1. What is an Angle?
- Definition: An angle is formed when two rays meet at a point.
- Representation: Often shown as an arc between the two rays.
- Measurement Unit: Degrees ((°)) or sometimes radians.
- Full rotation: (360°).
- Half-circle: (180°).
- Quarter-circle: (90°) (a right angle).
2. Properties of Angles
- Angles on a straight line: Always sum to (180°).
- Angles around a point: Always sum to (360°).
3. Types of Angles
- Acute Angle: Less than (90°).
- Right Angle: Exactly (90°).
- Obtuse Angle: Greater than (90°) but less than (180°).
- Straight Angle: Exactly (180°).
- Reflex Angle: Greater than (180°) but less than (360°).
- Complete Rotation: (360°).
Example: A clock's hands form different angles throughout the day.
4. Opposite Angles: Intersecting Lines
- When two lines intersect:
- Opposite angles are equal.
- Adjacent angles sum to (180°).
5. Parallel Lines and Transversals
When a transversal intersects parallel lines:
- Corresponding Angles: Equal (e.g., (a = a)).
- Alternate Angles: Equal (e.g., (c = c)).
- Co-interior Angles: Sum to (180°).
Special Shapes:
- Z-angles: Formed by alternate angles.
- F-angles: Formed by corresponding angles.
6. Measuring Angles
- Tool: A protractor is commonly used.
- Steps:
- Align the protractor’s center with the angle’s vertex.
- Ensure one ray aligns with the base line.
- Read the correct scale based on the angle's size.
Example:
- Triangle with angles: (90°), (45°), and (45°) sums to (180°).
7. Degrees vs. Radians
- Degrees: Most common unit for measuring angles.
- Radians: Used in advanced math and science.
- (2\pi) radians = (360°).
- (1) radian = approximately (57.3°).
8. Applications
- Angles are fundamental in geometry, trigonometry, and real-world problem-solving (e.g., navigation, architecture, engineering).
Tip: Always double-check measurements when working with angles for precision!