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Chapter 1 Section 9-1 Points, Lines, Planes, and Angles 9-1-1 © 2008 Pearson Addison-Wesley. All rights reserved Points, Lines, Planes, and Angles • The Geometry of Euclid • Points, Lines, and Planes • Angles 9-1-2 © 2008 Pearson Addison-Wesley. All rights reserved The Geometry of Euclid A point has A line has A plane is 9-1-3 © 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 9-1-4 © 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 __________ is a half-line including an initial point. A _____________ includes two endpoints. 9-1-5 © 2008 Pearson Addison-Wesley. All rights reserved Half-Line, Ray, and Line Segment Name Figure Symbol Line AB or BA Half-line AB Half-line BA Ray AB Ray BA Segment AB or segment BA 9-1-6 © 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. 9-1-7 © 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 9-1-8 © 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. 9-1-9 © 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° 9-1-10 © 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 9-1-11 © 2008 Pearson Addison-Wesley. All rights reserved Protractor A tool called a protractor can be used to measure angles. 9-1-12 © 2008 Pearson Addison-Wesley. All rights reserved Intersecting Lines When two lines intersect to form right angles they are called perpendicular. 9-1-13 © 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. 9-1-14 © 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Angle Measure Find the measure of each marked angle below. (3x + 10)° (5x – 10)° Solution 9-1-15 © 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 _________________, 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 9-1-16 © 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Angle Measure Find the measure of each marked angle below. (2x + 45)° (x – 15)° Solution 9-1-17 © 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. 1 3 5 7 2 4 6 8 9-1-18 © 2008 Pearson Addison-Wesley. All rights reserved Angles Formed When Parallel Lines are Crossed by a Transversal Name 5 4 (also 3 and 6) 1 8 (also 2 and 7) 9-1-19 © 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 9-1-20 Example: Finding Angle Measure Find the measure of each marked angle below. (x + 70)° (3x – 80)° Solution 9-1-21 © 2008 Pearson Addison-Wesley. All rights reserved