Inscribed Angle

An inscribed angle is an angle
whose vertex is on the circle
and whose sides are the chords of the circle.

θ = [1/2] m[arc AB]

θ: Inscribed angle
arc AB: Intercepted arc

See this blue arc.
The central angle of the arc is 80º.

So the measure of the arc is 80º.

The blue arc is 80º.
The inscribed angle is xº.

So x = [1/2]⋅80.

[1/2]⋅80 = 40

So x = 40.

∠APC is an inscribed angle
whose intercepted arc is arc ABC.

∠APC is 100º.

So [1/2] m[arc ABC] = 100.

Multiply 2 to both sides.

Then m[arc ABC] = 200.

So 200 is the answer.

(7x + 1)º is the inscribed angle
whose intercepted arc is the blue arc.

(3x + 29)º is the inscribed angle
whose intercepted arc is the same blue arc.

So (7x + 1)º and (3x + 29)º
are the inscribe angles
whose intercepted arc is the same blue arc.

So [7x + 1] = [3x + 29].

Move 3x to the left side.
And move +1 to the right side.

Then, 7x - 3x, 4x is equal to, 29 - 1, 28.

Divide both sides by 4.

Then x = 7.

So x = 7 is the answer.