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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 Learn to classify and name figures. Pre-Algebra 5-1 Points, Lines, Planes, and Angles Vocabulary point 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 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 that extends forever in all directions. Pre-Algebra P D 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 1A & 1B: 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 1C: 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 1D & 1E: 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 2A & 2B 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 2C 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 2D & 2E 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. m1 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 m1 = 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 3A & 3B: 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 3C: Classifying Angles C. Name two obtuse angles in the figure. SQP, RQT Pre-Algebra 5-1 Points, Lines, Planes, and Angles Additional Example 3D: Classifying Angles D. Name a pair of complementary angles. TQP, RQS mTQP + m RQS = 47° + 43° = 90° Pre-Algebra 5-1 Points, Lines, Planes, and Angles Additional Example 3E: Classifying Angles E. Name two pairs of supplementary angles. TQP, RQT mTQP + mRQT = 47° + 133° = 180° SQP, RQS mSQP + mRQS = 137° + 43° = 180° Pre-Algebra 5-1 Points, Lines, Planes, and Angles Try This: Example 4A 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 4B & 4C 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 4D D. Name a pair of complementary angles. AEB, CED mAEB + mCED = 15° + 75° = 90° C B A Pre-Algebra 15° 90° E 75° D 5-1 Points, Lines, Planes, and Angles Try This: Example 4E E. Name two pairs of supplementary angles. AEB, BED mAEB + mBED = 15° + 165° = 180° CED, AEC mCED + mAEC = 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 5A: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. A. If m1 = 37°, find m 3. The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 37° = 143°. The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 180° – 143° = 37°. Pre-Algebra 5-1 Points, Lines, Planes, and Angles Additional Example 5B: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. B. If m4 = y°, find m2. m3 = 180° – y° m2 = 180° – (180° – y°) = 180° – 180° + y° = y° Pre-Algebra Distributive Property m2 = m4 5-1 Points, Lines, Planes, and Angles Try This: Example 6A In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. A. If m1 = 42°, find m3. 2 3 1 4 The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 42° = 138°. The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 180° – 138° = 42°. Pre-Algebra 5-1 Points, Lines, Planes, and Angles Try This: Example 6B In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. B. If m4 = x°, find m2. m3 = 180° – x° 2 3 1 4 m2 = 180° – (180° – x°) = 180° –180° + x° Distributive Property m2 = m4 = 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 m1 = 47°, then find m 3. 47° Pre-Algebra