To prove :-
CQ = CA.
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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)
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