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
Take triangle ADB and ADC.
∠ADB = ∠ADC = 90 degree
AB = AC { isosceles triangle }
AD = DA { common side }
By RHS congruency, both triangles are congruent.
Hence, ▵ADB \mathtt{\cong } ▵ADC
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