Congruence among Right-Angled Triangles: A triangle is a closed polygon formed by three line segments (sides). A triangle is said to be a right-angled triangle if any one of its angles is equal to the right angle. Two triangles are said to be congruent if they superimpose each other. Thus, two congruent triangles have equal size and the same shape. Two right triangles are said to be congruent if they are equal or the same in all aspects (shape, size, etc.).

In this article, we will study the theorems related to congruence among right-angled triangles and solved examples.

Right-Angled Triangle

A triangle is a closed polygon formed by three line segments (sides). Triangles are classified into different types based on their angles and their sides. Some of such classified triangles are isosceles, acute, equilateral, obtuse, and right-angled triangles, etc.

A triangle has three vertices and three angles, and three sides. 

A triangle is said to be a right-angled triangle if one of its angles measures a right angle. The three sides of the right-angled triangle are named altitude or height, base, and the hypotenuse. The side that lies opposite to the right angle \({90^ \circ }\) is called the hypotenuse. The hypotenuse is the largest side in length.

A right triangle is said to be a right-angled isosceles triangle if the lengths of two sides (altitude, base) other than the hypotenuse have equal length. The two angles in the right-angled isosceles triangle other than the right angle are equal to \({45^ \circ }\)

Congruence and Congruent Triangles

Two triangles are said to be congruent if they superimpose each other. Thus, two congruent triangles have equal size and the same shape. In the congruency of two triangles

1. Corresponding sides of two triangles are equal in length
2. Corresponding angles of two triangles are equal in measure

In order to check the congruency of two triangles, we need to know about the length of all sides and the measures of all angles. We have certain rules (congruence criterion) to prove the congruence among the triangles.

They are listed below:

1. S.S.S
2. A.S.A
3. S.A.S
4. R.H.S

In this article, we are going to discuss the R.H.S congruence rule. The R.H.S congruence rule in full form is “Right angle-Hypotenuse-Side.”  

R.H.S Congruence Rule

The full form of the R.H.S congruence rule is “Right angle-Hypotenuse-Side.”  

The R.H.S congruence rule states that if the right-angle and the length of the hypotenuse and the length of any side (base or altitude) equals or matches exactly with the corresponding right angle, and the hypotenuse and the side of another triangle, then those triangles are said to be congruent triangles.

The R.H.S congruence rule is applicable to only right-angled triangles.

Here, in two right triangles \(ABC\) and \(XYZ,\)

1. \(AC=XZ\) (Length of the hypotenuse are equal, shown by a single line on the sides)

2. \(BC=YZ\) (Length of the base (side) are equal, shown by a double line on the sides)

3. \(\angle ABC = \angle XYZ = {90^ \circ }\) (Which are shown by the red color portion)

Thus, the corresponding side, hypotenuse, and the right angle of the two triangles are equal. So, by R.H.S congruence rule, they are said to be congruent.

The congruence among the triangles can be represented mathematically as follows:

\(\Delta A B C \cong \triangle X Y Z\)

Pythagoras Theorem

Interestingly, when the hypotenuse and the length of one side of the right triangle equal the corresponding hypotenuse and the sides, we may also say that the other sides of the triangle are also equal.

All sides in the right-angled triangle are related to each other by the Pythagoras theorem, which states that the sum of squares of two sides of the right triangle is equal to the square of the hypotenuse.

\({\rm{Hy}}{{\rm{p}}^{\rm{2}}}{\rm{ = Sid}}{{\rm{e}}^{\rm{2}}}{\rm{ + Sid}}{{\rm{e}}^{\rm{2}}}\)

In a right triangle of sides \(a, b, c,\) they are related to each other by \(c^{2}=a^{2}+b^{2}\)

Congruence among Right-Angled Triangles

In this section, we shall discuss the theorems that describe the congruence among the right-angled triangles. In two right triangles, already one right angle is equal in both. Based on this fact, we have mainly two theorems that help to prove the congruence among the right-angled triangles. They are

1. The LA (Leg-acute angle) Theorem
2. The LL (Leg-leg) Theorem

The LA (Leg-Angle) Theorem:

This theorem is the application or special A.S.A congruence rule in right-angled triangles as two right-angled triangles have one equal angle (Right angle). The leg-Angle (LA) theorem states that the length of any one side (leg) and the acute angle of the right triangle are equal to the corresponding side (leg) and the angle of another triangle, then those two right triangles are said to be congruent.

In the given triangles \(TIW\) and \(FUN,\)

The sides \(IW\) and \(FU\) and the angles \(∠IWT\) and \(∠UFN\) are equal, as shown in the below figure.

Therefore, by LA theorem, \(\Delta T I W \cong \Delta F U N\)

The LL (Leg-Leg) Theorem

This theorem is the application or special S.A.S congruence rule in right-angled triangles as two right-angled triangles have one equal angle (Right angle). The leg-Leg (LL) theorem states that the length of two sides (legs) of the right triangle is equal to the corresponding sides (legs) of another triangle, then those two right triangles are said to be congruent.

In the two right triangles, \(MOP, RGA,\) sides \(OP\) and \(AG\) are equal, and the other sides (Hypotenuse) \(PM\) and \(AR\) are equal, as shown in the below figure.

So, by LL (Leg-Leg) theorem, \(\Delta MOP \cong \Delta RGA\)

Solved Examples – Congruence among Right-Angled Triangles

Q.1. Prove that the line PO of the triangle PQR bisects the side QR by using the R.H.S congruence rule.

Ans: Here we need to show that the line \(OP\) bisects the side \( QR. (OQ=OR.)\)
In the two triangles, \(POQ\) and \(POR,\)
\(PQ=PR\)(Given)
\(OP=OP\) (Common)
\(\angle POQ = \angle POR = {90^ \circ }\) (Given)
By the R.H.S congruency rule, the two right triangles \(POQ\) and \(POR,\) are congruent to each other.
\(\Delta POQ \cong \Delta POR\)
We know that for the congruent triangles, the corresponding sides are equal.
\(OQ=OR\)
Therefore, the line \(PO\) bisects the side \(QR\) of the given isosceles triangle \(PQR.\)
Hence, proved.

Q.2. Check the congruence among the below shown right triangles.

Ans: Given two triangles \(ABC, PQR\) are right triangles.
So, \(\angle BAC = \angle QPR = {90^ \circ }\)
In \(∆ABC,\) the length of the hypotenuse is found by using Pythagoras theorem as follows:
\(B C^{2}=A C^{2}+A B^{2}=3^{2}+4^{2}=9+16=25\)
Thus, the length of the hypotenuse \(BC=5\) unis.
In two triangles, \(ABC, PQR,\)
\(BC=QR=5\) units
\(AB=PQ=3\) units
\(\angle BAC = \angle QPR = {90^ \circ }\)
By using the R.H.S congruency rule, two triangles are congruent to each other.
\(\Delta ABC \cong \Delta PQR\)

Q.3. Prove that the two pieces of sandwiches shown in the figure in the right triangle shape are congruent.

Ans: Given the two pieces of two sandwiches are in the shape of right triangles.
Given, triangles \(ABC\) and \(PQR\) are right triangles.
Thus, \(\angle ABC = \angle PQR = {90^ \circ }\)
In two right triangles, \(ABC, PQR,\)
\(BC=QR\) (Given in the figure)
\(AC=PR\) (Given in the figure)
\(\angle ABC = \angle PQR = {90^ \circ }\)
By R.H.S congruence rule, the two right triangles shown in the given pictures are congruent.
\(\Delta ABC \cong \Delta PQR\)

Q.4. Prove that the two triangles given are congruent by using the LA theorem.

Ans: In the two triangles \(RST\) and \(UVW,\)
\(\angle S R T=\angle V W U\) (Given in the figure)
\(RS=WU\) (Given in the figure)
Here, the length of one side and the angle of one triangle is equal to the corresponding side and the angle of another triangle.
So, by the LA theorem, the two triangles given are congruent.
\(\Delta R S T \cong \Delta W V U\)

Q.5. Prove that the diagonal of the rectangle divides it into two congruent right-angled triangles by using the R.H.S congruence rule.
Ans: Consider the rectangle \(ABCD,\) and the diagonal \(AC\) is drawn as shown in the figure.

In triangles \(ABC\) and \(ADC,\)
\(AB=CD\) (Opposite sides of the rectangles)
\(BC=AD\) (Opposite sides of the rectangles)
\(\angle ABC = \angle ADC = {90^ \circ }\) (Angles in a rectangle)
By R.H.S congruence rule,\(\Delta A B C \cong \Delta A D C\)
Therefore, the diagonal \(AC\) divides the rectangle \(ABCD\) into two congruent right triangles.

Summary

In this article, we have discussed the definitions of the right triangle and its properties. We also discussed the Pythagoras theorem. In this article, we also studied congruence and congruent triangles. 

This article also discussed the R.H.S congruence rule, the LA (Leg-angle) theorem, and the LL (Leg-leg) theorem to prove the congruence among right-angled triangles.

This article also gives the solved examples, which help us to understand the concepts and solve them easily.

Frequently Asked Questions (FAQs) – Congruence among Right-Angled Triangles:

Q.1. What are the five rules of congruency used in triangles?
Ans: The five congruence rules used in the triangles to prove that they are congruent triangles are listed below:
1. A.S.A (angle-side-angle)
2. S.A.S (Side-angle-side)
3. R.H.S (Right angle-hypotenuse-side)
4. S.S.S (side-side-side)
5. A.A.S (Angle-angle-side)

Q.2. What are all the right triangle congruence theorems?
Ans: The congruence theorems in the right triangle are
1. The LL (Leg-Leg) theorem
2. The LA (Leg- Acute angle) theorem

Q.3. What is congruence among the right-angled triangle?
Ans: If the length of the hypotenuse and the length of any one side of the right triangle are equal to the corresponding length of the hypotenuse and the side of the other triangle, and then those triangles are said to be congruent. This process is called the congruence among the right-angled triangles.

Q.4. How do you prove that the right triangle is congruent?
Ans: We can prove the right triangle is congruent by equating the length of one side and the hypotenuse of two triangles.

Q.5. What does R.H.S stand in the congruence of triangles?
Ans: The R.H.S stands in the congruence triangle is “Right angle-Hypotenuse-Side.”