similarity of triangles are
a) AA (or AAA) or Angle-Angle Similarity Theorem
b) SAS or Side-Angle-Side Similarity Theorem
c) SSS or Side-Side-Side Similarity Theorem
What are the properties of similar triangles?
The important properties of two similar triangles are
a) The shape of two similar triangles is the same but their sizes might be different.
b) Corresponding angles are equal in similar triangles.
c) In similar triangles, the corresponding sides are in the same ratio.
Conclusion
In geometry, similar triangles are triangles that are the same in shape, but may not be equal in size. When two triangles are similar, then their corresponding angles are equal and the corresponding sides are in the same ratio. The three rules that are used to prove the similarity of two triangles are AA, SAS, and SSS similarity rules.
Recommended Reading
- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
- How to Construct a Triangle(With Steps, Diagrams & Examples)
- Median of a Triangle(Definition & Properties)
- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
- Pair of Angles – Definition, Diagrams, Types, and Examples
- Construction of Angles(Using Protractor & Compass)
- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Tangent of a Circle – Meaning, Properties, Examples
- Angles in a Circle – Meaning, Properties & Examples
- Chord of a Circle – Definition, Properties & Examples
- How to Draw a Circle(With Steps & Pictures)
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
