RHS Congruency Criteria

To prove two triangles congruent,

We use RHS criteria when

• R ight angle
• H ypotenuse
• S ide are equal

In RHS congruence criteria,

• Both triangle will have a right angle.
• Hypotenuse of both triangles are equal.
• Anyone of other two sides of both triangle are equal.

For example:

Here,

Both of these triangles have

• right angle,
• have their hypotenuse equal
• and another side equal.
  •  

So, they are congruent

Let’s see more examples

Are these triangles congruent?

In ∆PQR & ∆NLM,

  ∠Q = ∠L  (    R ight angle, both 90°)

  PR = NM    ( H ypotenuse, both 13 cm)

  QR = LM    ( S ide, both 5 cm)

∴ ∆PQR ≅ ∆NLM  (RHS Congruence Rule)

Are these triangles congruent?

In ∆IJK ≅ ∆ONM,

  ∠J = ∠N     ( R ight angle, both 90°)

  IK = OM     ( H ypotenuse, both 5 cm)

  IJ = ON        ( S ide, both 4 cm)

∴ ∆IJK ≅ ∆ONM     (RHS Congruence Rule)

Are these triangles congruent?

In ∆ABC and ∆BAD,

  ∠C = ∠D           ( R ight angle, both 90°)

  AB = BA           ( H ypotenuse, common)

  AC = BD           ( S ide, both 2 cm)

∴ ∆ABC ≅ ∆BAD       (RHS Congruence Rule)

Are these triangles congruent?

In ∆ABC and ∆ADC,

  ∠B = ∠D           ( R ight angle, both 90°)

  AC = AC           ( H ypotenuse, common)

  AB = AD           ( S ide, both 3.6 cm)

∴ ∆ABC ≅ ∆ADC     (RHS Congruence Rule)

Are these triangles congruent?

In ∆PSQ and ∆PSR,

     ∠PSQ = ∠PSR            ( R ight angle, both 90°)

    PQ = PR                     ( H ypotenuse, both 3 cm)

    PS = PS                      ( S ide, common)

  ∴ ∆PSQ ≅ ∆PSR          (RHS Congruence Rule)

In Fig 7.33, BD and CE are altitudes of ∆ABC such that BD = CE.

(i) State the three pairs of equal parts in ∆CBD and ∆BCE.

(ii) Is ∆CBD ≅ ∆BCE? Why or why not?

(iii) Is ∠DCB = ∠EBC? Why or why not?

(i)In ∆CBD and ∆BCE

       ∠CDB = ∠BED        ( R ight angle, both 90°)

       CB = BC                  ( H ypotenuse, common)

       BD = CE                  ( S ide, given)

(ii) ∴ ∆CBD ≅ ∆BCE      (RHS Congruence Rule )

(iii) Now,

  ∠DCB = ∠EBC             (By CPCT )

ABC is an isosceles triangle with AB = AC and AD is one of its altitudes (i) State the three pairs of equal parts in ∆ADB and ∆ADC.

(ii) Is ∆ADB ≅ ∆ADC? Why or why not?

(iii) Is ∠B = ∠C? Why or why not?

(iv) Is BD = CD? Why or why not?

(i) In ∆ADB and ∆ADC

      ∠ADB = ∠ADC     ( R ight angle, both 90°)

       AB = AC              ( H ypotenuse, given)

       AD = AD             ( S ide, common )

(ii) ∴ ∆ADB ≅ ∆ADC  (RHS Congruence Rule )

(iii) Now, ∠B = ∠C      (By CPCT )

(iv) Also, BD = CD     (By CPCT)