Problem 23 - Prealgebra

Question

Classify each angle as acute, obtuse, right, or straight. $$110^{\circ}$$

Step-by-Step Solution

Answer

The angle \(110^{ ext{°}}\) is an obtuse angle.

Step 1: Understand the Types of Angles

To classify angles, let's first understand the categories:- **Acute Angle:** An angle less than \(90^{ ext{°}}\).- **Right Angle:** An angle exactly \(90^{ ext{°}}\).- **Obtuse Angle:** An angle greater than \(90^{ ext{°}}\) but less than \(180^{ ext{°}}\).- **Straight Angle:** An angle exactly \(180^{ ext{°}}\).

Step 2: Identify the Given Angle

In this exercise, the given angle is \(110^{ ext{°}}\). We need to determine which category this angle falls into based on its measure.

Step 3: Compare the Angle to Definitions

To classify \(110^{ ext{°}}\), compare it with the angle definitions:- \(110^{ ext{°}}\) is greater than \(90^{ ext{°}}\) but less than \(180^{ ext{°}}\).- Therefore, \(110^{ ext{°}}\) is an obtuse angle because it meets the criteria for being greater than \(90^{ ext{°}}\) but less than \(180^{ ext{°}}\).

Key Concepts

Acute Angle

An acute angle is a charming type of angle defined by its measure of being less than 90 degrees. While it may be small, it plays a significant role in various geometric contexts. Think of it as the angle of choice when something needs to be sharp or concise without being overly steep or flat.

• An acute angle is always less than a right angle.
• Examples include angles like 30°, 45°, and 60°.

These angles are commonly seen in triangles, particularly in equilateral and right-angled triangles, where they complement other angles. Whenever you hear 'acute,' think sharp and small, just like the word sounds.

Obtuse Angle

An obtuse angle is broader and more expansive, being larger than a right angle. It measures greater than 90 degrees but less than 180 degrees. This gives obtuse angles a more open appearance, often suggesting a relaxed and wide-opening.

• Obtuse angles are always greater than acute angles.
• Examples include angles like 110°, 120°, and 150°.

This angle type is less common in regular polygons but incredibly useful in design and aesthetics where a wide spread or embrace is needed. In art and architecture, these angles can create striking and elegant features.

Right Angle

The right angle is iconic with its exact measure of 90 degrees. It is the hallmark of perpendicular intersections, forming perfectly square corners. When anything is at a right angle, it signifies uprightness and stability.

• Right angles are found in many everyday objects like books, doors, and windows.
• It is a cornerstone in geometry, often used as a reference to ensure accuracy in constructions.

In mathematics, the right angle marks a point of balance between smaller acute angles and larger obtuse or straight angles. It is as if it stands guard, reminding us of precision and balance in our calculations and designs.

Straight Angle

A straight angle stands apart with its measure of exactly 180 degrees. It is the angle that creates a line, representing a perfect and unchanging direction. Think of a straight angle as the seamless flow from one point to another along a flat line.

• Straight angles divide a plane into two equal halves.
• They are the result of aligning two rays in opposite directions.

In daily life, straight angles communicate continuation or progression, and in geometry, they are foundational in understanding parallel lines and transformations. Whenever you look at a flat door or a level horizon, you are witnessing the essence of a straight angle.