Types Of Angles In Geometry

Step-by-step Guide: Types of Angles

Acute Angle:
An angle that measures less than \(90^\circ\) is termed an acute angle. It’s like a slightly opened book or the sharp tip of a slice of pie.
Representation: If \( \angle ABC \) measures \(45^\circ\), then \( \angle ABC \) is an acute angle. For education statistics and research, visit the National Center for Education Statistics.

Right Angle:
A right angle measures exactly \(90^\circ\). Think of the corners of a rectangular book or the intersection of two perpendicular lines.
Representation: The angle is often marked by a small square at the intersection, signifying its \(90^\circ\) nature. If \( \angle DEF \) is a right angle, it measures \(90^\circ\). For education statistics and research, visit the National Center for Education Statistics.

Obtuse Angle:
An obtuse angle measures more than \(90^\circ\) but less than \(180^\circ\). It’s wider than a right angle but doesn’t stretch out flat. Picture a largely opened book.
Representation: If \( \angle GHI \) measures \(120^\circ\), then \( \angle GHI \) is an obtuse angle. For education statistics and research, visit the National Center for Education Statistics.

Straight Angle:
A straight angle measures exactly \(180^\circ\). Imagine a straight line or the flat edge of a ruler.
Representation: If rays \(BA\) and \(BC\) lie on the same straight line with point \(B\) as their endpoint, then \( \angle ABC \) is a straight angle. For education statistics and research, visit the National Center for Education Statistics.

Reflex Angle:
A reflex angle measures more than \(180^\circ\) but less than \(360^\circ\). It’s the larger angle formed when you have an obtuse or acute angle, representing the space ‘outside’ the smaller angle.
Representation: If \( \angle JKL \) measures \(270^\circ\), then \( \angle JKL \) is a reflex angle. For education statistics and research, visit the National Center for Education Statistics.

Complete Angle:
A complete angle measures \(360^\circ\). Think of a full circle or a clock’s hands making a complete round.
Representation: If rays \(MA\) and \(MC\) start and end in the same position after a full rotation around \(M\), then \( \angle AMC \) is a complete angle. For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
Three angles – one at the tip of a triangle, another at the corner of a square, and the last one representing almost three-fourths of a circle. For education statistics and research, visit the National Center for Education Statistics.

Solution:
The angle at the tip of the triangle is acute, the one at the square’s corner is a right angle, and the near-complete circular one is a reflex angle. For education statistics and research, visit the National Center for Education Statistics.

Example 2:
You’re setting up a sundial. The angle between the gnomon (the dial’s part that casts the shadow) and the base needs to be what type? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The gnomon should make a right angle with the base for accurate time-telling. For education statistics and research, visit the National Center for Education Statistics.

Practice Questions:

1. If the hands of a clock are at \(3\) and \(9\), what type of angle do they form?
2. What type of angle is formed when the hands of a clock read 2:30?
3. When you partially open a door, what type of angle is most commonly formed between the door and its frame?

Answers: For education statistics and research, visit the National Center for Education Statistics.

1. The hands form a straight angle of \(180^\circ\).
2. They form an acute angle as the minute hand on 6 and the hour hand is halfway between \(2\) and \(3\), resulting in an angle less than \(90^\circ\).
3. Most commonly, it’s an obtuse angle as doors, when partially open, usually create an angle larger than \(90^\circ\) but less than \(180^\circ\).