Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 9: Geometry 9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles 9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5Space Figures, Volume, and Surface Area 9.6Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry

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

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

© 2008 Pearson Addison-Wesley. All rights reserved 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.

© 2008 Pearson Addison-Wesley. All rights reserved 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

© 2008 Pearson Addison-Wesley. All rights reserved 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.

© 2008 Pearson Addison-Wesley. All rights reserved Half-Line, Ray, and Line Segment NameFigureSymbol Line AB or BAAB or BA Half-line ABAB Half-line BABA Ray ABAB Ray BABA Segment AB or segment BA AB or BA A B

© 2008 Pearson Addison-Wesley. All rights reserved 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. Parallel IntersectingSkew

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

© 2008 Pearson Addison-Wesley. All rights reserved 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. VertexB A C Side

© 2008 Pearson Addison-Wesley. All rights reserved 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°

© 2008 Pearson Addison-Wesley. All rights reserved Angles Angles are classified and named with reference to their degree measure. MeasureName Between 0° and 90°Acute Angle 90°Right Angle Greater than 90° but less than 180° Obtuse Angle 180°Straight Angle

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

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

© 2008 Pearson Addison-Wesley. All rights reserved 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.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Angle Measure Find the measure of each marked angle below. (3x + 10)°(5x – 10)° Solution 3x + 10 = 5x – 10 2x = 20 x = 10 So each angle is 3(10) + 10 = 40°. Vertical angels are equal.

© 2008 Pearson Addison-Wesley. All rights reserved 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

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Angle Measure Find the measure of each marked angle below. (2x + 45)° (x – 15)° Solution 2x x – 15 = 180 3x + 30 = 180 3x = 150 x = 50 Evaluating each expression we find that the angles are 35° and 145°. Supplementary angles.

© 2008 Pearson Addison-Wesley. All rights reserved Angles Formed When Parallel Lines are Crossed by a Transversal The 8 angles formed will be discussed on the next few slides.

© 2008 Pearson Addison-Wesley. All rights reserved Angles Formed When Parallel Lines are Crossed by a Transversal Alternate interior angles Alternate exterior angles Angle measures are equal (also 3 and 6) (also 2 and 7) Name

© 2008 Pearson Addison-Wesley. All rights reserved 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° (also 3 and 5) (also 1 and 5, 3 and 7, 4 and 8) Name

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Angle Measure Find the measure of each marked angle below. (x + 70)° (3x – 80)° Solution Evaluating we find that the angles are 145°. Alternating interior angles.x + 70 = 3x – 80 2x = 150 x = 75