Question

In the given figure, AE bisects exterior angle CAD and AE is parallel to BC, then AB = AC.

Answer 1

Hint:
First use the given data to get some important relations and then apply the exterior angle theorem in the triangle ABC, which gives:
$\angle DAC = \angle ACB + \angle ABC$
Substitute the values obtained by the given data to get the desired result.

Complete step by step answer:
It is given in the problem that AE bisects exterior angle CAD and AE is parallel to BC, then AB = AC.
We have to analyze that the given statement is true or false.
We know that AE is parallel to BC, then using the property of alternate angles we have the conclusion that:
\[\angle CAE = \angle ACB\] … (1)
We have also given in the problem that, AE bisects exterior angle CAD, then we have the conclusion that:
$\angle DAE = \angle EAC$ … (2)
Now we can apply the exterior angle theorem in triangle ABC. Then according to the Exterior angle theorem,
$\angle DAC = \angle ACB + \angle ABC$
$ \Rightarrow \angle ABC = \angle DAC - \angle ACB$
From equation (1),
$\angle ACB = \angle EAC$
Then after the substitution, we have
$\angle ABC = \angle DAC - \angle EAC$
We can see from the figure:
$\angle DAC - \angle EAC = \angle DAE$
Then the above equation gives:
$\angle ABC = \angle DAE$
From equation (1) and (2), we have
$\angle ABC = \angle DAE = \angle EAC = \angle ACB$
$ \Rightarrow \angle ABC = \angle ACB$
We know that if the two angles of the triangle are the same then the corresponding sides of the triangle are also the same. Thus, we have
$AB = AC$
Thus, the given statement is true.

Note: A bisector always divides the angles into equal parts. Take the correct interior angles while applying the exterior angle theorem.The exterior angle theorem says that:
An exterior angle of a triangle is equal to the sum of the two opposite interior angles of the same triangle.