1. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-1 Chapter 1 Section 9-1 Points, Lines, Planes, and Angles

2. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-2 Points, Lines, Planes, and Angles The Geometry of Euclid Points, Lines, and Planes Angles

3. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-3 The Geometry of Euclid A point has A line has A plane is

4. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-4 Points, Lines, and Planes A D E l A capital letter usually represents a point. A line may named by two capital letters representing points that lie on the line or by a single letter such as l. A plane may be named by three capital letters representing points that lie in the plane or by a letter of the Greek alphabet such as

5. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-5 Half-Line, Ray, and Line Segment A point divides a line into two half-lines, one on each side of the point. A __________ is a half-line including an initial point. A _____________ includes two endpoints.

6. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-6 Half-Line, Ray, and Line Segment NameFigureSymbol Line AB or BA Half-line AB Half-line BA Ray AB Ray BA Segment AB or segment BA

7. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-7 Parallel and Intersecting Lines Parallel lines lie in the same plane and never meet. Two distinct intersecting lines meet at a point. Skew lines do not lie in the same plane and do not meet.

8. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-8 Parallel and Intersecting Planes Parallel planes never meet. Two distinct intersecting planes meet and form a straight line. ParallelIntersecting

9. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-9 Angles An angle is the union of two rays that have a common endpoint. An angle can be named with the letter marking its vertex, and also with three letters: - the first letter names a point on the side; the second names the vertex; the third names a point on the other side.

10. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-10 Angles Angles are measured by the amount of rotation. 360° is the amount of rotation of a ray back onto itself. 45° 90° 10° 150° 360°

11. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-11 Angles Angles are classified and named with reference to their degree measure. MeasureName Between 0° and 90° 90° Greater than 90° but less than 180° 180°

12. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-12 Protractor A tool called a protractor can be used to measure angles.

13. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-13 Intersecting Lines When two lines intersect to form right angles they are called perpendicular.

14. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-14 Vertical Angles In the figure below the pair are called vertical angles. are also vertical angles. A C B D E Vertical angles have equal measures.

15. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-15 Example: Finding Angle Measure Find the measure of each marked angle below. (3x + 10)°(5x – 10)° Solution

16. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-16 Complementary and Supplementary Angles If the sum of the measures of two acute angles is 90°, the angles are said to be _________________, and each is called the ________________ of the other. For example, 50° and 40° are complementary angles If the sum of the measures of two angles is 180°, the angles are said to be _________________, and each is called the ____________ of the other. For example, 50° and 130° are supplementary angles

17. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-17 Example: Finding Angle Measure Find the measure of each marked angle below. (2x + 45)° (x – 15)° Solution

18. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-18 Angles Formed When Parallel Lines are Crossed by a Transversal 1 2 3 4 5 6 7 8 The 8 angles formed will be discussed on the next few slides.

19. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-19 Angles Formed When Parallel Lines are Crossed by a Transversal 1 5 4 8 (also 3 and 6) (also 2 and 7) Name

20. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-20 Angles Formed When Parallel Lines are Crossed by a Transversal Interior angles on same side of transversal Corresponding angles Angle measures are equal. Angle measures add to 180°. 4 6 2 6 (also 3 and 5) (also 1 and 5, 3 and 7, 4 and 8) Name

21. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-21 Example: Finding Angle Measure Find the measure of each marked angle below. (x + 70)° (3x – 80)° Solution