In each of the given pairs of triangles of Fig. 6.42, applying only ASA congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.

Question

Philoid · Accepted Answer

Formula Used/Theory:-

ASA congruence criterion is in which 2 angles and a side between them are equal in both the triangles

(a) ∠A = ∠Q

But ∠B≠∠P

∴ Δ ABC and Δ PQR are not congruent

Result:- Δ ABC and Δ PQR are not congruent

(b) ∠ABD = ∠BDC

∠ADB = ∠DBC

BD = BD (common in both triangle)

∴ Δ ADB and Δ CBD are congruent by ASA

∆ADB ≅ ∆CBD

Result:- Δ ADB and Δ CBD are congruent by ASA

(c) ASA congruence criterion is in which 2 angles and a side between them are equal in both the triangles

∠X = ∠L

∠Y = ∠M

XY = ML

∴ Δ XYZ and Δ LMN are congruent by ASA

∆XYZ ≅ ∆LMN

Result:- Δ XYZ and Δ LMN are congruent by ASA

(d) → Angle sum property

Sum of all angles of triangle is 180°

By angle sum property

∠A + ∠B + ∠C = 180° ∠D + ∠E + ∠F = 180°

Equating both

We get;

∠A + ∠B + ∠C = ∠D + ∠E + ∠F

As ∠B = ∠F

∠A = ∠D

Cancelling out we get, ∠C = ∠E

∠C = ∠E

∠B = ∠F

BC = FE

∴ Δ ABC and Δ DFE are congruent by ASA

∆ABC ≅ ∆DFE

Result:- Δ ABC and Δ DFE are congruent by ASA

(e) In Δ PNO and Δ MNO

∠PNO = ∠MON

∠MNO≠∠PON

ON = ON (common in both triangles)

∴ Δ MNO and ΔPON are not congruent by ASA

Result:- ∴ Δ MNO and ΔPON are not congruent by ASA

(f) ∠D = ∠C

∠AOD = ∠COB

OD = CO

∴ Δ ADO and Δ BCO are congruent by ASA

∆ADO ≅ ∆BCO

Result:- Δ ADO and Δ BCO are congruent by ASA