R.H.S. and S.S.S. Criteria for Congruence: Criteria, Examples

R.H.S. and S.S.S. Criteria for Congruence: The measurements of the sides and angles of the two triangles determine congruence. There are a few criteria that can be used to verify whether two given triangles are congruent or not. The congruence of triangles defines the given triangle and its mirror image.

The S.S.S criteria for congruence state that the triangles are congruent if three sides of one triangle are equal to another. The R.H.S. criteria for congruence state that right triangles are congruent if one triangle’s hypotenuse and one side are the same as another triangle’s hypotenuse and one side. In this article, we will discuss about R.H.S and S.S.S. criteria for Congruence in detail. Scroll down to find out more!

Congruence of Triangle: Overview

Congruency means that the two given triangles have the same size and shape. Both triangles can be superimposed on one another. A triangle’s position or appearance appears to have changed when it is rotated, reflected, and translated. The six parts of a triangle and their equal parts in the opposite triangle must be identified in this case.

Two triangles are congruent if their three angles and three sides are equal to another triangle’s corresponding angles and sides.

In the \(\Delta A B C\) and \(\Delta P Q R\), as shown below, we can identify that \(A C=P R, C B=R Q\), and \(B A=Q P\) and \(\angle A=\angle P, \angle C=\angle R\) and \(\angle B=\angle Q\).

Then we can say that \(\Delta A B C \cong \Delta P Q R\).

We can confirm that \(\triangle A B C \cong \triangle P Q R\) by identifying the corresponding parts of the given triangles.

What is CPCT?

When we study the congruent triangle, we come across the word CPCT. The acronym CPCT stands for “Corresponding Parts of Congruent Triangles.” As we know, the matching parts of congruent triangles are equal. We typically utilize the abbreviation CPCT in short form instead of the full form while dealing with triangle concepts and solving problems.

Conditions for Congruence of Two Triangles

If two triangles are of the same size and shape, they are said to be congruent. It is unnecessary to find all six corresponding elements of both triangles to be equal to determine that they are congruent. There are five conditions for two triangles to be congruent, as per studies and experiments. The congruence properties are S.S.S., S.A.S., A.S.A., A.A.S., and R.H.S.

Congruence Rules

Two triangles are congruent if they satisfy any of the five congruence rules/conditions. They are given below:

1. Side-Side-Side (S.S.S.): The triangles are congruent if three sides of one triangle are equal to another.
2. Side-Angle-Side (S.A.S.): The triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.
3. Angle-Side-Angle (A.S.A.): The triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
4. Angle-Angle-Side (A.A.S.): The triangles are congruent if two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle.
5. Right-Angle-Hypotenuse-Side (R.H.S.): Two right triangles are congruent if one triangle’s hypotenuse and one side are the same as another triangle’s hypotenuse and one side.

SSS Rule of Congruence

The S.S.S. criterion stands for the postulate of side-by-side congruence.

Statement: If all three sides of one triangle are equal to the three equal sides of another triangle, the two triangles are congruent according to the same criterion.

In the \(\triangle A B C\) and \(\Delta Y Z X\), as shown above, we can identify that \(A C=Y Z, A B=X Y\), and \(B C=X Z\).

Then, we can say that \(\Delta A C B \cong \Delta Y Z X\).

R.H.S. Criteria for Congruence of Two Triangles

Statement: The R.H.S. congruence theorem states that if the right-angled triangle’s hypotenuse and anyone side are equal to another triangle’s hypotenuse and corresponding side, then the two triangles are congruent.

Only right-angled triangles are subject to this rule.

It’s important to note that if we keep the hypotenuse and any of the other two sides of two right triangles equal, we’ll automatically get three same sides because all three sides of a right triangle are related to each other, known as the Pythagoras theorem.

\(\text {hypotenuse}^{2}= \text {base}^{2}+\text {perpendicular} ^{2}\)

Example:

From the given figure,
\(\text {Hypotenuse} =5 \mathrm{~cm}\), \(\text {base} =3 \mathrm{~cm}\), \(\text {perpendicular} =4 \mathrm{~cm}\)
\(\text {hypotenuse}^{2}= \text {base}^{2}+ \text {perpendicular}^{2}\)
\(\Rightarrow 5^{2}=3^{2}+4^{2}\)
\(\Rightarrow 25=9+16\)
\(\Rightarrow 25=25\)
\(\Rightarrow L \cdot H \cdot S=R \cdot H \cdot S\)
So, it is proved.
In the above figure, \(\triangle A B C \cong \triangle X Y Z\)
Hence, \(\triangle A B C \cong \triangle X Y Z\) using R.H.S. congruency rule.

Solved Examples on R.H.S. and S.S.S. Criteria for Congruence

Q.1. State and prove whether the given triangles are congruent or not.

Ans: Given triangles are \(\Delta Z X Y\) and \(\triangle A B C\).
Here, \(X Z=A B\) (Corresponding sides),
\(Y Z=C A\) (Hypotenuse),
and \(\angle Z X Y=\angle A B C=90^{\circ}\) (Right angle)
Therefore, \(\Delta Z X Y \cong \triangle A B C\).
It is proved.
Hence, the given triangles are congruent by the RHS congruence criterion.

Q.2. In the given isosceles triangle \(\triangle A B C\), prove that the altitude \(AD\) bisects the base of the triangle \(BC\).

Ans: In the given triangle \(\triangle A B C\), two small right-angled triangles are formed, and those are \(\triangle A D B\) and \(\triangle A D C\).
Altitude \(A D\) bisects \(B C\) when \(B D=D C\).
So, let us prove that \(\triangle A D B \cong \triangle A D C\).
\(A B=A C\) (Given as equal, because it is an isosceles triangle),
\(A D=A D\) (common),
and, \(\angle A D B=\angle A D C=90^{\circ}\) (Right angle)
So, by the RHS congruence criterion,
\(\triangle A D B \cong \triangle A D C\).
\(B D=D C\) (By CPCT)
\(\therefore\) The altitude of triangle \(\triangle A B C\) bisects the base \(B C\) of the triangle.
Hence, it is proved.

Q.3. The two points, \(X\) and \(Y\), are on the opposite sides of the line segment \(A B\). The points \(X\) and \(Y\) are equidistant from points \(A\) and \(B\).

Can you prove that \(\triangle X A Y\) is congruent to the \(\triangle X B Y\) ?
Ans: As the two points \(X\) and \(Y\) are equidistant from the endpoints of the line segment \(A B\).
Therefore, \(A X=B X, A Y=B Y\)
Now the side \(X Y\) is common in both the triangles \(\triangle X A Y\) and \(\triangle X B Y\).
Therefore, by using the SSS congruence rule, the two triangles are congruent.
Hence, \(\triangle X A Y \cong \Delta X B Y\) is proved.

Q.4. In the given figure, \(\triangle P Q R \cong \triangle N L M, \angle Q=\angle L=90^{\circ}\), then find the length of the third side.

Ans: We know, the RHS congruence theorem states that if one right-angled triangle’s hypotenuse and one side are equal to another right-angled triangle’s hypotenuse and corresponding side, the two triangles are congruent.
From the given figure,
\(\text {Hypotenuse}=13 \mathrm{~cm}\), \(\text {base}=5 \mathrm{~cm}\), \(\text {perpendicular}=?\)
\(\text {hypotenuse}^{2}= \text {base}^{2}+ \text {perpendicular}^{2}\)
\(\Rightarrow \text {perpendicular}^{2}= \text {hypotenuse}^{2}- \text {base}^{2}\)
\(\Rightarrow \text {perpendicular}^{2}=13^{2}-5^{2}\)
\(\Rightarrow \text {perpendicular}^{2}=169-25\)
\(\Rightarrow \text {perpendicular}^{2}=144\)
\(\Rightarrow \text {perpendicular}=\sqrt{144}=12\)
\(\Rightarrow \text {perpendicular}=12 \mathrm{~cm}\)
Therefore, the lengths of the third side of congruent right-angled triangles are \(12 \,\text {cm}\) each.

Q.5. Check if the given triangles are congruent or not, as mentioned below.
Ans:

Ans: In the given triangles \(\Delta X Y Z, \Delta R Q P\)
\(\angle X Y Z=\angle P Q R=90^{\circ}\)
\(X Z=P R=5 \mathrm{~cm}\) (Hypotenuse of the given triangles)
\(Y Z=P Q=4 \mathrm{~cm}\) (Side of the given triangles)
Hence, by the R.H.S rule of congruence, given triangles are congruent.
Therefore, \(\Delta X Y Z \cong \Delta R Q P\).

Q.6. State whether the two triangles are congruent. Give a reason to support your answer.

Ans: We know, if all three sides of one triangle are equal to the three equal sides of another triangle, the two triangles are congruent according to the same criterion.
Given triangles are \(\triangle A B C\) and \(\triangle P Q R\).
We can identify that \(A C=P R, A B=P Q\), and \(B C=Q R\)
Then we can say that by the SSS congruence rule
\(\Delta A B C \cong \Delta P Q R\)
Hence, the given triangles are congruent triangles by the SSS congruence rule.

Summary

If all three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent per the SSS criterion. Two right triangles are congruent if one triangle’s hypotenuse and one side are the same as another triangle’s hypotenuse and the corresponding side. This article includes the definition of congruence of triangles, conditions of congruence of triangles. We explained in detail about RHS and SSS criteria for congruence.

FAQs on R.H.S. and S.S.S. Criteria for Congruence

Q.1. What is the RHS congruence rule?
Ans: Right Angle-Hypotenuse-Side is the RHS criterion of congruence (full form of RHS congruence). The RHS congruence theorem states that two right-angled triangles are congruent if their hypotenuse and anyone side are equal.

Q.2. Is SSS a congruence criterion?
Ans: Yes, the SSS Criterion stands for the postulate of side-by-side congruence. If all three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent as per this criterion.

Q.3. How do you prove triangles are congruent by RHS?
Ans: Two right triangles are congruent if one triangle’s hypotenuse and anyone side are the same as another triangle’s hypotenuse and the corresponding side.

Q.4. What are the criteria for RHS congruence?
Ans: By RHS criteria, two right-angled triangles are said to be congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other triangle.

Q.5. What are the 5 congruence conditions?
Ans: Two triangles are congruent if they satisfy any one of the five congruence conditions. They are as follows:
1. Side-Side-Side (SSS)
2. Side-Angle-Side (SAS) 
3. Angle-Side-Angle (ASA) 
4. Angle-Angle-Side (AAS)  
5. Right-Angle-Hypotenuse-Side (RHS).

Q.6. What is CPCT in a triangle?
Ans: We come across the term CPCT when studying the congruent triangle. “CPCT” is an abbreviation that stands for “Corresponding Parts of Congruent Triangles.” As we all know, the matching elements of congruent triangles are equal. In working with concepts of triangles and solving problems, we usually use the abbreviation CPCT in short form instead of the full form.

Now you are provided with all the necessary information on the RHS and SSS criteria for congruence and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.