Question

In ∆ABC, AE is the bisector of the exterior ∠CAD and AE || BC. Prove that AB = AC.​

Answer 1

Answer:

To prove:- AB = AC

Solution:

That's we've to prove ∆ABC is isosceles,as we've to prove two of their sides as equal.

Given AE || BC

then

∠CAE=∠ACB (alternate interior angles)

also

∠CEA=∠EAD (AE is the bisector)

then

∠EAD=∠ACB

Also

∠EAD=∠ABC ( corresponding angles)

If

∠CAE=∠ACB

∠CEA=∠EAD

∠EAD=∠ACB

∠EAD=∠ABC

Then

∠ACB=∠ABC

therefore

AB=AC (if two opposite angles of a triangle are equal then their opposite sides are also equal)

Hence,proved

Answer 2

QuEsTiOn,

• In ∆ABC, AE is the bisector of the exterior ∠CAD and AE || BC. Prove that AB = AC.

To FiNd,

• $\color{red} \boxed{ \sf \: To \: prove: \: AB = AC}$

SoLuTiOn,

That's we've to prove AABC is isosceles, as we've to prove two of their sides as equal.

Given AE || BC

then

$\color{blue} \sf \: ∠CAE=∠ACB \\ \sf(alternate interior angles)$

also

$\color{blue} \sf ∠CEA=∠EAD \\ \sf (AE \: is \: the \: bisector)$

then

∠EAD=∠ACB

Also

$\color{blue} \sf ∠EAD-∠ABC \\ \sf( corresponding \: angles)$

If

∠CAE=∠ACB

∠CEA=∠EAD

∠EAD=∠ACB

∠EAD=∠ABC

$\color{red}\sf \: Then,$

∠ACB=∠ABC

$\color{blue}\boxed{\sf \: AB-AC (if\: two\: opposite \:angles \:of\: a \:triangle \:are \:equal \:then \:their \:opposite \:sides \:are\: also \:equal)}$

Hope you get your AnSwEr.