1.2 points, lines and planes
- 1. 5-1 Points, Lines, Planes, and Angles 5-1 Points, Lines, Planes, and Angles Warm Up Problem of the Day Lesson Presentation Pre-Algebra Pre-Algebra
- 2. 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
- 3. 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
- 4. 5-1 Points, Lines, Planes, and Angles Learn to classify and name figures. Pre-Algebra
- 5. 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
- 6. 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
- 7. 5-1 Points, Lines, Planes, and Angles A point names a •A Point A location. Pre-Algebra
- 8. 5-1 Points, Lines, Planes, and Angles A line is perfectly C straight and l extends forever in B line l, or BC both directions. Pre-Algebra
- 9. 5-1 Points, Lines, Planes, and Angles A plane is a perfectly flat P E surface that D plane P, or F plane DEF extends forever in all directions. Pre-Algebra
- 10. 5-1 Points, Lines, Planes, and Angles A segment, or H line segment, is the part of a line GH between two G points. Pre-Algebra
- 11. 5-1 Points, Lines, Planes, and Angles A ray is a part of J a line that starts at one point and KJ extends forever in K one direction. Pre-Algebra
- 12. 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 Any 2 points on a line can be used. Pre-Algebra
- 13. 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 , plane JKL Any 3 points in the plane that form a triangle can be used. Pre-Algebra
- 14. 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
- 15. 5-1 Points, Lines, Planes, and Angles Try This: Example 1A & 1B 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 B D C Pre-Algebra
- 16. 5-1 Points, Lines, Planes, and Angles Try This: Example 1C C. Name a plane in the figure. Plane , plane ABC, Any 3 points in the plane BCD, plane CDA, plane that form a or plane DAB triangle can be used. A B D C Pre-Algebra
- 17. 5-1 Points, Lines, Planes, and Angles Try This: Example 1D & 1E D. Name four segments in the figure AB, BC, CD, DA E. Name four rays in the figure DA, AD, BC, CB A B D C Pre-Algebra
- 18. 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 1 m∠1 = 50° Y Z Pre-Algebra
- 19. 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 H F K J Pre-Algebra
- 20. 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 R Q M Pre-Algebra
- 21. 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
- 22. 5-1 Points, Lines, Planes, and Angles Reading Math A right angle can be labeled with a small box at the vertex. Pre-Algebra
- 23. 5-1 Points, Lines, Planes, and Angles Additional Example 2A & 2B: Classifying Angles A. Name a right angle in the figure. ∠TQS B. Name two acute angles in the figure. ∠TQP, ∠RQS Pre-Algebra
- 24. 5-1 Points, Lines, Planes, and Angles Additional Example 2C: Classifying Angles C. Name two obtuse angles in the figure. ∠SQP, ∠RQT Pre-Algebra
- 25. 5-1 Points, Lines, Planes, and Angles Additional Example 2D: Classifying Angles D. Name a pair of complementary angles. ∠TQP, ∠RQS m∠TQP + m∠ RQS = 47° + 43° = 90° Pre-Algebra
- 26. 5-1 Points, Lines, Planes, and Angles Additional Example 2E: 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
- 27. 5-1 Points, Lines, Planes, and Angles Try This: Example 2A A. Name a right angle in the figure. ∠BEC C B A 90° D 15° 75° E Pre-Algebra
- 28. 5-1 Points, Lines, Planes, and Angles Try This: Example 2B & 2C B. Name two acute angles in the figure. ∠AEB, ∠CED C. Name two obtuse angles in the figure. ∠BED, ∠AEC C B A 90° D 15° 75° E Pre-Algebra
- 29. 5-1 Points, Lines, Planes, and Angles Try This: Example 2D D. Name a pair of complementary angles. ∠AEB, ∠CED m∠AEB + m∠CED = 15° + 75° = 90° C B A 90° D 15° 75° E Pre-Algebra
- 30. 5-1 Points, Lines, Planes, and Angles Try This: Example 2D & 2E 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 90° D 15° 75° E Pre-Algebra
- 31. 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
- 32. 5-1 Points, Lines, Planes, and Angles Additional Example 3A: 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
- 33. 5-1 Points, Lines, Planes, and Angles Additional Example 3B: 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° Distributive Property = y° m∠2 = m∠4 Pre-Algebra
- 34. 5-1 Points, Lines, Planes, and Angles Try This: Example 3A In the figure, ∠1 and ∠3 are vertical angles, and ∠2 and ∠4 are vertical angles. 2 A. If m∠1 = 42°, find m∠3. 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
- 35. 5-1 Points, Lines, Planes, and Angles Try This: Example 3B In the figure, ∠1 and ∠3 are vertical angles, and ∠2 and ∠4 are vertical angles. 2 B. If m∠4 = x°, find m∠2. 3 1 4 m∠3 = 180° – x° m∠2 = 180° – (180° – x°) = 180° –180° + x° Distributive Property = x° m∠2 = m∠4 Pre-Algebra
- 36. 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