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)
