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Angles and Intercepted Arcs




 
In these lessons, 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 Inscribed Quadrilaterals.

Related Topics: More Geometry Lessons

Central Angles and their Arcs


What is a Central Angle?
A central angle is an angle with its vertex is at the center of the circle and its sides are the radii of the circle.
What is the relationship between central angles and their arcs?
The measure of a central angle is equal to the measure the intercepted arc.
The formula is
Measure of central angle = measure of intercepted arc

Example:

Find the value of x

Solution:

x = m∠AOB = minor arc from A to B = 120°


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.
Central angles = arcs they intercept



Inscribed Angles and their Arcs

What is an Inscribed Angle?
An inscribed angle is an angle with its vertex on the circle.
What is the relationship between inscribed angles and their arcs?
The measure of an inscribed angle is half the measure the intercepted arc.
The formula is
Measure of inscribed angle = \(\frac{1}{2}\) × measure of intercepted arc

Example:

Find the value of x

Solution:

x = m∠AOB = \(\frac{1}{2}\) × 120° = 60°

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


 
Examples of how to use the property that inscribed angles are 1/2 the measure of their intercepted arcs to find missing angles.

Angles with vertex inside the circle and their Arcs

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

The formula is
Measure of angle with vertex inside circle = \(\frac{1}{2}\) × (sum of intercepted arcs)

Example:

Find the value of x

Solution:

\(\frac{1}{2}\) × (160° + 25°) = 97.5°

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


Angles with vertex outside the circle and their Arcs

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

The formula is
Measure of angle with vertex outside the circle = \(\frac{1}{2}\) × (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 Relationship 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 Examples of Arc and Angle Relationship 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

A quadrilaterals inscribed in a circle if and only if its opposite angles are supplementary. How to use this property to find missing angles?
Circles - Inscribed Quadrilaterals
How to find missing angles inside inscribed quadrilaterals.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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