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Section 14-1

Section 14-1. Angles and Their Measures. Angles and Their Measures. Basic Terminology Degree Measure Angles in a Coordinate System. Basic Terminology.

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Section 14-1

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  1. Section 14-1 • Angles and Their Measures

  2. Angles and Their Measures • Basic Terminology • Degree Measure • Angles in a Coordinate System

  3. Basic Terminology A line may be drawn through the two distinct points A and B. This line is called line AB. The portion of the line between A and B, including points A and B themselves, is segment AB. The portion of the line AB that starts at A and continues through B, and on past B, is called ray AB. Point A is the endpoint of the ray. A B A B A B Line AB Segment AB Ray AB

  4. Basic Terminology An angle is formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle, while the ray in its location after rotation is the terminal side of the angle. The endpoint of the ray is the vertex of the angle. Terminal side Vertex A Initial side

  5. Basic Terminology If the rotation of the terminal side is counterclockwise, the angle measure is positive. If the rotation of the terminal side is clockwise, the angle measure is negative. Negative angle Positive angle

  6. Basic Terminology An angle can be named by using the name of its vertex. Alternatively, an angle can be named using three letters, with the vertex in the middle. A C B Name: angle C, angle ABC, or angle BCA.

  7. Degree Measure The most common unit of measure for angles is the degree. (The other common unit of measure is called the radian.) We assign 360 degrees to a complete rotation of a ray. 360°

  8. Degree Measure One degree, written 1°, represents 1/360 of a rotation. Therefore, 90° represents 1/4 of a rotation, and 180° represents 1/2 of a rotation 1° angle 90° 180°

  9. Special Angles 70° 90° 135° 180°

  10. Complement and Supplement If the sum of the measures of two angles is 90°, the angles are called complementary. Two angles with measures whose sum is 180° are supplementary.

  11. Example: Finding Complement and Supplement Give the complement and supplement of 60°. Solution The complement of 60° is 90° – 60° = 30°. The supplement of 60° is 180° – 60° = 120°.

  12. Angle Measurement Portions of a degree have been measured with minutes and seconds. One minute, written One second, written

  13. Angle Measurement The measure represents 42 degrees, 13 minutes, 24 seconds. Angles can be measured in decimal degrees. For example 12.345° represents

  14. Example: Calculating with Degree Measure Perform each calculation. Solution

  15. Example: Converting Degrees, Minutes, and Seconds to Decimal Degrees Convert to decimal degrees. Round to the nearest thousandth of a degree. Solution

  16. Example: Converting Degrees, Minutes, and Seconds to Decimal Degrees Convert to degrees, minutes and seconds. Round to the nearest second. Solution

  17. Angles in a Coordinate System An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis. Terminal side 0 0 Vertex Initial side

  18. Angles in a Coordinate System An angle in standard position is said to lie in the quadrant in which its terminal side lies. Angles in standard position having their terminal side along the x-axis or y-axis (90°, 180°, 270°, …) are called quadrantal angles. 90° Q II Q I 0° 360° 180° Q III Q IV 270°

  19. Coterminal Angles Two angles can have the same initial side and same terminal side but different amounts of rotation. Angles that have the same initial side and same terminal side are called coterminal angles. 410° The angles 50° and 410° shown are coterminal angles. 50° 0

  20. Example: Finding Measures of Coterminal Angles Find the angle of smallest possible positive measure coterminal with each angle a) 770° b) –88° Solution a) 770° – 2(360°) = 50° b) –88° + 360° = 272°

  21. Generating Coterminal Angles Sometimes it may be necessary to find an expression that will generate all angles coterminal with a given angle. Coterminal angles can be represented by adding integer multiples of 360° to the angle. For example, for an angle measure of 50°, we let n represent any integer and then the expression 50° + n(360°) will represent all the coterminal angles.

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