Question

Triangle ABC and triangle ABD are such that AD=BC, angle 1= angle 2 and angle 3=angle 4. prove that BD=AC.

Answer 1

angle 1=angle 2.............................................1

angle 3 = angle 4...........................................2

on adding equation 1 and 2 angle DAB=CBA

in triangle ABC and triangle ABD

AD=BC [GIVEN]

AB=AB [COMMON BASE]

DAB=CBA [proved above]

hence triangle ABC congruent triangle ABD [ by SAS criteria]

therefore BD= AC [ by CPCT]

Answer 2

GIVEN,

AD = BC

∠1 = ∠2

∠3 = ∠4

TO PROVE : BD = AC

PROOF :

HERE,

IN Δ A0D AND Δ BOC

AD = BC  (GIVEN)

∠1 = ∠2    (GIVEN)

∠AOD = ∠BOC (VERTICALLY OPP. ANGLES)

∴ Δ A0D ≅ Δ BOC (BY AAS)

SO, OD = OC (CPCT)--------------------------------------1

NOW,

IN Δ AOB

∠3 =∠4

OB = OA (ISOSCELES  Δ PROP.)-----------------------------2

THEN,

OD + OB = OC + OA (ADDING 1 AND 2)

∴ BD = AC

HENCE PROVED