Proof of RHS Congruence Rule
Let’s consider two right angle ΔABC and ΔDEF,
Where,
- ∠B = 90° and ∠E = 90°,
- Hypotenuse is equal i.e. AC = DF, and
- One side is equal i.e. BC = EF.
This, can be illustrated as follows.
To Proof: We need to prove that ΔABC and ΔDEF are congruent.
Proof:
In right ΔABC, By Pythagoras theorem,
AC2 = AB2 + BC2
⇒ AB2 = AC2 – BC2 . . . (1)
In right ΔDEF, By Pythagoras theorem,
DF2 = DE2 + EF2
⇒ DE2 = DF2 – EF2 . . . (2)
From (1),
AB2 = AC2 – BC2
⇒ AB2 = DF2 – EF2 (∵AC = DF and BC= EF (given))
⇒ AB2 = DE2 (From (2))
⇒ AB = DE . . .(3)
In ΔABC and ΔDEF,
- AB = DE (From Equation3)
- BC= EF (Given)
- AC = DF (Given)
ΔABC ≅ ΔDEF (By SSS congruence rule) [Hence proved.]
The above proves the RHS Congruence Rule.
RHS Congruence Rule
RHS Congruence Rule is also known as the HL (Hypotenuse-Leg) Congruence Theorem. It states the criteria for any two right-angle triangles to be congruent.
This rule states that if in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other triangle, then the triangles are congruent. In this article, we will discuss the criteria of congruence of right-angle triangles in detail including proof and examples.
Table of Content
- RHS Congruence Rule
- Proof
- Steps to apply RHS Congruence Rule
- RHS and SSS Congurence Rule
- Solved Examples
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