Question

7. In the given figure,
angleABE = angleCBE and angleADE = angleCDE.
Prove that AD = CD.​

Answer 1

In the given figure, EB is a line.

So, ∠ADE + ∠ADB = 180°, and

∠CDE + ∠CDB = 180°

Let's take ∠ADE = X

It is given that ∠ADE = ∠CDE = X

So, ∠X + ∠ADB = ∠X + ∠CDB

Thus, ∠ADB = ∠CDB

∠ABE = ∠ABD (∵ it is on the same line)

∠CBE = ∠CBD (∵ it is on the same line)

Now,

In ∆ADB and ∆CDB,

∠ABD = ∠CBD (Proved)

DB = DB (Common Line)

∠ADB = ∠CBD (Proved)

∴ ∆ADB ≅ ∆CDB (by ASA Congruence Rule)

AD = CD (By CPCT)

Hence Proved