Geometry

Introduction

Geometry is a branch of mathematics that deals with points, lines, planes and solids, and examines their properties, measurement, and mutual relationship in space.

Geometry was used by ancient Egypt to measure land and build pyramids. The Greeks used a straightedge and a compass to draw lines and circles. They discovered ways to calculate distances without actually measuring them.

People in many fields use geometry. For example, engineers, rocket scientists, surveyors, designers, graphic artists, carpenters, astronomers, plumbers, laboratory technologists, advertising copywriters, construction workers, chemists, astronauts, and many more.

Geometry is used by anyone who needs to draw a circle, line, or line segment. The name geometry comes from two Greek words: geo, meaning “land,” and metry, meaning “to measure.”

Applications

A Greek mathematician named Euclid brought together most of the knowledge of geometry. He set down general rules that he called axioms, postulates, and theorems. He published a 13-volume work called The Elements. Euclid’s system of geometry is called Euclidean Geometry. Students study Euclidean Geometry in high school.

Other mathematicians who contributed to the formation of geometry are Pythagoras, Gauss, Lewis Carroll, Johannes Kepler, and many others.

Much of the reasoning in geometry consists of three stages.

1. Look for a pattern2. Make a conjecture3. verify the conjecture

1-2 Points, Lines, and Planes

In geometry, some words, such as point, line, and plane, are undefined terms.

A point has no dimension. It is usually represented by a small dot.

A line extends in one dimension. It is represented by a straight line with two arrowheads to indicate that the line extends without end in two directions.

Space is a boundless, three dimensional set of all points.

A plane extends in two dimensions. It is usually represented by a shape that looks like a tabletop or wall.

Collinear points are points that lie on the same line.

Non-collinear points are points that do not lie on the same line.

Coplanar points are points that lie on the same plane.

Noncoplanar points are points that do not lie on the same plane.

A line segment or segment is a part on a line that consists of two endpoints and all points between those endpoints.

A ray is part of a line that consists of a point, called an initial point, and all points on the line that extend in one direction.

Intersect-Two or more geometric figures intersect if they have one or more points in common.

Intersection-The intersection of figures is the set of points that the figures have in common.

A point has no dimension.

It is usually represented by a small dot.

•A

Point A

USING UNDEFINED TERMS AND DEFINITIONS

A line extends in one dimension. It is usually representedby a straight line with two arrowheads to indicate that the line extends without end in two directions.

Line or AB

Collinear points are points that lie on the same line.

USING UNDEFINED TERMS AND DEFINITIONS

A plane extends in two dimensions. It is usually represented by a shape that looks like a table or a wall, however you must imagine that the plane extends without end.

USING UNDEFINED TERMS AND DEFINITIONS

Coplanar points are points that lie on the same plane.

Plane M or plane ABC

AC

MB

USING UNDEFINED TERMS AND DEFINITIONS

Naming Collinear and Coplanar Points

SOLUTION

• Points D, E, F lie on the same line, so they are collinear.

• There are many correct answers. For instance, points H, E, and G do not lie on the same line.

• Points D, E, F, and G lie on the same plane, so they are coplanar.Also, D, E, F, and H are coplanar.

G

H

FED

• Name three points that are collinear.

• Name four points that are coplanar.

• Name three points that are not collinear.

The line segment or segment AB

(symbolized by AB) consists of the

endpoints A and B, and all points on AB

that are between A and B.

USING UNDEFINED TERMS AND DEFINITIONS

Consider the line AB (symbolized by AB).

Drawing Lines, Segments, and Rays

1

2

3

4

Draw three noncollinear points J, K, L.

Then draw JK, KL and L J.

SOLUTION

Draw J, K, and L

Draw JK.

Draw LJ.

K

LJDraw KL.

Drawing Opposite Rays

Draw two lines. Label points on the lines and name twopairs of opposite rays.

SOLUTION Points M, N, and X are collinear and X is betweenM and N.

So, XM and XN areopposite rays.

Points P, Q, and X are collinear and X is between P and Q.

So, XP and XQ are opposite rays.

SKETCHING INTERSECTIONS OF LINES AND PLANES

Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common.

Sketch two planes that intersect in a line.

Sketching Intersections

SOLUTION

Draw two planes.

Emphasize the line wherethey meet.

Dashes indicate whereone plane is hiddenby the other plane