# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/circle/properties-of-angle-in-circle
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/circle/properties-of-angle-in-circle/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Properties of Angle in Circle",
  description: "Explore angle properties in circles including inscribed angles, Thales' theorem, and cyclic quadrilaterals. Master theorems with clear proofs.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Circle",
};

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

## Inscribed Angles Subtending the Same Arc

Inscribed angles that subtend the same arc have equal measures, regardless of where the vertex is positioned on the circle.

<LineEquation
  title="Inscribed Angles Subtending the Same Arc"
  description="All inscribed angles subtending 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}
/>

**Property:** <InlineMath math="\angle ACB = \angle ADB" />

Both inscribed angles ACB and ADB subtend the same arc AB, so they have equal measures.

## Central Angle and Inscribed Angle

The central angle is twice the inscribed angle that subtends the same arc.

<LineEquation
  title="Relationship Between Central Angle and Inscribed Angle"
  description="Central angle = 2 × inscribed angle for 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: 16 }, (_, i) => {
        const angle = (((i * 45) / 15) * Math.PI) / 180;
        return {
          x: 1.2 * Math.cos(angle),
          y: 1.2 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "2θ", at: 7, offset: [0.5, 0.3, 0] }],
    },
    {
      points: (() => {
        const C = {
          x: 4 * Math.cos((5 * Math.PI) / 4),
          y: 4 * Math.sin((5 * Math.PI) / 4),
        };
        const A = { x: 4, y: 0 };
        const B = {
          x: 4 * Math.cos(Math.PI / 4),
          y: 4 * Math.sin(Math.PI / 4),
        };

        // Calculate angle ACB at point C
        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("AMBER"),
      showPoints: false,
      labels: [{ text: "θ", at: 7, offset: [0.5, 0.5, 0] }],
    },

]}
cameraPosition={[0, 0, 12]}
showZAxis={false}
/>

**Property:** <InlineMath math="\angle AOB = 2 \times \angle ACB" />

## Inscribed Angle Subtending a Diameter

Every inscribed angle that subtends a diameter of the circle is always a right angle (90°).

<LineEquation
  title="Inscribed Angle Subtending a Diameter"
  description="Inscribed angle subtending a diameter is always 90°."
  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: [-0.3, -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("TEAL"),
      showPoints: true,
      labels: [{ text: "D", at: 1, offset: [-0.5, -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("TEAL"),
      showPoints: false,
    },

]}
cameraPosition={[0, 0, 12]}
showZAxis={false}
/>

**Property:** If AB is a diameter, then <InlineMath math="\angle ACB = \angle ADB = 90°" />

This is known as **Thales' Theorem**.

## Angles in a Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose four vertices lie on a circle. The sum of opposite angles is 180°.

<LineEquation
  title="Cyclic Quadrilateral"
  description="Sum of opposite angles = 180°."
  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: 0, y: 4, z: 0 },
        { x: -4, y: 0, z: 0 },
        { x: 0, y: -4, z: 0 },
        { x: 4, y: 0, z: 0 },
      ],
      color: getColor("ORANGE"),
      smooth: false,
      showPoints: true,
      labels: [
        { text: "A", at: 0, offset: [0.5, 0, 0] },
        { text: "B", at: 1, offset: [0, 0.5, 0] },
        { text: "C", at: 2, offset: [-0.5, 0, 0] },
        { text: "D", at: 3, offset: [0, -0.5, 0] },
      ],
    },
    {
      points: Array.from({ length: 16 }, (_, i) => {
        const angle = (((i * 90) / 15) * Math.PI) / 180;
        return {
          x: 0.8 * Math.cos(angle),
          y: 0.8 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("CYAN"),
      showPoints: false,
      labels: [{ text: "α", at: 7, offset: [0.3, 0.2, 0] }],
    },
    {
      points: Array.from({ length: 16 }, (_, i) => {
        const angle = ((180 + (i * 90) / 15) * Math.PI) / 180;
        return {
          x: 0.8 * Math.cos(angle),
          y: 0.8 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "γ", at: 7, offset: [-0.3, -0.2, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

**Property:** <InlineMath math="\alpha + \gamma = 180°" /> and <InlineMath math="\beta + \delta = 180°" />

## Exterior Angle Equals Opposite Interior Angle

In a cyclic quadrilateral, an exterior angle at a vertex equals the interior angle at the opposite vertex.

<LineEquation
  title="Exterior Angle of Cyclic Quadrilateral"
  description="Exterior angle = opposite interior 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: 4, y: 0, z: 0 },
        { x: 0, y: 4, z: 0 },
        { x: -4, y: 0, z: 0 },
        { x: 0, y: -4, z: 0 },
        { x: 4, y: 0, z: 0 },
      ],
      color: getColor("ORANGE"),
      smooth: false,
      showPoints: true,
      labels: [
        { text: "A", at: 0, offset: [0.5, 0, 0] },
        { text: "B", at: 1, offset: [0, 0.5, 0] },
        { text: "C", at: 2, offset: [-0.5, 0, 0] },
        { text: "D", at: 3, offset: [0, -0.5, 0] },
      ],
    },
    {
      points: [
        { x: 4, y: 0, z: 0 },
        { x: 6, y: 0, z: 0 },
      ],
      color: getColor("CYAN"),
      showPoints: true,
      labels: [{ text: "E", at: 1, offset: [0.5, 0, 0] }],
    },
    {
      points: Array.from({ length: 16 }, (_, i) => {
        const angle = ((270 + (i * 90) / 15) * Math.PI) / 180;
        return {
          x: 4 + 0.8 * Math.cos(angle),
          y: 0 + 0.8 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("PINK"),
      showPoints: false,
      labels: [{ text: "∠DAE", at: 7, offset: [0.8, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 16 }, (_, i) => {
        const angle = ((90 + (i * 90) / 15) * Math.PI) / 180;
        return {
          x: -4 + 0.8 * Math.cos(angle),
          y: 0 + 0.8 * Math.sin(angle),
          z: 0,
        };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      labels: [{ text: "∠BCD", at: 7, offset: [-0.8, 0.5, 0] }],
    },

]}
cameraPosition={[0, 0, 12]}
showZAxis={false}
/>

**Property:** <InlineMath math="\angle DAE = \angle BCD" /> (exterior angle at A = interior angle at C)

## Applications of Angle Properties in Circles

### Determining Angle Measures

In a circle with center O, the central angle AOB = 100°. Find the measure of inscribed angle ACB!

**Solution:**

Using the property of central and inscribed angles:

<div className="flex flex-col gap-4">
  <BlockMath math="\angle ACB = \frac{1}{2} \times \angle AOB" />
  <BlockMath math="\angle ACB = \frac{1}{2} \times 100°" />
  <BlockMath math="\angle ACB = 50°" />
</div>

### Finding Opposite Angles

In cyclic quadrilateral ABCD, given <InlineMath math="\angle A = 75°" />. Find the measure of <InlineMath math="\angle C" />!

**Solution:**

Using the property of cyclic quadrilaterals:

<div className="flex flex-col gap-4">
  <BlockMath math="\angle A + \angle C = 180°" />
  <BlockMath math="75° + \angle C = 180°" />
  <BlockMath math="\angle C = 180° - 75°" />
  <BlockMath math="\angle C = 105°" />
</div>

### Using Thales' Theorem

Point C lies on a circle with AB as the diameter. Find the measure of <InlineMath math="\angle ACB" />!

**Solution:**

Since AB is a diameter and C lies on the circle, by Thales' Theorem:

<BlockMath math="\angle ACB = 90°" />

## Practice Problems

1. In a circle with center O, central angle AOB = 140°. If C and D are two different points on the circle, find:

   - The measure of angle ACB
   - The measure of angle ADB

2. In cyclic quadrilateral PQRS, given:

   <div className="flex flex-col gap-4">
     <BlockMath math="\angle P = 85°" />
     <BlockMath math="\angle Q = 110°" />
   </div>

   Find the measures of <InlineMath math="\angle R" /> and <InlineMath math="\angle S" />!

3. Points A, B, and C lie on a circle. If AB is a diameter and BC = AC, find the measure of <InlineMath math="\angle BAC" />!

4. In a circle, inscribed angle APB = 35°. Find the measure of central angle AOB!

5. In cyclic quadrilateral KLMN, the exterior angle at vertex K is 65°. Find the measure of the interior angle at vertex M!

### Answer Key

1. **Finding inscribed angles subtending the same arc**

   <LineEquation
     title="Visualization"
     description="Central angle AOB = 140° and inscribed angles subtending arc AB."
     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((140 * Math.PI) / 180),
             y: 4 * Math.sin((140 * Math.PI) / 180),
             z: 0,
           },
         ],
         color: getColor("ORANGE"),
         showPoints: true,
         labels: [{ text: "B", at: 1, offset: [-0.3, 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((140 * Math.PI) / 180),
             y: 4 * Math.sin((140 * 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: [
           { x: 4, y: 0, z: 0 },
           {
             x: 4 * Math.cos((7 * Math.PI) / 4),
             y: 4 * Math.sin((7 * Math.PI) / 4),
             z: 0,
           },
         ],
         color: getColor("TEAL"),
         showPoints: true,
         labels: [{ text: "D", at: 1, offset: [0.3, -0.5, 0] }],
       },
       {
         points: [
           {
             x: 4 * Math.cos((140 * Math.PI) / 180),
             y: 4 * Math.sin((140 * Math.PI) / 180),
             z: 0,
           },
           {
             x: 4 * Math.cos((7 * Math.PI) / 4),
             y: 4 * Math.sin((7 * Math.PI) / 4),
             z: 0,
           },
         ],
         color: getColor("TEAL"),
         showPoints: false,
       },
       {
         points: Array.from({ length: 16 }, (_, i) => {
           const angle = (((i * 140) / 15) * Math.PI) / 180;
           return {
             x: 1.2 * Math.cos(angle),
             y: 1.2 * Math.sin(angle),
             z: 0,
           };
         }),
         color: getColor("PINK"),
         showPoints: false,
         labels: [{ text: "140°", at: 7, offset: [0.3, 0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

   **Solution:**

   Using the relationship between central and inscribed angles:

   <div className="flex flex-col gap-4">
     <BlockMath math="\text{Inscribed angle} = \frac{1}{2} \times \text{Central angle}" />
     <BlockMath math="\angle ACB = \frac{1}{2} \times \angle AOB" />
     <BlockMath math="\angle ACB = \frac{1}{2} \times 140°" />
     <BlockMath math="\angle ACB = 70°" />
   </div>

   Since inscribed angles subtending the same arc have equal measures:

   <BlockMath math="\angle ADB = \angle ACB = 70°" />

2. **Finding angles in a cyclic quadrilateral**

   **Solution:**

   In a cyclic quadrilateral, the sum of opposite angles = 180°

   For angle R (opposite to P):

   <div className="flex flex-col gap-4">
     <BlockMath math="\angle P + \angle R = 180°" />
     <BlockMath math="85° + \angle R = 180°" />
     <BlockMath math="\angle R = 180° - 85°" />
     <BlockMath math="\angle R = 95°" />
   </div>

   For angle S (opposite to Q):

   <div className="flex flex-col gap-4">
     <BlockMath math="\angle Q + \angle S = 180°" />
     <BlockMath math="110° + \angle S = 180°" />
     <BlockMath math="\angle S = 180° - 110°" />
     <BlockMath math="\angle S = 70°" />
   </div>

3. **Finding angle in an isosceles triangle with diameter**

   <LineEquation
     title="Visualization"
     description="AB is a diameter, BC = AC (isosceles triangle)."
     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: [-0.3, -0.5, 0] }],
       },
     ]}
     cameraPosition={[0, 0, 12]}
     showZAxis={false}
   />

   **Solution:**

   Since AB is a diameter, by Thales' Theorem:

   <BlockMath math="\angle ACB = 90°" />

   Since BC = AC, triangle ABC is a right isosceles triangle.

   In a right isosceles triangle, both base angles are equal:

   <div className="flex flex-col gap-4">
     <BlockMath math="\angle BAC + \angle ABC + \angle ACB = 180°" />
     <BlockMath math="\angle BAC + \angle ABC + 90° = 180°" />
     <BlockMath math="\angle BAC + \angle ABC = 90°" />
   </div>

   Since <InlineMath math="\angle BAC = \angle ABC" /> (base angles of isosceles triangle):

   <div className="flex flex-col gap-4">
     <BlockMath math="2 \times \angle BAC = 90°" />
     <BlockMath math="\angle BAC = 45°" />
   </div>

4. **Finding central angle from inscribed angle**

   **Solution:**

   Using the relationship between central and inscribed angles:

   <div className="flex flex-col gap-4">
     <BlockMath math="\text{Central angle} = 2 \times \text{Inscribed angle}" />
     <BlockMath math="\angle AOB = 2 \times \angle APB" />
     <BlockMath math="\angle AOB = 2 \times 35°" />
     <BlockMath math="\angle AOB = 70°" />
   </div>

5. **Finding interior angle from exterior angle in cyclic quadrilateral**

   **Solution:**

   In a cyclic quadrilateral, an exterior angle at a vertex equals the interior angle at the opposite vertex.

   If the exterior angle at K = 65°, then:

   <BlockMath math="\angle \text{interior at M} = \angle \text{exterior at K} = 65°" />

   This is because vertices K and M are opposite in cyclic quadrilateral KLMN.
