Question

for #1 – 9, find the value of each variable.

1. circle with center o, points a, b, c; arc ac is 82°, angle at b is x°
2. circle with points a, b, c; angle at a is 55°, arc bc is 200°
3. circle with a central angle 98°, inscribed angle a°, arc b°
4. circle with center o, arc ac is 92°, angle at a is x°, arc bc is y°
5. circle with center o, angle at b is 62°, arc bc is 100°, arc ac is w°, angle at o is x°, arc ab is y°
6. circle with arc 84°, angle g°, arc h°
7. circle with inscribed angle 73°, arc (5x + 2)°
8. circle with arc 200°, angle 4x°
9. circle with arc (30x - 120)°, inscribed angle (10x)°

Explanation:

Step1: Analyze Problem 1

We have a circle with arc \( AC = 82^\circ \). The angle \( x \) is an inscribed angle subtended by arc \( AC \). The formula for an inscribed angle is \( \text{Inscribed Angle}=\frac{1}{2}\times\text{Subtended Arc} \). So, \( x=\frac{82^\circ}{2} \).
\( x = 41^\circ \)

Step2: Analyze Problem 2

We have a circle with an inscribed angle of \( 55^\circ \). The arc \( b \) is subtended by a central angle equal to twice the inscribed angle (since central angle is twice inscribed angle for the same arc). So, \( b = 2\times55^\circ \).
\( b = 110^\circ \)

Step3: Analyze Problem 3

We have a central angle of \( 98^\circ \). The angle \( a \) is an inscribed angle subtended by the same arc as the central angle \( 98^\circ \). So, \( a=\frac{98^\circ}{2} \).
\( a = 49^\circ \)

Step4: Analyze Problem 4

Arc \( AC = 92^\circ \), and \( AB \) is a diameter (since \( O \) is the center). The arc \( BC \) (which is \( y \)) and arc \( AC \) should add up to \( 180^\circ \) (since \( AB \) is diameter, semicircle is \( 180^\circ \)). Wait, actually, angle \( x \): \( AC \) is \( 92^\circ \), the inscribed angle \( x \) subtended by arc \( BC \). First, arc \( BC = 180^\circ - 92^\circ=88^\circ \)? Wait, no, \( AB \) is diameter, so arc \( AB = 180^\circ \). Arc \( AC = 92^\circ \), so arc \( BC = 180 - 92 = 88^\circ \). Then angle \( x \) is inscribed angle subtended by arc \( BC \), so \( x=\frac{88^\circ}{2}=44^\circ \)? Wait, maybe better: The angle \( x \) and arc \( AC \): Wait, \( AC \) is \( 92^\circ \), the inscribed angle over arc \( BC \). Wait, maybe I made a mistake. Alternatively, since \( AB \) is diameter, \( \angle ACB = 90^\circ \) (Thales' theorem), but maybe not. Wait, the problem is to find \( x \) and \( y \). Arc \( AC = 92^\circ \), so the inscribed angle \( x \) subtended by arc \( BC \). Arc \( BC = y \), and arc \( AC + arc BC = 180^\circ \) (since \( AB \) is diameter). So \( y = 180 - 92 = 88^\circ \), and \( x=\frac{y}{2}=\frac{88^\circ}{2}=44^\circ \)? Wait, no, maybe \( x \) is subtended by arc \( AC \)? Wait, the diagram: \( A \) to \( C \) is \( 92^\circ \), \( A \) to \( B \) is diameter. So angle at \( A \), \( x \), subtended by arc \( BC \). So arc \( BC = 180 - 92 = 88 \), so \( x = 44^\circ \), \( y = 88^\circ \)?

Step5: Analyze Problem 5

Angle \( \angle ABC = 62^\circ \) (inscribed angle), and \( \angle AOC = x \) (central angle) subtended by arc \( AC \). The central angle is twice the inscribed angle for the same arc. So \( x = 2\times62^\circ = 124^\circ \). Then, the sum of arcs: arc \( AB = y \), arc \( BC = 100^\circ \), arc \( AC = w \), and arc \( AB + arc BC + arc AC = 360^\circ \)? Wait, no, triangle \( ABC \): \( \angle ABC = 62^\circ \), central angle \( x = 124^\circ \). Then arc \( AB \) and arc \( BC \): Wait, maybe arc \( AB = y \), arc \( BC = 100^\circ \), arc \( AC = w \). The inscribed angle \( \angle ABC = 62^\circ \) subtended by arc \( AC \), so arc \( AC = 2\times62^\circ = 124^\circ \) (wait, no, central angle is \( x = 124^\circ \), which is arc \( AC \)). Then, arc \( AB + arc BC + arc AC = 360^\circ \)? No, a circle is \( 360^\circ \), but if \( O \) is center, then arcs: arc \( AB = y \), arc \( BC = 100^\circ \), arc \( AC = 124^\circ \), so \( y + 100 + 124 = 360 \)? \( y = 360 - 224 = 136^\circ \)? Wait, maybe.

Step6: Analyze Problem 6

We have a circle with arc \( 84^\circ \). The angle \( g \) is an inscribed angle subtended by arc \( 84^\circ \)? Wait, the diameter: the arc opposite to \( h \) and \( 84^\circ \). Wait, the diameter divides the circle into two semicircles (\( 180^\circ \)). So \( h + 84^\circ = 180^\circ \)? No, \( h \) is an arc, and the angle \( g \): Wait, the angle \( g \) is inscribed angle subtended by arc \( 84^\circ \), so \( g=\frac{84^\circ}{2}=42^\circ \). And \( h = 180^\circ - 84^\circ = 96^\circ \)? Wait, maybe.

Step7: Analyze Problem 7

We have an inscribed angle of \( 73^\circ \) and an arc of \( (5x + 2)^\circ \). The inscribed angle is half the subtended arc. So \( 73^\circ=\frac{1}{2}\times(5x + 2)^\circ \). Multiply both sides by 2: \( 146 = 5x + 2 \). Subtract 2: \( 144 = 5x \). Wait, no, \( 146 - 2 = 5x \)? \( 144 = 5x \)? Wait, \( 5x + 2 = 2\times73 \). So \( 5x + 2 = 146 \). Then \( 5x = 144 \), \( x=\frac{144}{5}=28.8 \)? Wait, the handwritten says \( x = 29.6 \)? Maybe miscalculation. Wait, \( 2\times73 = 146 \), so \( 5x + 2 = 146 \), \( 5x = 144 \), \( x = 28.8 \).

Step8: Analyze Problem 8

We have an arc of \( 200^\circ \), and the angle \( 4x^\circ \) is an inscribed angle? Wait, no, the sum of arcs: a circle is \( 360^\circ \), so the remaining arc is \( 360 - 200 = 160^\circ \). The angle \( 4x^\circ \) is an inscribed angle subtended by the \( 200^\circ \) arc? Wait, no, inscribed angle subtended by arc \( 200^\circ \) would be \( \frac{200}{2}=100^\circ \), but here \( 4x \): Wait, maybe the angle \( 4x \) and the arc \( 200^\circ \): Wait, the handwritten says \( 4x = 200 \)? No, \( 4x = \frac{200}{2} \)? Wait, no, the handwritten has \( \frac{4x}{4}=\frac{200}{4} \)? Wait, no, the handwritten says \( x = 50 \), so \( 4x = 200 \)? Wait, maybe the angle \( 4x \) is a central angle? No, if it's inscribed, \( 4x=\frac{200}{2}=100 \), \( x = 25 \), but handwritten says \( x = 50 \). Maybe the arc is \( 200^\circ \), and the angle \( 4x \) is an inscribed angle subtended by the arc \( 200^\circ \), so \( 4x=\frac{200}{2}=100 \), \( x = 25 \). But handwritten says \( x = 50 \), maybe mistake.

Step9: Analyze Problem 9

We have a central angle \( (30x - 120)^\circ \) and an inscribed angle \( (10x)^\circ \) subtended by the same arc. So central angle is twice inscribed angle: \( 30x - 120 = 2\times10x \). So \( 30x - 120 = 20x \). Subtract \( 20x \): \( 10x - 120 = 0 \), \( 10x = 120 \), \( x = 12 \). Which matches the handwritten.

Answer:

1. \( x = 41^\circ \)
2. \( b = 110^\circ \)
3. \( a = 49^\circ \)
4. \( x = 44^\circ \), \( y = 88^\circ \) (assuming)
5. \( x = 124^\circ \), \( y = 136^\circ \) (assuming)
6. \( g = 42^\circ \), \( h = 96^\circ \) (assuming)
7. \( x = 28.8 \) (or \( 29.6 \) if miscalculation)
8. \( x = 25 \) (or \( 50 \) if handwritten is correct)
9. \( x = 12 \)

(Note: Some answers may vary based on diagram interpretation and handwritten notes.)