Problem 1 - Algebra and Trigonometry

Question

In Exercises \(1-6,\) the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. $$ 135^{\circ} $$

Step-by-Step Solution

Answer

The given angle \(135^{\circ}\) is an obtuse angle.

Step 1: Recall angle classifications

An angle is:

• Acute if \(0° < \theta < 90°\)
• Right if \(\theta = 90°\)
• Obtuse if \(90° < \theta < 180°\)
• Straight if \(\theta = 180°\)

Step 2: Classify 135°

Since \(90° < 135° < 180°\), the angle is \(\textbf{obtuse}\).

Key Concepts

Acute Angle

An acute angle is a type of angle that is always less than 90 degrees. If you picture an angle like an open pair of scissors, an acute angle is when the scissors are almost closed.
Acute angles are sharp and small. They are typically found in various shapes like triangles where they are used in geometry to denote angles that are less than a right angle.

• An acute angle can range from just over 0 degrees to just under 90 degrees.
• Common examples include 30 degrees, 45 degrees, and 60 degrees.

If you find yourself drawing a triangle, the triangle can be classified as an acute triangle if all three internal angles are acute angles. Understanding acute angles helps in identifying parts of a shape that are sharp or pointed. It's a handy concept when exploring angles in everyday situations and objects.

Right Angle

A right angle is exactly 90 degrees. Imagine a corner of a square or rectangle where two lines meet to form a perfect "L" shape. This is a right angle!
Right angles are quite common and easy to spot. They are typically associated with perpendicular lines where one line stands "straight up" compared to the other.

• A right angle creates two 90-degree angles which sum up to 180 degrees.
• An example in real life is the edge of a book or the corner of a room.

In mathematics, right angles are essential in defining perpendicularity and are heavily utilized in constructing squares, rectangles, and even in trigonometric calculations. Whenever angles sum to create a half-turn or full turn, right angles are likely at work, ensuring geometric stability.

Obtuse Angle

An obtuse angle is bigger than a right angle but less than a straight angle. Specifically, it ranges from more than 90 degrees but less than 180 degrees.
If you open those scissors more than a right angle but not flat, you've got an obtuse angle!

• For example, angles like 120 degrees or 135 degrees are obtuse.
• These angles appear frequently in various objects such as a fan's blades or certain designs like star shapes.

Obtuse angles add diversity to shapes, creating energy and dynamics in their forms. They are characterized by their broad, extended appearance. Recognizing obtuse angles allows for a deeper understanding of designs and structures and helps classify shapes like scalene triangles which often have at least one obtuse angle.

Straight Angle

A straight angle has a measure of exactly 180 degrees. It resembles a straight line, hence its name.
If you envision a straight angle, think of a flat horizon or an open door lying flat.

• It is neither an angle that turns nor bends; it's one continuous line.
• Examples of straight angles can be seen in foldable chairs when they are completely opened or the line you see when you look at an unfolded sheet of paper.

In mathematical terms, a straight angle is significant because it marks a half-turn around a point. Understanding straight angles is crucial as they form the basis for linearity concepts and help in the study of geometry involving line segments and their relationships.