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3-1 Arcs and Angles

3-1 Arcs and Angles. Every angle has a measure that is a number in the interval from 0 to 180, the number 180 being half the measure in degrees of a full circle. C. R. L. C. R. L. 3-1 Arcs and Angles. Consider. 360 o arc. 180 o arc. E. 90 o arc. 90 o arc. 90 o angle. C. A. F.

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3-1 Arcs and Angles

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  1. 3-1 Arcs and Angles • Every angle has a measure that is a number in the interval from 0 to 180, the number 180 being half the measure in degrees of a full circle.

  2. C R L

  3. C R L

  4. 3-1 Arcs and Angles • Consider 360o arc

  5. 180o arc E 90o arc 90o arc 90o angle C A F minor arcs D B O X W G 120o arc 180o arc Y Z major arc 3-1 Arcs and Angles

  6. 3-1 Arcs and Angles • An angle is the union of two rays that have the same endpoint. • The sides of the angle are the two rays. • The intersection of the two rays is the vertex of the angle.

  7. 3-1 Arcs and Angles • If the vertex of the angle is the center of a circle, then the angle is a central angle of the circle. • The portion of the circle inscribed (cut off) by the central angle is called an arc.

  8. 3-1 Arcs and Angles • is a straight angle. Its measure is 180o. • is a zero angle. Its measure is 0o. C B A

  9. 3-1 Arcs and Angles • Angles are measures in a counter-clockwise direction. • is measured counter-clockwise. Its measure is 45o. • is measured clockwise. Its measure is -45o. C B A

  10. 3-1 Arcs and Angles • Angle Measure Postulate • Unique Measure Assumption • Every angle has a unique measure from 0o to 180o. • Unique Angle Assumption • Given any ray VB and a real number r between 0 and 180, there is a unique angle BVA on each side of VB such that • Straight Angle Assumption • If VA and VB are opposite rays, then • Zero Angle Assumption • Is VA and VB are the same ray, then

  11. 3-1 Arcs and Angles • If m is the measure of an angle, then the angle is: • a zero angle if and only if m = 0; • an acute angle if and only if 0 < m < 90; • a right angle if and only if m = 90; • an obtuse angle if and only if 90 < m < 180; • a straight angle if and only if m = 180.

  12. 3-1 Arcs and Angles • You CAN assume these from a figure: • Collinearity and betweenness of points drawn on the same line; • Intersections of lines at a given point; • Points in the interior of an angle, on an angle, or in the exterior of an angle.

  13. 3-1 Arcs and Angles • You CANNOT assume these from a figure: • Collinearity of three or more points that are not drawn on the same line; • Parallel lines; • Exact measures of angles of segments.

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