Angles | Geometry Learning

∠What is an Angle?

Definition:

An angle is formed when two rays share a common endpoint. The common endpoint is called the vertex, and the two rays are called the sides of the angle.

Parts of an Angle:

•Vertex: The common endpoint where the two rays meet
•Sides: The two rays that form the angle
•Interior: The space between the two rays

Angle Measurement:

Angles are measured in degrees (°) using a protractor. The size of an angle depends on how much one ray has been rotated from the other, not on the length of the rays.

Key Point: The measure of an angle is independent of the length of its sides. Only the amount of rotation between the rays matters.

Types of Angles

📐

Acute Angle

Range: Greater than 0° and less than 90°

Description: Sharp, pointed angles that are smaller than a right angle.

Examples: Corner of a slice of pizza, hands of a clock at 1:00

⊥

Right Angle

Measure: Exactly 90°

Description: Forms a perfect corner, like the corner of a square.

Examples: Corner of a book, hands of a clock at 3:00 or 9:00

📏

Obtuse Angle

Range: Greater than 90° and less than 180°

Description: Wide, open angles that are larger than a right angle.

Examples: Opening of a book, hands of a clock at 4:00

—

Straight Angle

Measure: Exactly 180°

Description: Forms a straight line; the rays point in opposite directions.

Examples: Horizon line, hands of a clock at 6:00

🔄

Reflex Angle

Range: Greater than 180° and less than 360°

Description: Very wide angles that "turn back" on themselves.

Examples: Opening a door more than halfway, most clock positions

⭕

Complete Angle

Measure: Exactly 360°

Description: One complete rotation; the ray returns to its starting position.

Examples: Full spin, complete circle, one full day rotation

Measuring Angles

Using a Protractor:

Place the center: Position the center point of the protractor on the vertex of the angle
Align the baseline: Line up one side of the angle with the 0° line on the protractor
Read the scale: Find where the other side of the angle crosses the protractor scale
Choose the correct scale: Use the inner or outer scale depending on the angle's direction

Tips for Accurate Measurement:

• Make sure the vertex is exactly at the center point
• Choose the correct scale (inner or outer) based on the angle opening
• Extend the angle sides if they're too short to reach the scale
• Read the measurement carefully to the nearest degree

Drawing Angles

Steps to Draw an Angle:

Draw the first ray: Start with a ray (line with one endpoint)
Position the protractor: Place the center on the endpoint of your ray
Align with 0°: Make sure your ray lines up with the 0° mark
Mark the angle: Find the desired degree measure and make a point
Draw the second ray: Connect the endpoint to your marked point
Label the angle: Mark the vertex and, if needed, points on each ray

Remember: When reading the protractor, be careful to use the correct scale. If your angle opens to the right, use the scale that starts from 0° on the right.

✂️Angle Bisector

Definition:

An angle bisector is a ray that starts at the vertex of an angle and divides the angle into two equal parts.

Properties:

•Equal division: Creates two angles of equal measure
•Unique: Every angle has exactly one angle bisector
•Starts at vertex: The bisector ray begins at the angle's vertex

Example:

If ray OC is the angle bisector of ∠AOB, then ∠AOC = ∠BOC = ½∠AOB

Applications:

• Construction of equal angles
• Solving geometric problems
• Creating symmetrical designs
• Finding angle relationships

Quick Reference Table

Angle Type	Measure	Visual Description	Common Examples
Acute	0° < angle < 90°	Sharp, pointed	Pizza slice, early clock times
Right	angle = 90°	Perfect corner	Book corner, 3:00 on clock
Obtuse	90° < angle < 180°	Wide, open	Open book, 4:00 on clock
Straight	angle = 180°	Straight line	Horizon, 6:00 on clock
Reflex	180° < angle < 360°	Very wide	Open door, most clock positions
Complete	angle = 360°	Full rotation	Complete circle, full spin