Given that ∠ABC = ∠DBC and ∠ACB = ∠DCB, show that ∠BAC = ∠BDC. Are the two triangles congruent? Class 7

Given that ∠ABC = ∠DBC and ∠ACB = ∠DCB, show that ∠BAC = ∠BDC. Are the two triangles congruent? Class 7

Question 1.

Given that ∠ABC = ∠DBC and ∠ACB = ∠DCB, show that ∠BAC = ∠BDC. Are the two triangles congruent? Class 7

Solution:

In ∆ABC and ∆DBC,

∠ABC = ∠DBC ∠ACB = ∠DCB

and BC = BC (Common)

So, ∆ABC ≅ ∆DBC (By ASA)

In ∆ABC,

∠ABC + ∠ACB + ∠BAC = 180° ...(1) (Angle sum properties)

In ∆DBC,

∠DBC + ∠DCB + ∠BDC = 180° ...(2)

So, we can equate eqn. (1) and (2),

∠ABC + ∠ACB + ∠BAC

= ∠DBC + ∠DCB + ∠BDC

We know that,

∠ABC = ∠DBC and ∠ACB = ∠DCB

∠ABC + ∠ACB + ∠BAC = ∠DBC + ∠DCB +∠BDC

∠BAC = ∠BDC

Hence, proved.