# Geometry

Geometry is the study of size, shape and position of 2 dimensional shapes and 3 dimensional figures.

Basic Terms

- Lines
- Line Segments
- Rays
- Intersecting Lines
- Parallel Lines
- Perpendicular Lines

Lines

A line extends forever in both directions. A line has no end points. Therefore, we cannot measure how long it is.

Line Segments

A line segment does not extend forever, but has two distinct endpoints. A line segment is a portion of a line.

Rays

A ray is a straight line that begins at a certain point and extends forever in one direction. Below are two rays going in opposite direction.

Intersecting Lines

The term intersect is considered when lines, rays, line segments or figures meet, this means, they share a common point.

Parallel Lines

Two lines in the same plane which never intersect are called parallel lines.

Perpendicular Lines

Two lines are perpendicular if they intersect at a 90 degree angle.

**Types of Angles**

When two rays share the same endpoint they form an angle. The point where the rays intersect is called the vertex of the angle.

**Acute Angle**

An acute angle measures more than 0^{o}but less than 90^{o}.

**Right Angle**

A right angle measures exatly 90^{o}.

**Obtuse Angle**

An obture angle measures more than 90^{o}but less than 180^{o}.

**Straight Line Angle**

A straight line angle measures exactly 180^{o}.

**Reflex Angle**

A reflex angle measures more than 180^{o}but less than 360^{o}.

**Complementary Angles**

Two angles are complementary angles if the sum of their angles equals exactly 90 degrees.

**Supplementary Angles**

Two angles are supplementary if the sum of their angles equals exactly 180 degrees.

**Types of Triangles**

The Basic Types of triangles

A triangle is one of the basic shapes in geometry. A triangle has 3 sides and 3 interior angles. The sum of interior angles is 180

^{o}.The Basic Types of triangles

**Right-Angled Triangle**

A right-angled triangle has one of its angles measures 90^{o}.

**Scalene triangle**

A scalene triangle has no sides and no angles equal.

**Isosceles Triangle**

An Isosceles triangle has two of is sides and two of it angles equal.

**Equilateral Triangles**

An equilateral triangle has all sides and all angles equal. Each angle in an equilateral triangle measures 60 degrees.

**An Acute Triangle**

An acute triangle has all its angles acute (measures less than 90 degrees).

**Obtuse Triangles**

An obtuse triangle has one of its angle obtuse (measures more than 90 degrees).

**Types of Quadrilaterals**

A regular polygon has all sides equal and all angles equal.

1. The sum of the interior angles of a regular polygon with "n" number of sides is given by

the formula: sum of interior = 180(n - 2)

A quadrilateral is a polygon with four sides and four interior angles. The sum of interior angles is 360^{o}

# Angle Properties of a Polygon

A regular polygon has all sides equal and all angles equal.

Name of polygon | Number of sides | Sum of interior angles |

Triangle | 3 | 180^{0} |

Quadrilateral | 4 | 360^{0} |

Pentagon | 5 | 540^{0} |

Hexagon | 6 | 720^{0} |

Heptagon | 7 | 900^{0} |

Octagon | 8 | 1080^{0} |

Nonagon | 9 | 1260^{0} |

Decagon | 10 | 1440^{0} |

1. The sum of the interior angles of a regular polygon with "n" number of sides is given by

the formula: sum of interior = 180(n - 2)

^{0}.2. The sum of the exterior angle is 360^{0}.

3. The sum of the interior and exterior angles at any vertex (corner) of a polygon is 180^{0}.

Sample Problems

1. A regular polygon has 7 sides. Calculate the sum of the interior angles.

Solution: Sum of interior

= 180(n - 2)^{0}

= 180(7 - 2)^{0}

= 180(5)^{0}

= 900^{0}

2. A regular polygon has 5 sides. Calculate the size of each interior angle.

Sum of interior angles

= 180(n - 2)^{0}

= 180(5 - 2)^{0}

= 180(3)^{0}

= 540^{0}

Each interior angle = 540^{0} ÷ 5 = 108^{0}

Another way to solve this problem

Since the sum of the exterior angles of a regular polygon is 360^{0}, the size of each exterior

angle of this polygon with 5 sides would be 360^{0} ÷ 5.

360^{0} ÷ 5 = 72^{0}.

Also, we know that

exterior angle + interior angle = 180^{0}

72^{0} + interior angle = 180^{0}

Each interior angle = 180^{0} - 72^{0}.

Each interior angle = 108^{0}.

Copyright © 2012 Peter Smith
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