1. GEOMETRY

2. Geometry
Introduction

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

4. Geometry was used by ancient Egypt to measure land and build pyramids
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.

5. People in many fields use geometry
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.

6. 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.”

7. Applications

8. 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.

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

10. Much of the reasoning in geometry consists of three stages.
1. Look for a pattern
2. Make a conjecture
3. verify the conjecture

11. 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.

12. A plane extends in two dimensions
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.

13. 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.

14. 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.

15. • A point has no dimension. It is usually represented by a small dot.
USING UNDEFINED TERMS AND DEFINITIONS
A point has no dimension.
It is usually represented by a small dot. A
Point •

16. A line extends in one dimension. It is usually represented
USING UNDEFINED TERMS AND DEFINITIONS
A line extends in one dimension. It is usually represented
by 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.

17. 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.

18. M Coplanar points are points that lie on the same plane. A C B
USING UNDEFINED TERMS AND DEFINITIONS
Coplanar points are points that lie on the same plane. A C M B
Plane M or plane ABC

19. • Name three points that are collinear.
Naming Collinear and Coplanar Points
• Name three points that are collinear. G H F E D
• Name four points that are coplanar.
• Name three points that are not collinear.
SOLUTION
• Points D, E, F lie on the same line, so they are collinear.
• Points D, E, F, and G lie on the same plane, so they are coplanar.
Also, D, E, F, and H are coplanar.
• There are many correct answers. For instance, points H, E, and G do
not lie on the same line.

20. USING UNDEFINED TERMS AND DEFINITIONS
Consider the line AB (symbolized by AB).
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.

21. Drawing Lines, Segments, and Rays
Draw three noncollinear points J, K, L. Then draw JK, KL and L J.
SOLUTION K L J
Draw J, K, and L 1 2
Draw JK. 3
Draw KL. 4
Draw LJ.

22. So, XP and XQ are opposite rays.
Drawing Opposite Rays
Draw two lines. Label points on the lines and name two
pairs of opposite rays.
SOLUTION
Points M, N, and X are
collinear and X is between
M and N.
So, XM and XN are
opposite rays.
Points P, Q, and X are collinear and X is between P and Q.
So, XP and XQ are opposite rays.

23. 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.

24. Sketch two planes that intersect in a line.
Sketching Intersections
Sketch two planes that intersect in a line.
SOLUTION
Draw two planes.
Emphasize the line where
they meet.
Dashes indicate where
one plane is hidden
by the other plane