Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rotation formalisms in three dimensions wikipedia , lookup
Integer triangle wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Line (geometry) wikipedia , lookup
Multilateration wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Perceived visual angle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Math with Measurements – Angle I Angles are seen in many figures. To a mathematician, an angle is a figure formed by two rays with the same endpoint. The endpoint is called the vertex of the angle. This angle is named as ∠ B (angle B) or ∠ ABC (with the vertex named in the middle). Angles are measured in degrees. The symbol for degrees is a little circle °. Angle Facts: The FULL CIRCLE is 360° (360 degrees). A half circle or a straight angle is 180°. A fourth of a circle or a right angle is 90°. A special instrument, called a protractor, is used to measure an angle. The instrument is half a circle, marked with 180 divisions that are called degrees. To measure an angle, place the center of the base of the protractor on the vertex of the angle and line up the base with one of the sides of the angle. Then read the number where the other side of the angle intersects the marked circle of the protractor. Below you see a picture of how to measure an angle with a protractor. Take care to read from the right set of numbers. A protractor has two sets of numbers: one set goes from 0 to 180, the other set from 180 to 0. Which one you read depends on how you place the protractor: place it so that one side of the angle lines up with one of the zeros, and read that set of numbers. In the examples above we lined up the one side of the angle with the zero of the lower set of numbers, so we need to read the lower set of numbers. Practice 1 1. Measure the angles. 35 72 18 50 Angles are classified according to their measures. Acute Measures less than 90° Right Measures 90° Obtuse Measures greater than 90°, but less than 180° Straight Measures 180° Special Angles: Two angles are complementary if the sum of their measures is 90°. Two angles are supplementary if the sum of their measures is 180°. These two angles (40° and 50°) are Complementary Angles, because they add up to 90° (a right angle). These two angles (140° and 40°) are Supplementary Angles, because they add up to 180° (a straight angle). "C" of Complementary stands for "Corner" (a Right Angle), and "S" of Supplementary stands for "Straight" (180 degrees is a straight line) 2 Additional angle facts The angles in a triangle add up to 180o A straight line is 180o Parallel lines have equal angles The last fact may require a picture. Notice the parallel lines create two groups of angles that are all the same. If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. corresponding angles = = Vertical Angles are the angles opposite each other when two lines cross. For example, angles 2 and 3 is a pair of vertical angles. Symbol for minutes: (') This is similar to the symbol for feet. Symbol for seconds: (") This is similar to the symbol for inches. The number of degrees of arc in a circle is 360. The sum of the measures in degrees of the internal angles of a triangle is 180. Two lines that intersect to form right angles are perpendicular. The symbol means is perpendicular to. Two lines in the same plane that never intersect are parallel. The symbol means is parallel to. 3 2. In this figure, PQ and RS are parallel (PQ||RS). Of the following, which pair of angles has the sum of 180°? a. ∠5 and ∠7 b. ∠3 and ∠6 c. ∠1 and ∠5 d. ∠1 and ∠7 e. ∠ 2 and ∠8 3. In the left figure above, x = 40 and y = 60, what is the value of z? 80 (180 – 40 – 60 = 80) 4. What is the measure of ∠ABD? ∠ABD = 180 – 145 = 35 5. What is the measure of the missing interior angle for the figure shown below? N = 120 4 6. What is the measure of ∠QON? ∠QON = 90 – 64 = 26 7. AB and CD are straight lines. Find the unknown angles. ∠AOC = 152 ∠DOB = 152 ∠COB = 28 8. If the measure of ∠3 is 80°, which of the following is NOT true? A. ∠4 measures 100° B. ∠7 measures 80° C. ∠5 measures 100° D. ∠1 measures 80° 5