# 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 of its 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°

Central Angles and Arcs
A chord is a segment that has is endpoints on a circle.
The diameter is the longest chord of a circle and it passes through the venter of a circle.
A line is called a straight angle and it forms a 180 degree angle.
A central angle is an angle with its vertex at the center of a circle and its sides are radii of the same circle.
Show that central angles = arcs they intercept.
Examples to show 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 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°

Angle with vertex on the circle (Inscribed angle)
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°

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

### 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

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

### 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 two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Am inscribe polygon is a polygon with all its vertices on the circle. The circle is then called a circumscribed circle.
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed triangle is a diameter, then the triangle a right triangle, and the angle opposite the diameter is a right angle. 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.

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
When a quadrilateral is inscribed in a circle: • The interior angles add up to 360°
• Both pairs of opposite angles are supplementary
How to find missing angles inside inscribed quadrilaterals.

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