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Angles Vertex Side 2 Two rays with a common endpoint. Model Item An angle is Two rays with a common end point. Notation ABC B 1 The parts are the sides ( rays ) , the vertex ( common point), interior space, and exterior space. Angles are classified by rotation of the rays. Zero degrees 90 degrees Straight Angle: 180 degrees. Obtuse angles: between 90 and 180 degrees. Acute angles are < 90 degrees Types of Angles Acute angles are less than 90 degrees Right angles are equal to 90 degrees. [ Looks like letter L ] Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles look like lines and are equal to 180 degrees. F N R C O A U T A T P T Types of Angles Angles are differentiated by the quantify of rotation of the rays as if they were hands of a clock. No rotation is zero degrees and totally straight is 180 degrees. S T 90 degrees R Zero degrees 45 degrees Types of Angles Smallest Zero Small Acute Middle Right Large Obtuse Largest Straight Measuring Angles The Protractor Measuring Angles The Protractor The smaller number is for the acute angles and the larger number is for the obtuse angles. Notice, the numbers add up to 180. 500 1400 0 40 0 25 0 35 600 600 350 0 35 0 57 530 Adjacent Angles 1 2 Same vertex, Common ray, and no common interior Non-Adjacent Angles 2 1 Not the same endpoint. Non-Adjacent Angles B A T G BAT and BAG Overlapping Interiors is not allowed. A 4 3 E 5 6 D 2 1 L 9 8 T 7 S How Many Angles ? 2+ 1=3 How Many Angles ? 3+ 2+ 1=6 How Many Angles ? 4 + 3 + 2 + 1 = 10 Did you see the pattern? 2+1=3 3+2+1=6 4 + 3 + 2 + 1 = 10 Total angles = sum of countdown of the smallest angle totals. 0 50 Vertex Position One ray must be horizontal. Reading a protractor Protractor Postulate For AB on a given plane, choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of A O AB. B Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: OA A is paired with 0 and 0 O OB with 180. 180 B Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: If and OQ is paired with y, OP is paired with x then m POQ x y AB P X Q AB Y OB A 0 O 180 B Protractor Postulate These rays can be paired with the real numbers from 0 to 180 in such a way that: If and OQ is paired with y, OP is paired with x then m POQ x y 100 150 = 50 P Example 100 Q AB AB 150 OB A 0 O 180 B Example 2 70 120 C Top Scale T A m CAT m CAT 110 60 or 60 110 500 Bottom Scale 70 120 or 120 70 500 Angle Addition Postulate A B O C If point B lies in the interior of AOC then m AOB m BOC m AOC And Angle Addition Postulate B A O C If AOC is a straight angle and B is any point not on AC then m AOB m BOC 180 Note: The angle addition postulate is just like the segment addition postulate. When the two angles form a straight line then they are called linear pairs. Euclid referred to this concept as … “The sum of the parts equals the whole.” Angle Addition Applications A B 310 O 220 C m AOB 530 Example 2 A 5x +13 m AOB 4 x 1 B m BOC 22 4x +1 O m AOC 5 x 13 220 C 4x +1 +22 = 5x +13 4x +23 = 5x +13 10 = x Find the values of the angles. m AOB 4(10) 1 m AOB 41 m AOC 5(10) 13 Substitute back into expressions. m AOC 63 Summary Angles are 2 rays with a common end point. There are 4 types of angles: Acute – less than 900 F N C T O P Right = 900 A R U T Obtuse – between 900 and 1800 Straight = 1800 A T Summary 2 Angles can be indicated by numbers, the vertex, or by 3 letter of which the middle letter is the vertex. Angles are measured with a protractor. The Protractor Postulate establishes measuring angles with a protractor. The Angle Addition Postulate establishes the sum of two adjacent angles is indeed the sum of the two angles. C’est fini. Good day and good luck.