5 1 Points Lines Planes and Angles 5

5 -1 Points, Lines, Planes, and. Angles 5 -1 Points, Warm Up Problem of the Day Lesson Presentation Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Warm Up Solve. 1. x + 30 = 90 x = 60 2. 103 + x = 180 x = 77 3. 32 + x = 180 x = 148 4. 90 = 61 + x x = 29 5. x + 20 = 90 x = 70 Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Problem of the Day Mrs. Meyer’s class is having a pizza party. Half the class wants pepperoni on the pizza, 1 of the class wants sausage on the 3 pizza, and the rest want only cheese on the pizza. What fraction of Mrs. Meyer’s class wants just cheese on the pizza? 1 6 Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Learn to classify and name figures. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles point Vocabulary line plane segment ray angle right angle acute angle obtuse angle complementary angles supplementary angles vertical angles congruent Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles A point names a location. Pre-Algebra • A Point A

5 -1 Points, Lines, Planes, and Angles A line is perfectly straight and extends forever in both directions. Pre-Algebra l B C line l, or BC

5 -1 Points, Lines, Planes, and Angles A plane is a perfectly flat surface P that extends forever D in all directions. Pre-Algebra E F plane P, or plane DEF

5 -1 Points, Lines, Planes, and Angles A segment, or line segment, is the part of a line between two points. Pre-Algebra H G GH

5 -1 Points, Lines, Planes, and Angles A ray is a part of a line that starts at one point and extends forever in K one direction. Pre-Algebra J KJ

5 -1 Points, Lines, Planes, and Angles Additional Example 1 A & 1 B: Naming Points, Lines, Planes, Segments, and Rays A. Name 4 points in the figure. Point J, point K, point L, and point M B. Name a line in the figure. KL or JK Pre-Algebra Any 2 points on a line can be used.

5 -1 Points, Lines, Planes, and Angles Additional Example 1 C: Naming Points, Lines, Planes, Segments, and Rays C. Name a plane in the figure. Plane Pre-Algebra , plane JKL Any 3 points in the plane that form a triangle can be used.

5 -1 Points, Lines, Planes, and Angles Additional Example 1 D & 1 E: Naming Points, Lines, Planes, Segments, and Rays D. Name four segments in the figure. JK, KL, LM, JM E. Name four rays in the figure. KJ, KL, JK, LK Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Try This: Example 1 A & 1 B A. Name 4 points in the figure. Point A, point B, point C, and point D B. Name a line in the figure. DA or BC Any 2 points on a line can be used. A D Pre-Algebra B C

5 -1 Points, Lines, Planes, and Angles Try This: Example 1 C C. Name a plane in the figure. Plane , plane ABC, plane BCD, plane CDA, or plane DAB Any 3 points in the plane that form a triangle can be used. A D Pre-Algebra B C

5 -1 Points, Lines, Planes, and Angles Try This: Example 1 D & 1 E D. Name four segments in the figure AB, BC, CD, DA E. Name four rays in the figure DA, AD, BC, CB A D Pre-Algebra B C

5 -1 Points, Lines, Planes, and Angles An angle ( ) is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees. 1 One degree, or 1°, is of a circle. m 1 360 means the measure of 1. The angle can be named XYZ, ZYX, 1, or Y. The vertex must be the middle letter. X Y Pre-Algebra 1 Z m 1 = 50°

5 -1 Points, Lines, Planes, and Angles The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°. G F Pre-Algebra H K J

5 -1 Points, Lines, Planes, and Angles The measures of angles that fit together to form a complete circle, such as MRN, NRP, PRQ, and QRM, add to 360°. P N M Pre-Algebra R Q

5 -1 Points, Lines, Planes, and Angles A right angle measures 90°. An acute angle measures less than 90°. An obtuse angle measures greater than 90° and less than 180°. Complementary angles have measures that add to 90°. Supplementary angles have measures that add to 180°. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Reading Math A right angle can be labeled with a small box at the vertex. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 2 A & 2 B: Classifying Angles A. Name a right angle in the figure. TQS B. Name two acute angles in the figure. TQP, RQS Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 2 C: Classifying Angles C. Name two obtuse angles in the figure. SQP, RQT Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 2 D: Classifying Angles D. Name a pair of complementary angles. TQP, RQS m TQP + m RQS = 47° + 43° = 90° Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 2 E: Classifying Angles E. Name two pairs of supplementary angles. TQP, RQT m TQP + m RQT = 47° + 133° = 180° SQP, RQS m SQP + m RQS = 137° + 43° = 180° Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Try This: Example 2 A A. Name a right angle in the figure. BEC C B A Pre-Algebra 15° 90° E 75° D

5 -1 Points, Lines, Planes, and Angles Try This: Example 2 B & 2 C B. Name two acute angles in the figure. AEB, CED C. Name two obtuse angles in the figure. BED, AEC C B A Pre-Algebra 15° 90° E 75° D

5 -1 Points, Lines, Planes, and Angles Try This: Example 2 D D. Name a pair of complementary angles. AEB, CED m AEB + m CED = 15° + 75° = 90° C B A Pre-Algebra 15° 90° E 75° D

5 -1 Points, Lines, Planes, and Angles Try This: Example 2 D & 2 E E. Name two pairs of supplementary angles. AEB, BED m AEB + m BED = 15° + 165° = 180° CED, AEC m CED + m AEC = 75° + 105° = 180° C B A Pre-Algebra 15° 90° E 75° D

5 -1 Points, Lines, Planes, and Angles Congruent figures have the same size and shape. • Segments that have the same length are congruent. • Angles that have the same measure are congruent. • The symbol for congruence is , which is read “is congruent to. ” Intersecting lines form two pairs of vertical angles. Vertical angles are always congruent, as shown in the next example. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 3 A: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. A. If m 1 = 37°, find m 3. The measures of 1 and 2 add to 180° because they are supplementary, so m 2 = 180° – 37° = 143°. The measures of 2 and 3 add to 180° because they are supplementary, so m 3 = 180° – 143° = 37°. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Additional Example 3 B: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. B. If m 4 = y°, find m 2. m 3 = 180° – y° m 2 = 180° – (180° – y°) = 180° – 180° + y° = y° Pre-Algebra Distributive Property m 2 = m 4

5 -1 Points, Lines, Planes, and Angles Try This: Example 3 A In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. A. If m 1 = 42°, find m 3. 2 3 1 4 The measures of 1 and 2 add to 180° because they are supplementary, so m 2 = 180° – 42° = 138°. The measures of 2 and 3 add to 180° because they are supplementary, so m 3 = 180° – 138° = 42°. Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Try This: Example 3 B In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. B. If m 4 = x°, find m 2. m 3 = 180° – x° 2 3 1 4 m 2 = 180° – (180° – x°) = 180° – 180° + x° Distributive Property m 2 = m 4 = x° Pre-Algebra

5 -1 Points, Lines, Planes, and Angles Lesson Quiz In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. 1. Name three points in the figure. Possible answer: A, B, and C 2. Name two lines in the figure. Possible answer: AD and BE 3. Name a right angle in the figure. Possible answer: AGF 4. Name a pair of complementary angles. Possible answer: 1 and 2 5. If m 1 = 47°, then find m 3. 47° Pre-Algebra