Question

If m( arc APC) = 60' and angle BAC = 80
find a) angle ABC b) arc (BQC)​

Answer 1

(1) angle ABC = 30°

(2) arc BQC = 160°

The inscribed angle theorem states that " The measure of an inscribed angle is half the measure of its intercepted arc.

So, we have, ∠ = 1/2 (m arc)

Given,

∠  BAC = 80°

m(arc APC) = 60°

(1) angle ABC

∠ = 1/2 (m arc)

∠ ABC = 1/2 (m arc APC)

= 1/2 × 60°

∴ ∠ ABC = 30°

(2) arc BQC

∠ = 1/2 (m arc)

∠ BAC = 1/2 (m arc BQC)

80° = 1/2 (m arc BQC)

2 × 80° = (m arc BQC)

∴ arc BQC = 160°

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Answer 2

Given:

m( arc APC) = 60

angle BAC = 80

Find angle ABC and arc (BQC)​ =?

(a) For angle ABC

∠ ABC is the inscribed angle of arc APC

∴ By Inscribed angle theorem

m∠ ABC = ½ m(arc APC)

$=\frac{}{2}\times60$

m∠ ABC = 30°

(b) For arc (BQC)​

Now, ∠ BAC is the inscribed angle of arc BQC,

∴ By Inscribed angle theorem,

m∠ BAC = ½ m(arc BQC)

80° = ½ m(arc BQC)

m(arc BQC) = 2 × 80°

m(arc BQC) = 160°

Hence  m∠ ABC = 30° and m(arc BQC) = 160°