Question

"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"

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.

 

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

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.

=========================================================

Answer 2

hi friend
here is your answer

(i) to show = ΔAPB congruent ΔAQB
answer-
In ΔAPB and ΔAQB
angleAPB = angleAQB = 90° (given)
anglePAB= angleQAB (given)
AB= AB (common)
therefore ΔAPB congruent ΔAQB (by AAS congruence rule)

(ii) BP=BQ
answer-
to show : BP=BQ
solution :
As ΔAPB congruent ΔAQB
BP=BQ (cpct)

hope that this will help you
if you like my way of answering please mark as brainliest
n let me know if I can help you in future

thank you
-Sneha