Problem 24 - Prealgebra

Question

Classify each angle as acute, obtuse, right, or straight. $$155^{\circ}$$

Step-by-Step Solution

Answer

The angle \(155^{\circ}\) is an obtuse angle.

Step 1: Understanding Angle Types

An acute angle is less than \(90^{\circ}\), a right angle is exactly \(90^{\circ}\), an obtuse angle is between \(90^{\circ}\) and \(180^{\circ}\), and a straight angle is exactly \(180^{\circ}\).

Step 2: Comparing to Angle Ranges

Check which category \(155^{\circ}\) falls under. Since \(90^{\circ} < 155^{\circ} < 180^{\circ}\), \(155^{\circ}\) is greater than a right angle but less than a straight angle.

Step 3: Classifying the Angle

Since \(155^{\circ}\) falls between \(90^{\circ}\) and \(180^{\circ}\), it is classified as an obtuse angle.

Key Concepts

Acute Angles

Acute angles are some of the smallest angles you'll encounter in geometry. They are always less than 90 degrees, which means they appear narrow and pointed.

• An angle like 45 degrees is a good illustration of an acute angle.
• It's sharper than half of a right angle.

When measuring angles, any degree amount less than 90 is considered acute. These angles often appear in various geometric shapes like triangles, specifically known as acute triangles where all three internal angles are acute.

Obtuse Angles

Obtuse angles span a broader range of measurements compared to acute angles. They measure between 90 and 180 degrees, so they look more spread out and open than a right angle.

• An example of an obtuse angle is 155 degrees.
• It's wider than a right angle but not quite a straight line.

Obtuse angles are found in various structures and are very common in everyday objects, such as the hands of a clock at certain times. Always remember, for an angle to be obtuse, it must be larger than 90 degrees but less than 180 degrees.

Right Angles

Right angles are one of the simplest and most important types of angles in geometry. They measure exactly 90 degrees and can be observed in many everyday objects, such as the corners of a book or a computer screen.

• They are easy to recognize because they form a perfect L shape.
• Right angles are a key feature in rectangles and squares, which have four right angles each.

Due to their precise nature, right angles are often used for ensuring perpendicularity in construction and design. Understanding right angles is crucial for both practical applications and theoretical geometry.

Straight Angles

Straight angles form a straight line and measure exactly 180 degrees. This type of angle doesn’t really look like an angle because it’s essentially a line.

• Think of it as a completely unfolded angle where the two rays point in directly opposite directions.
• It splits a circle into two equal halves.

Straight angles are fundamental in mathematics as they help with understanding linear relationships and transformations. You often see straight angles when observing horizons or in architecture where beams may align in a straight path. Knowing how straight angles work allows one to appreciate the beauty of symmetry and alignment in design and nature.