# Nakafa Learning Content

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URL: https://nakafa.com/en/subjects/mathematics/circle/central-angle-and-inscribed-angle
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle/central-angle-and-inscribed-angle/en.mdx

Learn central and inscribed angles in circles. Learn the key relationship, theorems, and solve problems with worked examples and proofs.

---

## Definition of Central Angle

A central angle is an angle formed by two radii of a circle with the vertex located at the center of the circle. The sides of the central angle are radii that connect the center to points on the circle.

Component: LineEquation
Props:
- title: Central Angle
- description: An angle whose vertex is located at the center of the circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
},
{
points: Array.from({ length: 46 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 1.2 * Math.cos(angle),
y: 1.2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("CYAN"),
showPoints: false,
labels: [{ text: "α", at: 22, offset: [0.3, 0.2, 0] }],
},
]
- cameraPosition: [0, 0, 12]
- showZAxis: false

In the figure above:

- Point $$O$$ is the center of the circle
- $$OA$$ and $$OB$$ are radii of the circle
- $$\angle AOB$$ is the central angle
- The measure of the central angle is denoted by $$\alpha$$

Visible text: - Point is the center of the circle
- and are radii of the circle
- is the central angle
- The measure of the central angle is denoted by

## Definition of Inscribed Angle

An inscribed angle is an angle formed by two chords with the vertex located on the circle. The sides of the inscribed angle are chords that connect the vertex to two other points on the circle.

Component: LineEquation
Props:
- title: Inscribed Angle
- description: An angle whose vertex is located on the circle.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 4, y: 0, z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [0.5, 0, 0] },
{ text: "C", at: 1, offset: [-0.5, -0.3, 0] },
],
},
{
points: [
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 0, offset: [0.3, 0.3, 0] }],
},
{
points: [{ x: 0, y: 0, z: 0 }],
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [0, -0.5, 0] }],
},
]
- cameraPosition: [0, 0, 12]
- showZAxis: false

In the figure above:

- Point $$C$$ is located on the circle
- $$CA$$ and $$CB$$ are chords
- $$\angle ACB$$ is the inscribed angle
- Point $$O$$ is the center of the circle

Visible text: - Point is located on the circle
- and are chords
- is the inscribed angle
- Point is the center of the circle

## Relationship Between Central Angle and Inscribed Angle

Central angles and inscribed angles that subtend the same arc have a special relationship. Let's observe this relationship.

Component: LineEquation
Props:
- title: Relationship Between Central Angle and Inscribed Angle
- description: Central angle and inscribed angle that subtend the same arc.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
},
{
points: [
{ x: 4, y: 0, z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
},
{
points: [
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
},
{
points: Array.from({ length: 46 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle ... [truncated; 1587 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false

### Theorem of Central Angle and Inscribed Angle Relationship

```math
\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}
```

If a central angle and an inscribed angle subtend the same arc, then:

- $$\text{measure of inscribed angle} = \frac{1}{2} \times \text{measure of central angle}$$
- $$\text{measure of central angle} = 2 \times \text{measure of inscribed angle}$$

Visible text: - 
-

## Proof of Central Angle and Inscribed Angle Relationship

Let's prove the relationship between central angle and inscribed angle by constructing auxiliary lines.

Component: LineEquation
Props:
- title: Proof with Auxiliary Lines
- description: Constructing auxiliary line from $$C$$ through{" "}
$$O$$ to prove the relationship.
  Visible text: Constructing auxiliary line from through{" "}
 to prove the relationship.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0.3, 0.3, 0] }],
},
{
points: [
{ x: 4, y: 0, z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
},
{
points: [
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
},
{
points: [
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math. ... [truncated; 1331 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false

**Proof steps:**

1. Construct line $$CD$$ that passes through point $$O$$ (center of the circle)
2. Note that $$OA = OB = OC = OD$$ (radii of the circle)
3. Triangles $$AOC$$ and $$BOC$$ are isosceles triangles
4. Let $$\angle ACO = x$$ and $$\angle BCO = y$$
5. Since they are isosceles triangles: $$\angle CAO = x$$ and $$\angle CBO = y$$
6. Exterior angles of triangles: $$\angle AOD = 2x$$ and $$\angle BOD = 2y$$
7. Therefore: $$\angle AOB = 2x + 2y = 2(x + y) = 2 \times \angle ACB$$

Visible text: 1. Construct line that passes through point (center of the circle)
2. Note that (radii of the circle)
3. Triangles and are isosceles triangles
4. Let and 
5. Since they are isosceles triangles: and 
6. Exterior angles of triangles: and 
7. Therefore:

## Properties of Central Angle and Inscribed Angle

1. **Inscribed Angle Subtending a Diameter**

   Every inscribed angle that subtends a diameter of a circle measures $$90^\circ$$ (right angle).

   <LineEquation
     title="Inscribed Angle Subtending a Diameter"
     description={<>Inscribed angle subtending a diameter is always $$90^\circ$$.</>}
     data={[
       {
         points: Array.from({ length: 361 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("PURPLE"),
         showPoints: false,
       },
       {
         points: [
           { x: -4, y: 0, z: 0 },
           { x: 4, y: 0, z: 0 },
         ],
         color: getColor("ORANGE"),
         showPoints: true,
         labels: [
           { text: "A", at: 0, offset: [-0.5, 0, 0] },
           { text: "B", at: 1, offset: [0.5, 0, 0] },
         ],
       },
       {
         points: [
           { x: -4, y: 0, z: 0 },
           { x: 0, y: 4, z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: true,
         labels: [{ text: "C", at: 1, offset: [0, 0.5, 0] }],
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 0, y: 4, z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: false,
       },
       {
         points: [{ x: 0, y: 0, z: 0 }],
         color: getColor("AMBER"),
         showPoints: true,
         labels: [{ text: "O", at: 0, offset: [0, -0.5, 0] }],
       },
       {
         points: (() => {
           const C = { x: 0, y: 4 };
           const A = { x: -4, y: 0 };
           const B = { x: 4, y: 0 };

           // Calculate angle ACB at point C (should be 90°)
           const angleCA = Math.atan2(A.y - C.y, A.x - C.x);
           const angleCB = Math.atan2(B.y - C.y, B.x - C.x);

           return Array.from({ length: 16 }, (_, i) => {
             const t = i / 15;
             const angle = angleCA + t * (angleCB - angleCA);
             return {
               x: C.x + 0.8 * Math.cos(angle),
               y: C.y + 0.8 * Math.sin(angle),
               z: 0,
             };
           });
         })(),
         color: getColor("PINK"),
         showPoints: false,
         labels: [{ text: "90°", at: 7, offset: [0.5, -0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

2. **Inscribed Angles Subtending the Same Arc**

   $$\angle ACB = \angle ADB$$, both angles subtend the same
   arc $$AB$$.

   <LineEquation
     title="Inscribed Angles Subtending the Same Arc"
     description="Inscribed angles that subtend the same arc have equal measures."
     data={[
       {
         points: Array.from({ length: 361 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("PURPLE"),
         showPoints: false,
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
         ],
         color: getColor("ORANGE"),
         showPoints: true,
         labels: [
           { text: "A", at: 0, offset: [0.5, 0, 0] },
           { text: "B", at: 1, offset: [0.3, 0.3, 0] },
         ],
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: true,
         labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
       },
       {
         points: [
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
           { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("CYAN"),
         showPoints: false,
       },
       {
         points: [
           { x: 4, y: 0, z: 0 },
           { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("TEAL"),
         showPoints: true,
         labels: [{ text: "D", at: 1, offset: [-0.5, 0.3, 0] }],
       },
       {
         points: [
           { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
           { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
         ],
         color: getColor("TEAL"),
         showPoints: false,
       },
       {
         points: Array.from({ length: 46 }, (_, i) => {
           const angle = (i * Math.PI) / 180;
           return {
             x: 4 * Math.cos(angle),
             y: 4 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("AMBER"),
         showPoints: false,
         labels: [{ text: "Arc AB", at: 22, offset: [1.5, 0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

Visible text: 1. **Inscribed Angle Subtending a Diameter**

 Every inscribed angle that subtends a diameter of a circle measures (right angle).

 <LineEquation
 title="Inscribed Angle Subtending a Diameter"
 description={<>Inscribed angle subtending a diameter is always .</>}
 data={[
 {
 points: Array.from({ length: 361 }, (_, i) => {
 const angle = (i * Math.PI) / 180;
 return {
 x: 4 * Math.cos(angle),
 y: 4 * Math.sin(angle),
 z: 0,
 };
 }),
 color: getColor("PURPLE"),
 showPoints: false,
 },
 {
 points: [
 { x: -4, y: 0, z: 0 },
 { x: 4, y: 0, z: 0 },
 ],
 color: getColor("ORANGE"),
 showPoints: true,
 labels: [
 { text: "A", at: 0, offset: [-0.5, 0, 0] },
 { text: "B", at: 1, offset: [0.5, 0, 0] },
 ],
 },
 {
 points: [
 { x: -4, y: 0, z: 0 },
 { x: 0, y: 4, z: 0 },
 ],
 color: getColor("CYAN"),
 showPoints: true,
 labels: [{ text: "C", at: 1, offset: [0, 0.5, 0] }],
 },
 {
 points: [
 { x: 4, y: 0, z: 0 },
 { x: 0, y: 4, z: 0 },
 ],
 color: getColor("CYAN"),
 showPoints: false,
 },
 {
 points: [{ x: 0, y: 0, z: 0 }],
 color: getColor("AMBER"),
 showPoints: true,
 labels: [{ text: "O", at: 0, offset: [0, -0.5, 0] }],
 },
 {
 points: (() => {
 const C = { x: 0, y: 4 };
 const A = { x: -4, y: 0 };
 const B = { x: 4, y: 0 };

 // Calculate angle ACB at point C (should be 90°)
 const angleCA = Math.atan2(A.y - C.y, A.x - C.x);
 const angleCB = Math.atan2(B.y - C.y, B.x - C.x);

 return Array.from({ length: 16 }, (_, i) => {
 const t = i / 15;
 const angle = angleCA + t * (angleCB - angleCA);
 return {
 x: C.x + 0.8 * Math.cos(angle),
 y: C.y + 0.8 * Math.sin(angle),
 z: 0,
 };
 });
 })(),
 color: getColor("PINK"),
 showPoints: false,
 labels: [{ text: "90°", at: 7, offset: [0.5, -0.5, 0] }],
 },
 ]}
 cameraPosition={[0, 0, 12]}
 showZAxis={false}
 />

2. **Inscribed Angles Subtending the Same Arc**

 , both angles subtend the same
 arc .

 <LineEquation
 title="Inscribed Angles Subtending the Same Arc"
 description="Inscribed angles that subtend the same arc have equal measures."
 data={[
 {
 points: Array.from({ length: 361 }, (_, i) => {
 const angle = (i * Math.PI) / 180;
 return {
 x: 4 * Math.cos(angle),
 y: 4 * Math.sin(angle),
 z: 0,
 };
 }),
 color: getColor("PURPLE"),
 showPoints: false,
 },
 {
 points: [
 { x: 4, y: 0, z: 0 },
 { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
 ],
 color: getColor("ORANGE"),
 showPoints: true,
 labels: [
 { text: "A", at: 0, offset: [0.5, 0, 0] },
 { text: "B", at: 1, offset: [0.3, 0.3, 0] },
 ],
 },
 {
 points: [
 { x: 4, y: 0, z: 0 },
 { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
 ],
 color: getColor("CYAN"),
 showPoints: true,
 labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
 },
 {
 points: [
 { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
 { x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
 ],
 color: getColor("CYAN"),
 showPoints: false,
 },
 {
 points: [
 { x: 4, y: 0, z: 0 },
 { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
 ],
 color: getColor("TEAL"),
 showPoints: true,
 labels: [{ text: "D", at: 1, offset: [-0.5, 0.3, 0] }],
 },
 {
 points: [
 { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4), z: 0 },
 { x: 4 * Math.cos((3 * Math.PI) / 4), y: 4 * Math.sin((3 * Math.PI) / 4), z: 0 },
 ],
 color: getColor("TEAL"),
 showPoints: false,
 },
 {
 points: Array.from({ length: 46 }, (_, i) => {
 const angle = (i * Math.PI) / 180;
 return {
 x: 4 * Math.cos(angle),
 y: 4 * Math.sin(angle),
 z: 0,
 };
 }),
 color: getColor("AMBER"),
 showPoints: false,
 labels: [{ text: "Arc AB", at: 22, offset: [1.5, 0.5, 0] }],
 },
 ]}
 cameraPosition={[0, 0, 12]}
 showZAxis={false}
 />

## Calculating Inscribed Angle

Given central angle $$\angle AOB = 80^\circ$$. Find the measure of inscribed angle $$\angle ACB$$ that subtends the same arc!

Visible text: Given central angle . Find the measure of inscribed angle that subtends the same arc!

Component: LineEquation
Props:
- title: Visualization
- description: Central angle $$AOB = 80^\circ$$, find inscribed angle $$ACB$$.
  Visible text: Central angle , find inscribed angle .
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 * Math.cos(angle),
y: 4 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "A", at: 1, offset: [0.5, 0, 0] },
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4 * Math.cos((80 * Math.PI) / 180), y: 4 * Math.sin((80 * Math.PI) / 180), z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "B", at: 1, offset: [0, 0.5, 0] }],
},
{
points: [
{ x: 4, y: 0, z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "C", at: 1, offset: [-0.5, -0.3, 0] }],
},
{
points: [
{ x: 4 * Math.cos((80 * Math.PI) / 180), y: 4 * Math.sin((80 * Math.PI) / 180), z: 0 },
{ x: 4 * Math.cos((5 * Math.PI) / 4), y: 4 * Math.sin((5 * Math.PI) / 4), z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
},
{
points: Array.from({ length: 18 }, (_, i) => {
const angle = ((i * 80) / 17) * Math.PI / 180;
return { ... [truncated; 1372 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false

**Solution:**

Component: MathContainer
Children:

```math
\angle ACB = \frac{1}{2} \times \angle AOB
```

```math
\angle ACB = \frac{1}{2} \times 80^\circ
```

```math
\angle ACB = 40^\circ
```

## Calculating Central Angle

Given inscribed angle $$\angle ACB = 35^\circ$$. Find the measure of central angle $$\angle AOB$$ that subtends the same arc!

Visible text: Given inscribed angle . Find the measure of central angle that subtends the same arc!

**Solution:**

Component: MathContainer
Children:

```math
\angle AOB = 2 \times \angle ACB
```

```math
\angle AOB = 2 \times 35^\circ
```

```math
\angle AOB = 70^\circ
```

## Practice Problems

1. If the central angle of a circle is $$120^\circ$$, what is the measure of the inscribed angle that subtends the same arc?

2. Inscribed angle $$\angle ABC = 45^\circ$$. Find the measure of central angle $$\angle AOB$$!

3. In a circle, inscribed angle $$PQR$$ subtends a diameter. What is the measure of angle $$PQR$$?

4. Two inscribed angles subtend the same arc. If one angle measures $$25^\circ$$, find the measure of the other angle!

Visible text: 1. If the central angle of a circle is , what is the measure of the inscribed angle that subtends the same arc?

2. Inscribed angle . Find the measure of central angle !

3. In a circle, inscribed angle subtends a diameter. What is the measure of angle ?

4. Two inscribed angles subtend the same arc. If one angle measures , find the measure of the other angle!

### Answer Key

1. Inscribed angle is $$\frac{1}{2} \times 120^\circ = 60^\circ$$

2. Central angle $$\angle AOB = 2 \times 45^\circ = 90^\circ$$

3. $$\angle PQR = 90^\circ$$ (an inscribed angle subtending a diameter is always $$90^\circ$$)

4. The other angle is $$25^\circ$$ (inscribed angles subtending the same arc have equal measures)

Visible text: 1. Inscribed angle is 

2. Central angle 

3. (an inscribed angle subtending a diameter is always )

4. The other angle is (inscribed angles subtending the same arc have equal measures)