An **inscribed angle** is formed when two secant lines intersect on a circle. In other words, the inscribed angle in a circle is formed when one of the end points of
two chords in a circle meet in a point.

In the above picture, *L* ABC is the of the circle.

From the picture above,

- Angle ABC is the inscribed angle.

- CB and BA are the chords.

- Arc CA is the intercepted arc.

- Formula: Angle CAB = Arc AC/2.

- Two or more inscribed angles intercepting a same arc will be equal.

BAC angle = BDC angle

- An inscribed angle is the measure of half the central angle intercepting the same arc.

CABangle = (1 / 2) BOC angle

CDB angle = (1 / 2) BOC angle

**Another Formula for inscribed angle:**

If we know the length of the minor arc, radius, the inscribed angle is found by:

Angle= `(90 L)/(pi* R)`

where,

L is the length of the shortest arc BA

R is defined as radius.

I am planning to write more post on **Inscribed Angle Theorem**,
**Consecutive Interior Angles** Keep checking my blog.

From the circle diagram below, the chord CA has a length of 12 cm and center at O. The circle has a radius of 14 cm. Find the measure of the inscribed angle ABC.

**Solution:**

**1. **First we calculate the central angle AOC. The triangle AOC is a isosceles triangle. Distance of OC = Distance of OA = radius = 14 cm. Here we use cosine law to find
value of cos (angle AOC).

AC^{2} = OC^{2} + OA^{2} - 2 OC OA cos (angle COA)

**2.** Substitute the value of the angles AC, OC and AO in cos (angle AOC) as follows

cos(angle COA) = `[ 14^2 + 14^2- 122 ] / [28 * 14 ]`

= `62 / 98`

**3. **The measure of the angle COA is given by.

Angle COA = `arccos (62 / 98)`

According to the theorem described before, the size of angle CBA will be equal to half the size of angle COA.

Angle CBA = `(1/ 2) arccos (62 / 98)`

= 25.38 degrees.

**Problem 1:**

** **Given that the **central angle** of circle is 50 degrees. Find the angle of
inscribed circle.

**Answer = 100 degree**

**Problem 2:**

** **Given that the central angle of circle is 120 degrees. Find the angle of inscribed circle.

**Answer = 240 degree.**