# RHS congruence rule

In this chapter we will learn the concept of RHS congruence with solved examples.

Let us first review the basics of congruence.

## What are congruent triangles?

Two triangles are said to be congruent when they have equal sides and angles.

When placed against each other, congruent triangles will overlap fully.

## What is RHS congruency rule ?

This rule is applicable only for right angled triangles (i.e. triangle with 90 degree angle )

RHS stands for Right angle – Hypotenuse – Side.

According to RHS rule, two right triangles are congruent when they have equal hypotenuse and side.

For example, consider the triangles below.

In triangle ABC and PQR

∠B = ∠Q { right angle }
BC = RQ = 4 cm { equal hypotenuse }
AC = PR = 5.3 cm { equal side}

By RHS congruency, both the triangles are congruent.
Hence, ▵ABC \mathtt{\cong } ▵PQR

I hope the concept is clear, let us solve some problems for better understanding.

## RHS congruency rule- Solved Problems

(01) Observe the below image and prove ▵AMB \mathtt{\cong } ▵CMB.

Solution
Taking triangle AMB and CMB.

∠AMB = ∠CMB = 90 degree
AB = BC = 4 cm { same hypotenuse }
BM = MB { same side }

By RHS congruency, both the triangles are congruent.
Hence, ▵AMB \mathtt{\cong } ▵CMB

(02) Study the below image and find ∠MON.

Solution
Taking triangle ABC and MNO.

∠ABC = ∠MNO { 90 degree }
AC = MO = 5.5 cm { hypotenuse }
BC = ON = 4.3 cm

By RHS congruency, both triangles are congruent.
Hence, ▵ABC \mathtt{\cong } ▵MNO.

We know that in congruent triangles, corresponding angles and sides are equal.

Hence, ∠ACB = ∠MON { 90 degree }

(03) Prove that the altitude AD bisect side BC of the below isosceles triangle.

Solution

AB = AC { isosceles triangle }
AD = DA { common side }

By RHS congruency, both triangles are congruent.

We know that congruent triangles have equal sides and angles.

Hence, BD = DC.
This means that line AD bisect side BC into two equal parts.

(04) Prove triangle ABC and ABM are congruent.

Solution
Taking triangle MAB and CBA.

∠MAB = ∠CBA = 90 degree
AC = AB = 4.3 cm { hypotenuse }
AM = BC = 3.4 cm

By RHS congruency, the triangles MAB and CBA are congruent.
Hence, ▵MAB \mathtt{\cong } ▵ CBA

(05) In the below triangle, MD = DN and BD = DC. Using RHS congruency theorem to prove AB = AC

Solution
Taking triangle MDB and NDC.

∠BMD = ∠DNC = 90 degree
BD = DC { same length hypotenuse }
MD = DN

By RHS congruency, ▵MDB \mathtt{\cong } ▵NDC.

We know that congruent triangles have same side length and angle measurement.

So, ∠MBD = ∠NCD.

We know that side opposite to equal angles are also equal.
Hence, AB = AC

Next chapter : Prove that angle bisector of triangle are concurrent