Question

In a quadrilateral ACBD, $AC = AD$ and AB bisect $\angle A$. Show that $\Delta ABC$ is congruent to $\Delta ABD$

Answer 1

Hint: For 2 triangles to be congruent any 3 measures including any one side of the 2 triangles must be equal. From the given information we can prove that 2 sides and one angle are equal. Hence, we can show the 2 triangles are congruent.

Complete step by step answer:

Consider $\Delta ABC$ and $\Delta ABD$,
It is given that $AC = AD$… (1)
It is given that line AB bisects $\angle A$,
$ \Rightarrow \angle CAB = \angle DAB$… (2)
As line AB is a common side of both the triangle, we can write,
$AB = AB$… (3)
From (1), (2), and (3), we can say that 2 sides and one angle of the 2 triangles are equal. So, by Side Angle Side (SAS) congruence criteria, $\Delta ABC$ and $\Delta ABD$ are congruent.

Note: Congruency of 2 triangles can be proved using either of the following criteria
SSS criteria – if all the corresponding sides are equal
SAS criteria – If the 2 sides and the angle between these sides are equal
ASA criteria – If the 2 angles and the side between them are equal.
AAS criteria – If the 2 angles and non-included side are equal.
Two right-angled triangles are congruent if they have one side and one acute angle are equal.
For any 2 congruent triangles, all its corresponding sides and angles will be equal and can be superimposed to each other.