## Introduction

- The angle formed by two radii of the circle and having its vertex at the center of the circle is called the
.**central angle** - In other words, a central angle is an angle subtended by an arc of a circle at the center of the circle, as shown below.

- It can be seen that the central angle divides a circle into sectors.
- The angle subtended an arc of a circle at any point on the circumference of the circle is called an
.**inscribed angle** - In the figure shown above, $\mathit{\theta}$ is the central angle and $\mathit{\alpha}$ is the inscribed angle.

## Important Points

- The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

- Congruent arcs of a circle subtend equal angles at the center.

- The angle subtended by diameter on any point of a circle is $90\xb0$.

- Angles in the same segment of a circle are equal.
- The measure of a central angle is equal to the measure of the arc forming the central angle.
- The measure of an inscribed angle is half the measure of the intercepted arc.
- In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary. â€¨

## Solved Examples

Example 1: Circle O has points A, B, and C on the circle, as shown in the figure. Given $m\angle AOB=50\xb0$, find the measure of $\angle ACB$.

Solution: $\angle ACB=\frac{1}{2}\angle AOB$ $=\frac{1}{2}\times 50\xb0$ $=25\xb0$

Example 2: In the figure shown below $m\stackrel{\u23dc}{SQ}=45\xb0$ and $\angle PQR=20\xb0$. Determine $m\angle POR$.

Solution: $OQ=OR\text{[radii}]$

$\angle OQR=\angle ORQ=20\xb0$

In $\u2206ROQ$,

$\angle ROQ+\angle OQR+\angle ORQ=180\xb0$

$\angle ROQ+20\xb0+20\xb0=180\xb0$

$\angle ROQ=140\xb0$

Also, $\angle ROQ+\angle POR=180\xb0$

$140\xb0+\angle POR=180\xb0$

$\angle POR=40\xb0$

Question 3: Points A, B, C, and D lie on Circle O. $\angle BAC$ is inscribed in the circle. Find x, given $m\stackrel{\u23dc}{BDC}=148\xb0$ and $m\angle BAC=\left(4x+24\right)\xb0$.

Solution: $\stackrel{\u23dc}{BDC}=148\xb0$, $\angle BAC=\left(4x+24\right)\xb0$

$\angle BAC=\frac{1}{2}\stackrel{\u23dc}{BDC}$

$\left(4x+24\right)\xb0=\frac{1}{2}\times 148\xb0$

$4x+24\xb0=74\xb0$

$4x=50\xb0$

$x=12.5\xb0$

## Cheat Sheet

- The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
- Congruent arcs of a circle subtend equal angles at the center.
- The angle subtended by diameter on any point of a circle is $90\xb0$.
- Angles in the same segment of a circle are equal.
- The measure of an inscribed angle is half the measure of the intercepted arc.
- In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary.

## Blunder Areas

- The angle subtended by an arc at the center of the circle is equal to the angle measure of the arc.

- Abhishek Tiwari
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