Question

5.
Line l is the bisector of an angle Z A and B is any
point on l. BP and BQ are perpendiculars from B
to the arms of Z A (see Fig. 7.20). Show that:
(1) A APB=AAQB
(ii) BP = BQ or B is equidistant from the arms
of ZA​

Answer 1

Step-by-step explanation:

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

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.

Hope it helps.....