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Transcript

5-1 Points, Lines, Planes, and Angles Warm Up Problem of the Day Lesson Presentation Pre-Algebra WARM-UP 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 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 Learn to classify and name figures. Plane Geometry Vs Solid Geometry Can you explane the difference? explain YOU NEED TO KNOW THESE WORDS…see 5-1 in book point line plane segment ray angle rightiangle acuteiiangle obtuseiiangle complementaryiiangles supplementaryiiangles vertical angles congruent If you didn’t have time to write them all down, don’t bang you head! Just look in your book and follow along with me! You should be doing this anyhow!...lesson 5-1 Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures. Let’s look more closely at these things! A point names a location. •A Point A A line is perfectly straight and extends forever in both directions. l B C line l, or BC A plane is a perfectly flat surface that extends forever in all directions. P D E F plane P, or plane DEF A segment, or line segment, is the part of a line between two points. H G GH A ray is a part of a line that starts at one point and extends forever in K one direction. J KJ 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. 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. 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 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 D B C Try This: Example 1C 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 B C 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 D B C 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 1 Z m1 = 50° The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°. G F H K J 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 R Q 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°. Reading Math A right angle can be labeled with a small box at the vertex. 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 Additional Example 2C: Classifying Angles C. Name two obtuse angles in the figure. SQP, RQT Additional Example 2D: Classifying Angles D. Name a pair of complementary angles. TQP, RQS mTQP + m RQS = 47° + 43° = 90° 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° Try This: Example 2A A. Name a right angle in the figure. BEC C B A 15° 90° E 75° D 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 15° 90° E 75° D Try This: Example 2D D. Name a pair of complementary angles. AEB, CED mAEB + m CED = 15° + 75° = 90° C B A 15° 90° E 75° D 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 AES = 75° + 105° = 180° C B A 15° 90° E 75° D 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. 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 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°. 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 m4 = y°, find m2. m3 = 180° – y° m2 = 180° – (180° – y°) = 180° – 180° + y° = y° Distributive Property m2 = m 4 Try This: Example 3A In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. A. If m1 = 42°, find m3. 1 2 4 3 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°. Try This: Example 3B In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. B. If m4 = x°, find m 2. 1 2 4 3 m3 = 180° – x° m2 = 180° – (180° – x°) = 180° –180° + x° Distributive Property m2 = m4 = x° 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°