Question

"Question 3 AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.

Class 9 - Math - Triangles Page 119"

Answer 1

Congruence of triangles:

Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:

There are 4 criteria for congruence of triangles.

ASA(angle side angle):

Two Triangles are congruent if two angles and the included side of One triangle are equal to two angles & the included side of the  other triangle.

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Given,
AD and BC are equal perpendiculars to AB.

To prove,
CD bisects AB i.e, OA=OB

Proof,

In ΔAOD and ΔBOC,
∠A = ∠B (Perpendicular)
∠AOD = ∠BOC (Vertically opposite angles)
AD = BC (Given)

Hence, ΔAOD ≅ ΔBOC (by ASA congruence rule.)

Then,
AO = OB (CPCT).

Thus,CD bisects AB.

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Answer 2

yess, I can help you
In ∆AOD and ∆BOC
AD=BC ( given )
<OAD=<OBC ( all are 90° )
<AOD = <BOC ( opposite angle )
so,
by ASA congruence rule
OA = OB ( CPCT )
so, CD bisects AB