Question

To prove :-
CQ = CA.​

Answer 1

Answer:

Given: ΔABC is an equilateral triangle.

BP is the bisector of ∠B.

To Prove : CQ = CA

Proof : ΔABC is an equilateral Δ (Given)

∠ABC=∠CAB=∠ACB=60

o

(Given)

∠ABP=∠CBP (BP is the bisector)

∠CBP=

2

1

×∠ABC

∠CBP=

2

1

×60

o

=30

o

∠CBP=∠CAP=30

o

..... (1) (∠s incircled in the same arc)

∠ACB=60

o

(Given)

∴∠ACQ=180

o

−60

o

=120

o

(Linear pair)

∴∠ACQ=120

o

..... (2)

In ΔACQ,

∴∠AQC=180

o

−(30

o

+120

o

) From (1) and (2)

∠AQC=180

o

−150

o

∴∠AQC=30

o

...... (3)

In ΔACQ, ∠CAQ=∠AQC=30

o

From (1) and (3)

∴CQ=CA (Converse of isosceles Δ theorem)