1 / 21

Section 9-1

Section 9-1. Points, Lines, Planes, and Angles. Points, Lines, Planes, and Angles. The Geometry of Euclid Points, Lines, and Planes Angles. The Geometry of Euclid. A point has no magnitude and no size. A line has no thickness and no width and it extends indefinitely in two directions.

sarah
Download Presentation

Section 9-1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 9-1 • Points, Lines, Planes, and Angles

  2. Points, Lines, Planes, and Angles • The Geometry of Euclid • Points, Lines, and Planes • Angles

  3. The Geometry of Euclid A point has no magnitude and no size. A line has no thickness and no width and it extends indefinitely in two directions. A plane is a flat surface that extends infinitely.

  4. Points, Lines, and Planes 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 l A E D

  5. Half-Line, Ray, and Line Segment A point divides a line into two half-lines, one on each side of the point. A ray is a half-line including an initial point. A line segment includes two endpoints.

  6. Half-Line, Ray, and Line Segment A B A B A B A B A B A B

  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. Intersecting Skew Parallel

  8. Parallel and Intersecting Planes Parallel planes never meet. Two distinct intersecting planes meet and form a straight line. Parallel Intersecting

  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. A Side Vertex B Side C

  10. Angles Angles are measured by the amount of rotation. 360° is the amount of rotation of a ray back onto itself. 90° 45° 10° 150° 360°

  11. Angles Angles are classified and named with reference to their degree measure.

  12. Protractor A tool called a protractor can be used to measure angles.

  13. Intersecting Lines When two lines intersect to form right angles they are called perpendicular.

  14. Vertical Angles In the figure below the pair are called vertical angles. are also vertical angles. A D B E C Vertical angles have equal measures.

  15. Example: Finding Angle Measure Find the measure of each marked angle below. (3x + 10)° (5x – 10)° Solution 3x + 10 = 5x – 10 Vertical angles are equal. 2x = 20 x = 10 So each angle is 3(10) + 10 = 40°.

  16. Complementary and Supplementary Angles If the sum of the measures of two acute angles is 90°, the angles are said to be complementary, and each is called the complement 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 supplementary, and each is called the supplement of the other. For example, 50° and 130° are supplementary angles

  17. Example: Finding Angle Measure Find the measure of each marked angle below. (2x + 45)° (x – 15)° Solution 2x + 45 + x – 15 = 180 3x + 30 = 180 Supplementary angles. 3x = 150 x = 50 Evaluating each expression we find that the angles are 35° and 145°.

  18. Angles Formed When Parallel Lines are Crossed by a Transversal The 8 angles formed will be discussed on the next few slides. 1 2 > 3 4 5 6 > 7 8

  19. Angles Formed When Parallel Lines are Crossed by a Transversal Name Angle measures are equal. Alternate interior angles 5 4 (also 3 and 6) 1 Angle measures are equal. Alternate exterior angles 8 (also 2 and 7)

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

  21. Example: Finding Angle Measure Find the measure of each marked angle below. (3x – 80)° (x + 70)° Solution x + 70 = 3x – 80 Alternating interior angles. 2x = 150 x = 75 Evaluating we find that the angles are 145°.

More Related