∠Unit 3 · GeometryAngles & Relationships

Lesson One · Part 1

Concept of an Angle and Types of Angles

Before comparing angles we must agree on what an angle is, how it is named, and how it is measured. Then every angle can be sorted into one of six families by its measure.

Concept of an Angle

An angle is the union of two rays with a common starting point.

  • • The common starting point of the two rays is called the vertex of the angle.
  • • Each ray is called a side of the angle.
BA ∪ BC = ∠ABCIt may be named: ∠ABC, ∠CBA, ∠B or ∠1 — the vertex letter always sits in the middle.
BAC∠ABC

Naming rule

Use the three-letter name (∠ABC) whenever several angles share the same vertex. The single-letter name (∠B) is only safe when exactly one angle is drawn at that vertex.

Angle Measure

A protractor is used to measure an angle, and angles are measured in degrees. For the figure opposite we write:

m (∠ABC) = 50°

Degrees are divided into smaller parts: minutes ( ′ ) and seconds ( ″ ).

1° = 60′
1′ = 60″
04590135180m (∠ABC) = 50°

Worked conversion

89° 60′ = 89° + 1° = 90°, so it is a right angle. Always convert minutes to degrees before you compare two measures.

Types of Angles according to their Measurements

1

Zero Angle

BAC0°

An angle that measures 0°, where the two sides coincide with each other.

2

Acute Angle

BAC0° < m < 90°

An angle that measures more than 0° but less than 90°.

3

Right Angle

BACm = 90°

An angle that measures exactly 90°. Its two sides are perpendicular.

4

Obtuse Angle

BAC90° < m < 180°

An angle that measures more than 90° but less than 180°.

5

Straight Angle

BACm = 180°

An angle that measures exactly 180°; its sides form a straight line in opposite directions.

6

Reflex Angle

BAC180° < m < 360°

An angle that measures more than 180° but less than 360°.

Note — the full circle

A full circle measures 360°, so for any angle ∠M: m(∠M) + m(reflex ∠M) = 360°

Example: if m(∠ABC) = 130°, then m(reflex ∠ABC) = 360° − 130° = 230°.

Try it yourself 1 — complete the table (answers shown in the second row)
m (∠ABC)135°85°165°100°89°52.5°89° 60′
m (reflex ∠ABC)225°275°195°260°271°307.5°270°

Ready to test this? The classify-the-angle game draws a random angle and asks you to name its type.