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What is the measure of a complete angle after one rotation?
A right angle measures 180°.
False
What is the term for directed angles with the same initial and terminal rays?
Co-terminal angles
The angle in standard position has its initial ray along the positive ______ axis.
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Match the types of angles with their corresponding measures:
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Which of the following correctly describes directed angles?
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The measure of the angle AOB is the same regardless of whether it is measured clockwise or anticlockwise.
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What is the measure of angle ABC if it is drawn with a measure of 40°?
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In a directed angle, the ray _____ is called the initial arm.
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Match the following angle characteristics:
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What is the total number of degrees in one complete rotation?
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Co-terminal angles can have the same measure.
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What is the term used for an angle whose terminal ray lies along the x-axis or y-axis?
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The unit of measurement of angles in the sexagesimal system is called a ______.
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Match the following terms with their correct descriptions:
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How many degrees are there in one radian?
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One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
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What is the formula to convert degrees into radians?
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An angle measuring _____ radians corresponds to a straight angle.
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Which of the following radian measures corresponds to 45 degrees?
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Match the following degree measures with their corresponding radian measures:
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The conversion factor from radians to degrees is 180°/π.
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To convert 1 degree into minutes, you multiply the fractional part by _____ minutes.
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What is the formula to convert degrees into radians?
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1° is equivalent to $57.3248$ radians.
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What is the radian measure of a 90° angle?
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To convert radians into degrees, multiply the radian measure by ______.
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Which of the following is the correct conversion of 70° to radians?
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Match the degree measures with their corresponding radian measures:
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The term 'minute' in time measurement is the same as the 'minute' in angular measurement.
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1R in a minute hand of a clock equals ______ degrees.
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Study Notes
Angle Basics
- An angle is formed by two rays (OA and OB) with a common endpoint (O).
- Directed angles designate orientation based on rotation: anticlockwise (positive) or clockwise (negative).
Types of Angles
- Zero Angle: No rotation, or $m(\angle AOB) = 0°$.
- Complete Angle: One full rotation, $m(\angle AOB) = 360°$.
- Straight Angle: Rays in opposite directions, $m(\angle AOB) = 180°$.
- Right Angle: $m(\angle AOB) = 90°$, one-fourth of a complete angle.
Angle Measurements
- Measured in degrees (sexagesimal system) or radians (circular system).
- Degrees: One complete rotation is 360°. Co-terminal angles differ by multiples of 360°.
- Radians: Defined such that an arc length equal to the radius subtends a central angle of 1 radian.
Radian Relations
- $1 \text{ radian} = \frac{\pi}{180°}$.
- Proportions for converting degrees to radians and vice versa:
- Degrees to Radians: Multiply by $\frac{\pi}{180}$.
- Radians to Degrees: Multiply by $\frac{180}{\pi}$.
Angles in Different Quadrants
- Quadrantal angles lie on the x-axis or y-axis (multiples of 90°).
- A directed angle exists in a quadrant if its terminal ray does.
Key Conversions
- Using $\pi \approx 3.14$ to approximate angles in degrees.
- Example: $1° \approx 57.3248°$ in decimal fractions.
Clock Angles
- The rotation of clock hands is analogous to angular measurements:
- 1 full rotation = 360° (both minute and hour hands).
- 1 minute hand rotation = 6°; 1 hour hand rotation = 30°.
Conversion Examples
- 70° to radians: $\frac{70}{180}\pi = \frac{7\pi}{18}$.
- -120° to radians: $\frac{-120}{180}\pi = \frac{-2\pi}{3}$.
- $\frac{1}{4}°$ to radians: $\frac{1}{4} \times \frac{\pi}{180} = \frac{\pi}{720}$.
Arc and Sector Areas
- The length of an arc and the area of a sector relate to the central angle and the circle's radius, applying the angular measurements directly.
Application Activities
- Practice drawing directed angles and converting between degrees and radians for reinforcement of concepts.
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Description
This quiz covers the fundamentals of angle measurement, including directed angles, and various units of measurement. It also explores arc lengths and the area of a sector in a circle. You will reinforce your understanding through drawing activities and learning key properties.
