Question

. In quadrilateral ACBD, AC = AD and AB bisect A (see Fig. 7.16). Show that ΔABC ΔABD. What can you say about BC and BD​

Answer 1

Step-by-step explanation:

Given :-

In a quadrilateral ACBD, AC = AD and AB bisect A.

To find :-

Show that ΔABC=~ ΔABD.?

What can you say about BC and BD?

Solution:-

Given that

ABCD is a quadrilateral

AC = AD

AB bisects A

=> < BAC = < BAD

And

AB divides the quadrilateral into two triangles they are ∆BAC and ∆BAD

In ∆BAC and ∆BAD,

AC = AD ( Given)

< BAC = < BAD ( AB bisects A )

AB = AB ( Common Side)

Therefore, ∆BAC and ∆BAD are congruent by SAS Property.

∆BAC =~ ∆BAD

=> BC = BD ( CPCT)

Hence, Proved.

Answer :-

∆BAC =~ ∆BAD

BC = BD

Used formulae:-

• An angle bisector bisects the angle into two equal parts.
• In two triangles , The two sides and the included angle of a first triangle are equal to the corresponding two sides and the included angle of the second triangle. Then they are congruent and this property is called Side-Angle-Side (SAS) property.
• Corresponding parts are equal in both congruent triangles
• CPCT - Corresponding parts in the congruent Triangles

Answer 2

Step-by-step explanation:

It is given that AC and AD are equal i.e. AC = AD and the line segment AB bisects A.

We will have to now prove that the two triangles ABC and ABD are similar i.e. ΔABC ΔABD

Proof:

Consider the triangles ΔABC and ΔABD,

(i) AC = AD (It is given in the question)

(ii) AB = AB (Common)

(iii) CAB = DAB (Since AB is the bisector of angle A)

So, by SAS congruency criterion, ΔABC ΔABD.

For the 2nd part of the question, BC and BD are of equal lengths by the rule of C.P.C.T.

hope it helps you