Types Of Angles In Geometry

Step-by-step Guide: Types of Angles

Acute Angle:
An angle that measures less than \(90^\circ\) is termed as 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.

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 \(90^\circ\).

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.

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.

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.

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.

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.

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.

Example 2:
You’re setting up a sun dial. The angle between the gnomon (the dial’s part that casts the shadow) and the base needs to be what type?

Solution:
The gnomon should make a right angle with the base for accurate time-telling.

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:

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 \(180^\circ\).