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# Angles of Polygon

## Key Concepts

• Find the sum of angles of a quadrilateral.
• Find the sum of interior angles of an n-sided polygon.
• Guess the measure of each angle of a regular polygon.
• Find the sum of exterior angles of n-sided polygon.

## Triangle

A triangle is a polygon with three edges and three vertices.

### Angle sum property of a triangle

Angle sum property of a triangle states that the sum of angles of a triangle is180°

In geometry, a four-sided polygon, having four edges (sides) and four corners (vertices), is a quadrilateral

The sum of angles of a quadrilateral is 360°.

### Pentagon

A figure that has five sides and five vertices is called a pentagon.

The sum of angles of a pentagon is 540°.

### Hexagon

A 6-sided polygon is called a hexagon.

The sum of angles of a hexagon is 720°

### Sum of interior angles of a 𝒏-sided polygon

There are n−2 triangles in each n sided polygon.

The sum of the measures of the interior angles of a convex 𝒏 -gon is 𝟏𝟖𝟎°×𝒏−𝟐.

### Regular polygon

A polygon that is equilateral (all sides have the same length) and whose angles are of equal measures is a regular polygon

The measure of interior angles of a regular 𝒏 -gon is 180°×(n−2)/n

### Exterior angle of a polygon

The angle between any side of a polygon and a line extended from the next side is called an exterior angle of a polygon

### Sum of exterior angles of a polygon

The sum of all the exterior angles of a polygon is 360°.

## Exercise

• What is the value of x ?
• What are the measures of the exterior angles of the polygon shown?
• The sum of the interior angles of a regular n-gon is 6, . What is the measure of each interior angle?
• What is the value of x? What is the measure of each exterior angle?
• Find the value of x and the measure of each interior angle.

### What we have learned

• In geometry, a four-sided polygon, having four edges (sides) and four corners (vertices) is a quadrilateral.
• The sum of the measures of the interior angles of a convex 𝒏-gon is 180° ( n-2 )
• A polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length) is a regular polygon.
• The measure of interior angles of a regular 𝒏-gon is 180° x (n-2)/n
• The angle between any side of a polygon and a line extended from the next side is called an exterior angle of a polygon.
• The sum of all the exterior angles of a polygon is 360°

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