Congruence of Triangles Criteria

December 14, 2022

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The word ‘congruent’ or ‘congruence’ means ‘in agreement’ or ‘harmony’. In math, two shapes are congruent if they have the same shape and size. We can also say if two shapes are congruent, then the mirror image of one shape is the same as the other.

Let’s understand what is the congruence of triangles and what is the congruence of triangles criteria with examples.

What is Congruence in Triangles?

Two triangles are said to be congruent if the three angles and the three sides of a triangle are equal to the corresponding angles and the corresponding sides of another triangle.

In the above figure, $\triangle \text{ABC}$ and $\triangle \text{PQR}$, we can identify that $\text{AB} = \text{PQ}$, $\text{BC} = \text{QR}$, and $\text{CA} = \text{RP}$, $\angle \text{A} = \angle \text{P}$, $\angle \text{B} = \angle \text{Q}$, and $\angle \text{C} = \angle \text{R}$.

Corresponding Parts of Triangles

The two triangles need to be of the same size and shape to be congruent. Both the triangles under consideration should superimpose on each other. When we rotate, reflect, and/ or translate a triangle, its position or appearance seems to be different. In that case, we need to identify the six parts of a triangle and their corresponding parts in the other triangle.

The sides that are equal in two triangles are called corresponding sides and the angles that are equal to each other are called the corresponding angles.

In the above figure,

Corresponding sides are $\text{AB}$ and $\text{PQ}$, $\text{BC}$ and $\text{QR}$, $\text{CA}$ and $\text{RP}$
Corresponding angles are $\angle \text{A}$ and $\angle \text{P}$, $\angle \text{B}$ and $\angle \text{Q}$, $\angle \text{C}$ and $\angle \text{R}$
Corresponding vertices are $\text{A}$ and $\text{P}$, $\text{B}$ and $\text{Q}$, $\text{C}$ and $\text{R}$

Mathematically it is written as

$\text{AB} \leftrightarrow \text{PQ}$, $\text{BC} \leftrightarrow \text{QR}$, $\text{CA} \leftrightarrow \text{RP}$
$\angle \text{A} \leftrightarrow \angle \text{P}$, $\angle \text{B} \leftrightarrow \angle \text{Q}$, $\angle \text{C} \leftrightarrow \angle \text{R}$
$\text{A} \leftrightarrow \text{P}$, $\text{B} \leftrightarrow \text{Q}$, $\text{C} \leftrightarrow \text{R}$

Two triangles are said to be congruent if they are of the same size and same shape. Necessarily, not all the six corresponding elements of both triangles must be found to determine that they are congruent. There are 4 criteria for two triangles to be congruent. They are SSS, SAS, ASA, and RHS congruence properties.

SSS Criterion for Congruence

SSS criterion stands for Side-Side-Side criterion. Under this criterion, two triangles are congruent if the three sides of a triangle are equal to the corresponding sides of the other triangle.

If $\triangle \text{ABC} \cong \triangle \text{XYZ}$ under SSS criterion, then the three angles of $\triangle \text{ABC}$ are bound to be equal to the corresponding angles of $\triangle \text{XYZ}$.

SAS Criterion for Congruence

SAS criterion stands for the Side-Angle-Side criterion. Under this criterion, two triangles are congruent if the two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle.

If $\triangle \text{ABC} \cong \triangle \text{XYZ}$ under SAS criterion, then the third side ($\text{AB}$) and the other two angles of $\triangle \text{ABC}$ are bound to be equal to the corresponding side ($\text{XY}$) and the angles of $\triangle \text{XYZ}$.

ASA Criterion for Congruence

ASA criterion stands for the Angle-Side-Angle criterion. Under the ASA criterion, two triangles are congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.

If $\triangle \text{ABC} \cong \triangle \text{XYZ}$ under ASA criterion, then the third angle $\left( \angle \text{BAC} \right)$ and the other two sides of $\triangle \text{ABC}$ is bound to be equal to the corresponding angle $\left( \angle \text{YXZ} \right)$ and the sides of $\triangle \text{XYZ}$.

RHS Criterion for Congruence

RHS criterion stands for the right angle-hypotenuse-side congruence criterion. Under this criterion, two triangles are congruent, if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle.

If $\triangle \text{ABC} \cong \triangle \text{XYZ}$ under RHS criterion, then the third side ($\text{AB}$) and the other two angles of $\triangle \text{ABC}$ are bound to be equal to the corresponding side ($\text{AB}$) and angles of $\triangle \text{XYZ}$.

Difference Between Congruent Triangles and Similar Triangles

Similarity and congruency are two different properties of triangles. The following are the differences between similar triangles and congruent triangles.

Congruent Triangles	Similar Triangles
Congruent triangles are the same in shape and size. They superimpose each other in their original shape.	Similar triangles have the same shape but may be different in size. They superimpose each other when magnified or demagnified.
They are represented using the symbol $’\cong’$. For example, congruent triangles $\text{ABC}$ and $\text{XYZ}$ will be represented as $\triangle \text{ABC} \cong \triangle \text{XYZ}$.	They are represented using the symbol $’\sim’$. For example, similar triangles $\text{ABC}$ and $\text{XYZ}$ will be represented as $\triangle \text{ABC} \sim \triangle \text{XYZ}$.
The ratio of corresponding sides is equal to 1 for congruent triangles.	The ratio of all the corresponding sides is equal in similar triangles. This common ratio is also called as ‘scale factor’ in similar triangles.

Key Takeaways

Two triangles are congruent if the six parts (3 sides and 3 angles) of one triangle are equal to the corresponding six parts of the other triangle.
There are four conditions to determine if two triangles are congruent. They are SSS, SAS, ASA, and RHS criteria.
Two triangles with equal corresponding angles may not be congruent to each other because one triangle might be an enlarged copy of the other. Hence, there is no AAA criterion for congruence.
We represent congruency by using the symbol $\left( \cong \right)$.
If two triangles are congruent then their perimeters and areas are equal.

Practice Problems

What is the meaning of congruence?
What is the meaning of congruent figures?
What is the meaning of congruent triangles?
Explain the SSS congruence criterion.
Explain the SAS congruence criterion.
Explain the ASA congruence criterion.
Explain the RHS congruence criterion.

FAQs

What are congruent triangles?

Two triangles are said to be congruent if they are of the same size and same shape. In other words, two triangles are congruent if all the sides and angles of a triangle are equal to the corresponding sides and angles of its congruent triangle.

What are the tests of congruence in triangles?

We can prove the congruency of any two triangles by using four different properties, which are – SSS, SAS, ASA, and RHS.

Can we prove congruence in triangles using the AAA criterion?

No, we cannot prove congruence in triangles using the AAA criterion. If two triangles are congruent then at least three parts of one triangle should be equal to the corresponding parts of the other triangle. However, if the three angles of one triangle are equal to the corresponding angles of the other triangle, then the triangles are similar but not congruent. For example, two equilateral triangles have all three angles equal, but their sides could be different, and hence they won’t be congruent but are similar.

Conclusion

Two triangles are said to be congruent if they are of the same size and same shape. Whenever two triangles are congruent, they have the same perimeter and area. There are four criteria to prove the congruence of two triangles and they are SSS, SAS, ASA, and RHS.

Congruence of Triangles Criteria – SSS, SAS, ASA, RHS

What is Congruence in Triangles?

Corresponding Parts of Triangles