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1.3 Segments, Rays and Angles Copyright © 2014 Pearson Education, Inc. Slide 1-1 Undefined Terms Description A line segment or simply segment is part of a line. It consists of two endpoints and all the points between them. How to Name It Name a segment by its end points: AB (segment AB) or BA (segment BA) Example AB BA Copyright © 2014 Pearson Education, Inc. Slide 1-2 Postulate 1.13 Segment Addition Postulate If point B is between points A and C, then AB + BC = AC. Also, if AB + BC = AC, then point B is between points A and C. Copyright © 2014 Pearson Education, Inc. Slide 1-3 Using the Segment Addition Example Postulate If MP = 47 units, find MN and NP. Solution Use the value of x to find MN and NP. MN = 5x + 1 = 5(3) + 1 = 16 NP = 11x – 2 = 11(3) – 2 = 31 Check: See that MP = 47. 16 + 31 = 47 Copyright © 2014 Pearson Education, Inc. Slide 1-4 Using the Segment Addition Example Postulate If MP = 47 units, find MN and NP. Solution MN + NP = MP (5x + 1) + (11x – 22) = 47 16x – 1 = 47 16x = 48 16 x 48 16 16 x3 Copyright © 2014 Pearson Education, Inc. Slide 1-5 Terms Description A ray is part of a line. It consists of an endpoint and all points of a line on one side of the endpoint. How to Name It Name a ray by its endpoint and any other point on the ray. Here, the order of points is important—list the endpoint first. AB (ray AB) Example BA AB Copyright © 2014 Pearson Education, Inc. Slide 1-6 Example Naming Segments and Rays Use the given figure to find the following. a. Name the segments in the figure. The three segments are DE or ED EF or FE DF or FD b. Name the rays in the figure. ED, EF DE or DF FD or FE c. Which of the rays in part b are opposite rays? ED, EF Copyright © 2014 Pearson Education, Inc. Slide 1-7 Example Drawing Segments, and Rays a. Draw NL b. Draw LM Copyright © 2014 Pearson Education, Inc. Slide 1-8 Defined Terms Definition An angle consists of two different rays with a common endpoint. The rays are the sides of the angle. The common endpoint is the vertex of the angle. Point A is the vertex. The sides are rays AC and AB. Ways to name the angle: A, 1, CAB, BAC Copyright © 2014 Pearson Education, Inc. Slide 1-9 Example Identifying and Naming Angles a. How many different angles are in the diagram? b. Write two other ways to name ∠1. Solution a. There are three different angles in the diagram: ∠1, ∠2, and ∠MPQ. b. ∠1 can also be named as ∠MPN or ∠NPM. Copyright © 2014 Pearson Education, Inc. Slide 1-10 Protractor The instrument shown is called a protractor. It can be used to measure angles in units called degrees (°). For example, the measure of ∠A (also denoted by m∠A) is 30°. Copyright © 2014 Pearson Education, Inc. Slide 1-11 Example Measuring and Classifying Angles Find m∠RQM, m∠RQS, and m∠RQN. Then classify each angle as acute, right, obtuse, or straight. Solution mRQM | 45 0 | 45 acute angle mRQS | 90 0 | 90 right angle mRQN |165 0 | 165 obtuse angle Copyright © 2014 Pearson Education, Inc. Slide 1-12 Classifying Angles Acute Angle 0° < m∠A < 90° Right Angle m∠B = 90° Copyright © 2014 Pearson Education, Inc. Slide 1-13 Classifying Angles Obtuse Angle 90° < m∠C < 180° Straight Angle m∠D = 180° Copyright © 2014 Pearson Education, Inc. Slide 1-14 Special Angle Pairs Definition Two angles are complementary if their measures have a sum of 90°. Each angle is called the complement of the other. ∠1 and ∠2 are complementary angles. ∠A and ∠B are complementary angles. Copyright © 2014 Pearson Education, Inc. Slide 1-15 Special Angle Pairs Definition Two angles are supplementary if their measures have a sum of 180°. Each angle is called the supplement of the other. ∠3 and ∠4 are supplementary angles. ∠C and ∠D are supplementary angles. Copyright © 2014 Pearson Education, Inc. Slide 1-16 Defined Terms The interior of an angle contains all points between the two sides of the angle. The exterior of an angle contains all points that are not in the interior of the angle and are not on the angle. Copyright © 2014 Pearson Education, Inc. Slide 1-17 Special Angle Pairs Definition Two angles are called adjacent angles if they share a common side and a common vertex, but have no interior points in common. ∠1 and ∠2 are adjacent angles. ∠3 and ∠4 are adjacent angles. Copyright © 2014 Pearson Education, Inc. Slide 1-18 Postulate 1.14 Angle Addition Postulate If P is in the interior of ∠ABC, then m∠ABP + m∠PBC = m∠ABC. Copyright © 2014 Pearson Education, Inc. Slide 1-19