Line l is the bisector of an angle

Asked by admin @ 29/08/2021 in Math viewed by 248 People

"Question 5 Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see the given figure). Show that: (i) ΔAPB ≅ ΔAQB (ii) BP = BQ or B is equidistant from the arms of ∠A.

Class 9 - Math - Triangles Page 119"

Answered by admin @ 29/08/2021

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.

-----------------------------------------------------------------------------------------------------

Given,

l is the bisector of an angle ∠A i.e, ∠BAP = ∠BAQ

BP and BQ are perpendiculars.

To prove:
i) ΔAPB ≅ ΔAQB

ii)
BP = BQ or B is equidistant from the arms of ∠A.

Proof:

(i)
In ΔAPB and ΔAQB,
∠P = ∠Q. (90°)
∠BAP = ∠BAQ (l is bisector)

AB = AB (Common)

Hence,
ΔAPB ≅ ΔAQB (by
AAS congruence rule).

(ii) since ΔAPB ≅
ΔAQB,

Then,

BP
BQ. (by CPCT.)

Hence,
B is equidistant from the arms of ∠A.

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Line l is the bisector of an angle

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