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Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved Chapter 9: Geometry 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Points, Lines, Planes, and Angles Curves, Polygons, and Circles Perimeter, Area, and Circumference The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem Space Figures, Volume, and Surface Area Transformational Geometry Non-Euclidean Geometry, Topology, and Networks Chaos and Fractal Geometry 9-1-2 © 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 9-1 Points, Lines, Planes, and Angles 9-1-3 © 2008 Pearson Addison-Wesley. All rights reserved Points, Lines, Planes, and Angles • The Geometry of Euclid • Points, Lines, and Planes • Angles 9-1-4 © 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. 9-1-5 © 2008 Pearson Addison-Wesley. All rights reserved 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 , , or . l A D E 9-1-6 © 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. 9-1-7 © 2008 Pearson Addison-Wesley. All rights reserved Half-Line, Ray, and Line Segment Name Line AB or BA Half-line AB Figure Symbol A AB or BA A Half-line BA B A Ray AB A Ray BA B A BA AB B A Segment AB or segment BA AB B B B BA AB or BA 9-1-8 © 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 Intersecting Skew 9-1-9 © 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. Parallel Intersecting 9-1-10 © 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, B , and also with three letters: ABC - the first letter names a point on the side; the second names the vertex; the third names a point on the other side. A Vertex B C 9-1-11 © 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° 9-1-12 © 2008 Pearson Addison-Wesley. All rights reserved Angles Angles are classified and named with reference to their degree measure. Measure Between 0° and 90° 90° Greater than 90° but less than 180° 180° Name Acute Angle Right Angle Obtuse Angle Straight Angle 9-1-13 © 2008 Pearson Addison-Wesley. All rights reserved Protractor A tool called a protractor can be used to measure angles. 9-1-14 © 2008 Pearson Addison-Wesley. All rights reserved Intersecting Lines When two lines intersect to form right angles they are called perpendicular. 9-1-15 © 2008 Pearson Addison-Wesley. All rights reserved Vertical Angles In the figure below the pair ABC and DBE are called vertical angles. DBA and EBC are also vertical angles. A D B E C Vertical angles have equal measures. 9-1-16 © 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 Vertical angels are equal. So each angle is 3(10) + 10 = 40°. 9-1-17 © 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 9-1-18 © 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 + 45 + x – 15 = 180 Supplementary angles. 3x + 30 = 180 3x = 150 x = 50 Evaluating each expression we find that the angles are 35° and 145° . © 2008 Pearson Addison-Wesley. All rights reserved 9-1-19 Angles Formed When Parallel Lines are Crossed by a Transversal The 8 angles formed will be discussed on the next few slides. 1 3 5 7 2 4 6 8 9-1-20 © 2008 Pearson Addison-Wesley. All rights reserved Angles Formed When Parallel Lines are Crossed by a Transversal Name Alternate interior angles 5 4 Angle measures are equal. (also 3 and 6) 1 Alternate exterior angles Angle measures are equal. 8 (also 2 and 7) 9-1-21 © 2008 Pearson Addison-Wesley. All rights reserved Angles Formed When Parallel Lines are Crossed by a Transversal Name Interior angles on same side of transversal 6 Angle measures add to 180°. 4 (also 3 and 5) Corresponding angles 2 6 (also 1 and 5, 3 and 7, 4 and 8) Angle measures are equal. © 2008 Pearson Addison-Wesley. All rights reserved 9-1-22 Example: Finding Angle Measure Find the measure of each marked angle below. (x + 70)° (3x – 80)° Solution x + 70 = 3x – 80 Alternating interior angles. 2x = 150 x = 75 Evaluating we find that the angles are 145°. 9-1-23 © 2008 Pearson Addison-Wesley. All rights reserved

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