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GEOMETRY

GEOMETRY. 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.

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GEOMETRY

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

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

  19. Naming Collinear and Coplanar Points H G F E D • Name three points that are collinear. • 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 lineAB (symbolized by AB). The line segment or segmentAB (symbolized by AB) consists of the endpointsA and B, and all points on AB that are between A and B.

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

  22. Drawing Opposite Rays So, XM and XN are opposite rays. So, XP and XQ are 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. Points P, Q, and X are collinear and X is between P and Q.

  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. 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

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