Question

Given: AB || DE , AD bisects BE.
Prove: ABC = DEC using the ASA postulate.

Answer 1

As per ASA postulate, the two triangles are congruent.

Explanation:

We are given two triangles:

`\triangle ABC` and
`\triangle DEC`.

AD bisects BE.

AB || DE.

Let us have a look at two properties.

1. When two lines are parallel and a line intersects both of them, then alternate angles are equal.

i.e. AB || ED and
`\angle B` and
`\angle E` are alternate angles
`\Rightarrow`
`\angle B = \angle E`.

2. When two lines are cutting each other, angles formed at the crossing of two, are known as Vertically opposite angles and they are are equal.

`\Rightarrow \angle ACB = \angle DCE`

Also, it is given that AD bisects BE.

i.e. EC = CB

1.
`\angle B = \angle E`

2. EC = CB

3.
`\angle ACB = \angle DCE`

So, we can in see that in
`\triangle ABC` and
`\triangle DEC`, two angles are equal and side between them is also equal to each other.

Hence, proved that
`\triangle ABC`
`\cong`
`\triangle DEC`.