Chapter 8 Geometry and Measurement Copyright 2011 Pearson

8. 1 Lines and Angles Copyright © 2011 Pearson Education, Inc. Publishing as Prentice

Planes A plane is a flat surface that extends indefinitely. Plane Space extends in

Points The most basic concept of geometry is the idea of a point in

Lines, Segments, and Rays A Line AB or AB B Line Segment AB or

Angles An angle is made up of two rays that share the same endpoint

Angles An angle can be measured in degrees. There are 360º (degrees) in a

Classifying Angles Name Angle Measure Acute Angle Between 0° and 90° Right Angle Exactly

Classifying Angles Two angles that have a sum of 90° are called complementary angles.

Parallel Lines Two lines in a plane can be either parallel or intersecting. Parallel

Perpendicular Lines Two lines are perpendicular if they form right angles when they intersect.

Vertical Angles When two lines intersect, four angles are formed. Two of these angles

Adjacent Angles Two angles that share a common side are called adjacent angles. Adjacent

Transversal Lines A line that intersects two or more lines at different points is

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

Corresponding angles have equal measures. a b d c f e g h m

Alternate Interior Angles Alternate interior angles are angles on opposite sides of the transversal

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Planes A plane is a flat surface that extends indefinitely. Plane Space extends in all directions indefinitely. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 3

Points The most basic concept of geometry is the idea of a point in space. A point has no length, no width, and no height, but it does have location. We will represent a point by a dot, and we will label points with letters. A Point A Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 4

Lines, Segments, and Rays A Line AB or AB B Line Segment AB or AB A B Ray AB or AB A B Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 5

Angles An angle is made up of two rays that share the same endpoint called the vertex. A Vertex x B C The angle can be named ABC , CBA , B The angle can be named , , , or x or . The vertex is the middle point. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 6

Angles An angle can be measured in degrees. There are 360º (degrees) in a full revolution or full circle. 360º Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 7

Classifying Angles Name Angle Measure Acute Angle Between 0° and 90° Right Angle Exactly 90° Obtuse Angle Straight Angle Examples Between 90° and 180° Exactly 180º Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 8

Classifying Angles Two angles that have a sum of 90° are called complementary angles. Two angles that have a sum of 180° are called supplementary angles. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 9

Parallel Lines Two lines in a plane can be either parallel or intersecting. Parallel lines never meet. Intersecting lines meet at a point. The symbol is used to denote “is parallel to. ” p q Parallel lines Intersecting lines p q Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 10

Perpendicular Lines Two lines are perpendicular if they form right angles when they intersect. The symbol is used to denote “is perpendicular to. ” Perpendicular lines m n n m Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 11

Vertical Angles When two lines intersect, four angles are formed. Two of these angles that are opposite each other are called vertical angles. Vertical angles have the same measure. a d b c m a = m c m d = m b Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 12

Adjacent Angles Two angles that share a common side are called adjacent angles. Adjacent angles formed by intersecting lines are supplementary. That is, they have a sum of 180 °. a d b c a and b b and c c and d d and a Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 13

Transversal Lines A line that intersects two or more lines at different points is called a transversal. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 14

Parallel Lines Cut by a Transversal If two parallel lines are cut by a transversal, then the measures of corresponding angles are equal and the measures of the alternate interior angles are equal. a b c d e f g h Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 15

Corresponding angles have equal measures. a b d c f e g h m a = m e m c = m g m b = m f m d = m h Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 16

Alternate Interior Angles Alternate interior angles are angles on opposite sides of the transversal between the two parallel lines. a b c f e g d h m c = m f m d = m e Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3 ed 17