what did the acorn say when he grew up? points lines planes circles polygons congruency similarity

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What did the acorn say when he grew up? Points Lines Planes Circl es Polygon s Congruency Similarity

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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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Page 1: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

What did the acorn say when he grew up?

Points Lines Planes

Circles PolygonsCongruency

Similarity

Page 2: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity
Page 3: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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

Page 4: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

• 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.

Page 5: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

• 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.

Page 6: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

• 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.”

Page 7: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

• 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.”

Page 8: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

• 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.

Page 9: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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

Page 10: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

Points, Lines, and Planes

• Space-A boundless, three-dimensional set of all points.

Page 11: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

Coordinate Plane, points, lines, and Planes

• Identify collinear points.• Identify and model points, lines, and planes.• Identify coplanar points and intersecting

lines and planes.

Page 12: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

A point has no dimension.

It is usually represented by a small dot.

•A

Point A

USING UNDEFINED TERMS AND DEFINITIONS

Page 13: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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

Page 14: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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

Page 15: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

Coplanar points are points that lie on the same plane.

Plane M or plane ABC

AC

MB

USING UNDEFINED TERMS AND DEFINITIONS

Page 16: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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.

Page 17: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

Another undefined concept in geometry is the idea thata point on a line is between two other points on the line.You can use this idea to define other important termsin geometry.

USING UNDEFINED TERMS AND DEFINITIONS

Page 18: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

The line segment or segment AB(symbolized by AB) consists of the endpoints A and B, and all points on ABthat are between A and B.

USING UNDEFINED TERMS AND DEFINITIONS

Consider the line AB (symbolized by AB).

Page 19: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

The ray AB (symbolized by AB) consists of the initial point A and all points on ABthat lie on the same side of A as point B.

Note that AB is the same as BA, and ABis the same as BA.

However, AB and BA are not the same.They have different initial points and extend in different directions.

USING UNDEFINED TERMS AND DEFINITIONS

Page 20: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

If C is between A and B, then CAand CB are opposite rays.

Like points, segments and rays are collinear if they lieon the same plane. So, any two opposite rays arecollinear. Segments, rays, and lines are coplanar if they lie on the same plane.

USING UNDEFINED TERMS AND DEFINITIONS

Page 21: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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.

Page 22: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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.

Page 23: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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.

Page 24: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

Sketching Intersections

Sketch a line that intersects a plane at one point.

SOLUTION Draw a plane and a line.

Emphasize the pointwhere they meet.

Dashes indicate wherethe line is hidden bythe plane.

Page 25: What did the acorn say when he grew up? Points Lines Planes Circles Polygons Congruency Similarity

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