Which of the following pairs of triangles are congruent ? In each case, state the condition of congruency :

(a) In △ ABC and △ DEF, AB = DE, BC = EF and ∠B = ∠E.

(b) In △ ABC and △ DEF, ∠B = ∠E = 90°; AC = DF and BC = EF.

(c) In △ ABC and △ QRP, AB = QR, ∠B = ∠R and ∠C = ∠P.

(d) In △ ABC and △ PQR, AB = PQ, AC = PR and BC = QR.

(e) In △ ABC and △ PQR, BC = QR, ∠A = 90°, ∠C = ∠R = 40° and ∠Q = 50°.

(a) Given,

In △ ABC and △ DEF,

⇒ AB = DE

⇒ BC = EF

⇒ ∠B = ∠E

∴ △ ABC ≅ △ DEF (By S.A.S. axiom)

Hence, △ ABC and △ DEF are congruent by S.A.S. axiom.

(b) Given,

In △ ABC and △ DEF,

⇒ AC = DF

⇒ BC = EF

⇒ ∠B = ∠E (Both equal to 90°)

∴ △ ABC ≅ △ DEF (By R.H.S. axiom)

Hence, △ ABC and △ DEF are congruent by R.H.S. axiom.

(c) Given,

In △ ABC and △ QRP,

⇒ AB = QR

⇒ ∠B = ∠R

⇒ ∠C = ∠P

∴ △ ABC ≅ △ QRP (By A.A.S. or A.S.A. axiom)

Hence, △ ABC and △ DEF are congruent by A.A.S. or A.S.A. axiom.

(d) Given,

In △ ABC and △ PQR,

⇒ AB = PQ

⇒ AC = PR

⇒ BC = QR

∴ △ ABC ≅ △ PQR (By S.S.S. axiom)

Hence, △ ABC and △ PQR are congruent by S.S.S. axiom.

(e) In △ ABC,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 90° + ∠B + 40° = 180°

⇒ ∠B + 130° = 180°

⇒ ∠B = 180° - 130° = 50°.

In △ ABC and △ PQR,

⇒ BC = QR (Given)

⇒ ∠B = ∠Q (Both equal to 50°)

⇒ ∠C = ∠R (Both equal to 40°)

∴ △ ABC ≅ △ PQR (By A.S.A. axiom)

Hence, △ ABC and △ PQR are congruent by A.S.A. axiom.