Theorem 7.5 Class 9 Maths Explanation with Proof

In this post, we will discuss theorem 7.5 class 9 Maths which is related to Chapter 7 Triangles Class 9 Mathematics. After understanding theorem 7.5, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

Theorem 7.5 (RHS congruence rule)

If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

Given 

Two right triangles ABC and Def in which

∠B=∠E=90° 

AC=DF

And, BC=EF

To prove

∆ABC=∆DEF 

                                

Construction 

Produce DE to G such that GE=AB. Join GF

Proof

In ∆ABC and ∆GEF, we have

AB=GE                                (by construction)

∠CBA=∠FEG                      (each 90°)

And BC=EF                        (given)

So, ∆ABC and ∆GEF (SAS congruence rule)

⟹ ∠A=∠G

And AC=GF                       (CPCT)

AC=GF

And AC=DF

⟹ DF=GF

⟹ ∠G=∠D (In ∆FDG, angles opposite to equal sides)

∠A=∠D                                    (∵∠G=∠A)

Now in, ∆ABC and ∆DEF, we have

∠A=∠D , ∠B=∠E

∴ ∠C=∠F 

In, ∆ABC and ∆DEF, we have

BC=EF                               (given)

∠C=∠F                              (proved)

 And,   AC=DF                  (given)

So, ∆ABC and ∆DEF(SAS congruenc rule)

Related Topics

1. Theorem 7.4

2. Theorem 7.6