Angles and Intercepted Arcs

In this lesson, we will learn some formulas relating the angles and the intercepted arcs of circles.

  • Measure of a central angle.
  • Measure of an inscribed angle - angle with its vertex on the circle
  • Measure of an angle with vertex inside a circle.
  • Measure of an angle with vertex outside a circle.

We will also learn about angles of Inscribed Triangles and Quadrilaterals.


Central Angle

A central angle is an angle with its vertex is at the center of the circle. The measure of a central angle is equal to the measure the intercepted arc.

The formula is

Central angle = intercepted arc

Example:

Find the value of x


Solution:



This video shows how to use the property that the measure of a central angle is equal to the measure of its intercepted arc to find the missing measures of arcs and angles in given figures.



Inscribed Angle - Angle with its vertex on the circle

An inscribed angle is an angle with its vertex on the circle. The meaure of an inscribed angle is half the measure the intercepted arc.

The formula is

Inscribed angle = (intercepted arc)

Example:

Find the value of x


Solution:

 

This video deals with angles formed with vertices on the circle. 






Inscribed angles are 1/2 the measure of their intercepted arcs.

Angle with vertex inside the circle

The measure of an angle with its vertex inside the circle is half the sum of the intercepted arcs.

The formula is

Angle = (sum of intercepted arcs)

Example:

Find the value of x


Solution:

 

The following video shows how to apply the formula for angles with vertex inside the circle.




Angle with vertex outside the circle

The measure of an angle with its vertex outside the circle is half the difference of the intercepted arcs.

The formula is

Angle = (difference of intercepted arcs)

Example:

Find the value of x

The following video shows how to apply the formula for angles with vertex outside circle.




Arc and Angle Relationships Problems

This video will go through a few examples of how to use the formulas involving Arc and Angle Relationships to find the measure of missing angles or missing arcs.



More Arc and Angle Relationships Problems




Inscribed Triangles

If an inscribed triangle is a right triangle, then the hypotenuse is the diameter. If an inscribed angle has a diameter as one of its sides, then its a right triangle.



Right Triangles Inscribed in Circles (Proof)
Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle




Inscribed Quadrilaterals

Quadrilaterals inscribed in a circle. Opposite angles are supplementary.



Circles - Inscribed Quadrilaterals
How to find missing angles inside inscribed quadrilaterals.






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