To prove two triangles congruent,
We use ASA Criteria –Angle Side Angle
In ASA Congruency Criteria,
- 2 angles of both the triangles are equal
- The side between these angles of both the triangles are equal.
For example
Here,
2 angles and the side between them are equal.
So, they are congruent
Let’s take some examples
Are these triangles congruent?
In ∆ABC and ∆RPQ,
∠A = ∠R ( Both are 30 °)
AB = RP ( Both are 3 cm )
∠B = ∠P ( Both are 40 °)
∴ ∆ABC ≅ ∆RPQ ( ASA Congruence Rule)
Here,
A ⟷ R
B ⟷ P
C ⟷ Q
Are these triangles congruent?
In ∆ABC and ∆RPQ,
∠J = ∠N (Both are 90 °)
JK = NM (Both are 7 cm )
∠K = ∠M (Both are 30 °)
∴ ∆IJK ≅ ∆LNM (ASA Congruence Rule)
Here,
J ⟷ N
K ⟷ M
I ⟷ L
Are these triangles congruent?
In these two triangles,
the side is not same
So, these two triangles are not congruent
∴ ∆QRP ≇ ∆DEF
Are these triangles congruent?
In ∆PRQ and ∆MLN,
∠R = ∠L ( Both are 60 °)
RQ = LN ( Both are 6 cm )
∠Q = ∠N (Both are 30 °)
∴ ∆PQR ≅ ∆MLN (ASA Congruence Rule)
Here,
R ⟷ L
Q ⟷ N
P ⟷ M
Are these triangles congruent?
Here, ∠BAD = ∠BAC + ∠CAD
= 30° + 45°
= 75° …(1)
And, ∠ABC = ∠ABD + ∠CBD
= 30° + 45°
= 75° …(2)
From (1) & (2)
∠BAD = ∠ABC = 75°
In ∆ABD and ∆ BAC,
∠BAD = ∠ABC (Proved above )
AB = BA (Common )
∠DBA = ∠CAB (Both are 30 °)
∴ ∆ABD ≅ ∆BAC (ASA Congruency)
Here,
A ⟷ B
B ⟷ A
D ⟷ C
Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by ASA congruence rule. In case of congruence, write it in symbolic form.
∆DEF | ∆DEF |
(a) ∠D = 60°, ∠F = 80°, DF = 5 cm |
∠Q = 60°, ∠R = 80°, QR = 5 cm |
In ∆DEF and ∆QPR,
∠D = ∠Q (Both are 60 °)
DF = QR (Both are 5 cm )
∠F = ∠R (Both are 80 °)
∴ ∆DEF ≅ ∆QPR (ASA Congruence Rule)
Here,
D ⟷ Q
F ⟷ R
E ⟷ P
Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by ASA congruence rule. In case of congruence, write it in symbolic form.
∆DEF | ∆DEF |
(a) ∠D = 60°, ∠F = 80°, DF = 6 cm |
∠Q = 60°, ∠R = 80°, QP = 6 cm |
In ∆DEF and ∆QPR,
∠D = ∠Q (Both are 60 °)
DF = QR (Both are 5 cm )
∠F = ∠R (Both are 80 °)
∴ ∆DEF ≅ ∆QPR (ASA Congruence Rule)
Here,
D ⟷ Q
F ⟷ R
E ⟷ P
Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by ASA congruence rule. In case of congruence, write it in symbolic form.
∆DEF | ∆DEF |
(a) ∠E = 80°, ∠F = 30°, EF = 5 cm |
∠P = 80°, PQ = 5 cm, ∠R = 30° cm |
Since PQ is not the side between ∠P and ∠R
Thus, side between two angles is not equal
∴ The triangles are not congruent
So, ∆DEF ≇ ∆QPR
In Fig 7.28, ray AZ bisects ∠DAB as well as ∠DCB.
State the three pairs of equal parts in triangles BAC and DAC.
In ∆BAC and ∆DAC,
∠BAC = ∠DAC (AZ bisects ∠DAB )
AC = AC (Common )
∠BCA = ∠DCA (AZ bisects ∠DCB )
∴ ∆BAC ≅ ∆DAC (ASA Congruence Rule)