Question

5. In figure 4, DC LAC, DB LAB, AC = AB and ADbisects A.
Show that DC-DB.
B​

Answer 1

Given: AB = AC, D is the point in the interior of ∆ABC such that ∠DBC = ∠DCB.

To Prove: AD bisects ∠BAC of ∆ABC.

Proof: In ∆DBC,

∵ ∠DBC = ∠DCB ...(1) | Given

∴ DB = DC    ...(2)

| Sides opposite to equal angles of a triangle are equal

In ∆ABC,

∵ AB = AC    | Given

∴ ∠ABC = ∠ACB

∴ ∠ABC = ∠ACB    ...(3)

| Angles opposite to equal sides of a triangle are equal

Subtracting (1) from (3), we get,

∠ABC - ∠DBC = ∠ACB - ∠DCB

⇒ ∠ABD = ∠ACD    ...(4)

In ∆ADB and ∆ADC,

AB = AC    | Given

DB = DC    | Proved in (2)

∠ABD = ∠ACD | Proved in (4)

∴ ∆ADB ≅ ∆ADC

| SAS congruence rule

∴ ∠DAB = ∠DAC    | CPCT

⇒ AD bisects ∠BAC of ∆ABC.