Chapter 3: Angles

Basics of Angles

Definition

• An angle is formed by two line segments from one center point.
• Example: If point O has two lines OA and OB, then OA and OB form an angle with vertex O.
• Represent angles using notation like ∠AOB or simply ∠A if unambiguous.

Measuring Angles

• Units: Degrees (°)
• A full circle is 360°.
• A right angle is 90°.
• Angles can be represented in a rotated manner (e.g., clockwise).
• Use a small 0 for degrees notation (e.g., 30°).
• Key partial measurements:
  • 1/4 circle = 90° (right angle, represented by a square symbol).
  • 1/2 circle = 180° (straight angle).
  • 3/4 circle = 270°.

Types of Angles

Acute Angle

• Between 0° and 90°.

Right Angle

• Exactly 90° (represented by a square).

Obtuse Angle

• Between 90° and 180°.

Straight Angle

• Exactly 180° (a straight line).

Reflex Angle

• Between 180° and 360°.
• Example: More than 270° but less than 360°.

Adjacent Angles

• Share a common arm and vertex.
• Example: ∠AOB and ∠BOC are adjacent if they share vertex O and arm OB.
• Properties:
  • Share a common arm.
  • Have a common vertex.
  • Located on opposite sides of the common arm.

Complementary Angles

• Two angles whose sum is 90°.
• Example: If one angle is 30°, its complement is 90° - 30° = 60°.
• Pairs: (60°, 30°), (50°, 40°), (85°, 5°), etc.

Supplementary Angles

• Two angles whose sum is 180°.
• Example: If one angle is 40°, its supplement is 180° - 40° = 140°.
• Pairs: (40°, 140°), (85°, 95°), (60°, 120°), etc.

Vertically Opposite Angles

• Formed when two straight lines intersect.
• Vertically opposite angles are always equal.
• Example: If angle AOC is 80°, then angle BOD must also be 80°.

Angles Around a Point

• Sum of angles around a point is 360°.
• Example: If angles around point O are X, Y, and Z, then X + Y + Z = 360°.

Problems and Examples

Finding Values of Missing Angles

• Use the properties of angle sum (e.g., straight angle = 180°, around a point = 360°).
• Calculate complementary and supplementary angles by subtraction.
• Use vertically opposite angles to find equal pairs.
• Identify adjacent angles and check their sum.

Example Problems

• Find angle values with given relationships (e.g., given one angle, find its complementary or supplementary pairs).
• Apply properties of vertically opposite angles and around a point sums to solve geometric problems.

SUMMARY

•If sum of a pair of acute angles add up to 90° it is a complementary angle .