Section 9 1 Points Lines Planes and Angles

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Section 9. 1 Points, Lines, Planes, and Angles Copyright 2013, 2010, 2007, Pearson, Education,

Section 9. 1 Points, Lines, Planes, and Angles Copyright 2013, 2010, 2007, Pearson, Education, Inc.

What You Will Learn Points Lines Planes Angles 9. 1 -2 Copyright 2013, 2010,

What You Will Learn Points Lines Planes Angles 9. 1 -2 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Basic Terms A point, line, and plane are three basic terms in geometry that

Basic Terms A point, line, and plane are three basic terms in geometry that are NOT given a formal definition, yet we recognize them when we see them. 9. 1 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Lines, Rays, Line Segments A line is a set of points. Any two distinct

Lines, Rays, Line Segments A line is a set of points. Any two distinct points determine a unique line. Any point on a line separates the line into three parts: the point and two half lines. A ray is a half line including the endpoint. A line segment is part of a line between two points, including the endpoints. 9. 1 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Basic Terms Description Diagram Line AB A Ray AB B B A Ray BA

Basic Terms Description Diagram Line AB A Ray AB B B A Ray BA B A Line segment AB 9. 1 -5 Symbol A B Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane We can think of a plane as a twodimensional surface that extends infinitely

Plane We can think of a plane as a twodimensional surface that extends infinitely in both directions. Any three points that are not on the same line (noncollinear points) determine a unique plane. 9. 1 -6 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane Two lines in the same plane that do not intersect are called parallel

Plane Two lines in the same plane that do not intersect are called parallel lines. 9. 1 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane A line in a plane divides the plane into three parts, the line

Plane A line in a plane divides the plane into three parts, the line and two half planes. 9. 1 -8 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane Any line and a point not on the line determine a unique plane.

Plane Any line and a point not on the line determine a unique plane. The intersection of two distinct, non-parallel planes is a line. 9. 1 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane Two planes that do not intersect are said to be parallel planes. 9.

Plane Two planes that do not intersect are said to be parallel planes. 9. 1 -10 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles An angle is the union of two rays with a common endpoint; denoted.

Angles An angle is the union of two rays with a common endpoint; denoted. 9. 1 -11 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles The vertex is the point common to both rays. The sides are the

Angles The vertex is the point common to both rays. The sides are the rays that make the angle. There are several ways to name an angle: 9. 1 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles The measure of an angle is the amount of rotation from its initial

Angles The measure of an angle is the amount of rotation from its initial to its terminal side. Angles can be measured in degrees, radians, or gradients. 9. 1 -13 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles are classified by their degree measurement. Right Angle is 90º Acute Angle is

Angles are classified by their degree measurement. Right Angle is 90º Acute Angle is less than 90º Obtuse Angle is greater than 90º but less than 180º Straight Angle is 180º 9. 1 -14 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles 9. 1 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles 9. 1 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Types of Angles Adjacent Angles - angles that have a common vertex and a

Types of Angles Adjacent Angles - angles that have a common vertex and a common side but no common interior points. Complementary Angles - two angles whose sum of their measures is 90 degrees. Supplementary Angles - two angles whose sum of their measures is 180 degrees. 9. 1 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Determining Complementary Angles In the figure, we see that 9. 1 -17

Example 3: Determining Complementary Angles In the figure, we see that 9. 1 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Determining Complementary Angles Solution 9. 1 -18 Copyright 2013, 2010, 2007, Pearson,

Example 3: Determining Complementary Angles Solution 9. 1 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Determining Supplementary Angles In the figure, we see that 9. 1 -19

Example 3: Determining Supplementary Angles In the figure, we see that 9. 1 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 3: Determining Supplementary Angles Solution 9. 1 -20 Copyright 2013, 2010, 2007, Pearson,

Example 3: Determining Supplementary Angles Solution 9. 1 -20 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions When two straight lines intersect, the nonadjacent angles formed are called Vertical angles

Definitions When two straight lines intersect, the nonadjacent angles formed are called Vertical angles have the same measure. 9. 1 -21 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions A line that intersects two different lines, at two different points is called

Definitions A line that intersects two different lines, at two different points is called a transversal. 9. 1 -22 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions Special names are given to the angles formed by a transversal crossing two

Definitions Special names are given to the angles formed by a transversal crossing two parallel lines. 9. 1 -23 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Special Names 9. 1 -24 Alternate interior angles 3 & 6; 4 & 5

Special Names 9. 1 -24 Alternate interior angles 3 & 6; 4 & 5 Interior angles on the opposite side of the transversal–have the same measure Alternate exterior angles 1 & 8; 2 & 7 Exterior angles on the opposite sides of the transversal–have the same measure Corresponding angles 1 & 5, 2 & 6, 3 & 7, 4 & 8 One interior and one exterior angle on the same side of the transversal–have the same measure Copyright 2013, 2010, 2007, Pearson, Education, Inc. 1 2 3 4 5 6 7 8 1 3 2 4 5 6 7 8 1 3 5 6 7 8 2 4

Parallel Lines Cut by a Transversal When two parallel lines are cut by a

Parallel Lines Cut by a Transversal When two parallel lines are cut by a transversal, 1. alternate interior angles have the same measure. 2. alternate exterior angles have the same measure. 3. corresponding angles have the same measure. 9. 1 -25 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 6: Determining Angle Measures The figure shows two parallel lines cut by a

Example 6: Determining Angle Measures The figure shows two parallel lines cut by a transversal. Determine the measure of through. 9. 1 -26 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 6: Determining Angle Measures Solution 9. 1 -27 Copyright 2013, 2010, 2007, Pearson,

Example 6: Determining Angle Measures Solution 9. 1 -27 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 6: Determining Angle Measures Solution 9. 1 -28 Copyright 2013, 2010, 2007, Pearson,

Example 6: Determining Angle Measures Solution 9. 1 -28 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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Section 1 Jeopardy Points Lines and Planes DistanceSection 1 Jeopardy Points Lines and Planes Distance