In Figure, D and E are points on side BC of a ΔABC such that BD = CE and AD = AE. Show that ΔABD ≅ ΔACE.

Question

In Figure, D and E are points on side BC of a ΔABC such that BD = CE and AD = AE. Show that ΔABD ≅ Δ ACE.

Philoid · Accepted Answer

Given: In ΔABC, BD = CE and AD = AE.

In ΔADE,

AD = AE

⇒ ∠ADE = ∠AED …(1) (opposite angles to equal sides are equal)

Now, ∠ADE + ∠ADB = 180° (linear pair)

∠ADB = 180° - ∠ADE ..(2)

Also, ∠AED + ∠AEC = 180° (linear pair)

∠AEC = 180° - ∠AED

∠AEC = 180° - ∠ADE .. (3) (∵∠ADE = ∠AED)

From (2) and (3)

∠ADB = ∠AEC ..(4)

Now, In ΔADB and ΔAEC,

AD = AE (given)

BD = EC (given)

∠ADB = ∠AEC (from (4)

Hence, ΔABD ≅ΔACE (by SAS)