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Understanding Angles and Arcs: Key Concepts and Calculations in Geometry

Dive into the essential concepts of angles and arcs in geometry with this comprehensive lesson. Learn about different types of angles such as straight, right, acute, obtuse, and co-terminal angles. Discover how to measure angles in both degrees and radians, including conversions between the two. Explore the relationships of complementary and supplementary angles and how to calculate arc length based on radius and central angle. Additionally, understand linear and angular speed in the context of rotating objects. Engage in exercises that reinforce these fundamental concepts.

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Understanding Angles and Arcs: Key Concepts and Calculations in Geometry

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  1. Angles and Arcs Lesson 2.1

  2. Angles Extends indefinitely • Ray • Angle • Formed of two rays • Have same endpoint Endpoint

  3. less than 90° greater than 90° Angles • Angles can be • Straight • Right • Acute • Obtuse • Obese View NSpireDemo

  4. Angles • Positive Angles • Measured counter-clockwise • Negative Angles • Measured clockwise

  5. Angles • Degree = of a full revolution • Complementary angles • a + b = 90° • Supplementary angles • a + b = 180°

  6. Co-terminal Angles • We measure angles in standard position on the coordinate axes • This same angle could be θ

  7. Radian Angle Measure • Consider the length of the radius laid out on the circumference • The resulting angle θ, is 1 radian • There are radians in a circle r θ r View NSpireDemo

  8. Radian Angle Measure • Conversion of radians to degrees • Use this proportion • Solve for desiredvalue (degrees orradians) • Try it: • is how many degrees? • 135° is how many radians?

  9. Arc Length S • Given a circle with radius r and central angle = θ radians • The arc length can be calculated by • Try it: For a circle with radius 3.5 and θ = what is the arc length? θ r View NSpireDemo

  10. Linear and Angular Speed • Angular speed is radians/unit time • Linear speed is linear units / unit time

  11. Linear and Angular Speed • If ω is the angular velocity of a round object with radius r • The linear velocity of apoint on the circle is • For a 20 ft radius windmill rotating at 10 rpm, what is the linear velocity of the tip of the blade?

  12. Assignment • Lesson 2.1 • Page 130 • Exercises 1 – 69 EOO71,79,83

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