2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 8 Geometry and Measurement

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.1 Lines and Angles

4. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Plane A plane is a flat surface that extends indefinitely. Space extends in all directions indefinitely. Planes

5. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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

6. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. AB A B Line AB or AB Ray AB or AB Line Segment AB or AB AB Lines, Segments, and Rays

7. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Vertex An angle is made up of two rays that share the same endpoint called the vertex. The angle can be named,,, or. The angle can be named  ABC,  CBA,  B, or  x. A B C x The vertex is the middle point. Angles

8. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. An angle can be measured in degrees. There are 360º (degrees) in a full revolution or full circle. 360º Angles

9. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Classifying Angles Acute Angle Name Angle MeasureExamples Between 0° and 90° Right Angle Exactly 90° Obtuse Angle Between 90° and 180° Straight Angle Exactly 180º

10. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Two angles that have a sum of 90° are called complementary angles. Two angles that have a sum of 180° are called supplementary angles. Classifying Angles

11. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 linesIntersecting lines p  q Parallel Lines

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

13. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 b c d  m  a  m  cm  d  m  b Vertical Angles

14. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 and  b  b and  c  c and  d  d and  a a b c d Adjacent Angles

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

16. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 gh

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

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