Problem 86 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Two complementary angles can be equal in measure.

Step-by-Step Solution

Answer

The statement 'Two complementary angles can be equal in measure' is true. They can be equal in measure (if each measures 45 degrees) or different (so long as they still add up to 90 degrees), as the definition of complementary angles is two angles that altogether complete a right angle, which measures 90 degrees.

Step 1: Analyze the Definition of Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. That is if one angle is \(a\) and the other angle is \(b\), then \(a + b = 90^\circ\). It's important to clarify that the two angles do not have to be the same.

Step 2: Determine if Complementary Angles can be Equal

If two angles are each 45 degrees then they add up to 90 degrees and are indeed complementary. In this case, they can be equal. However, it is not a requirement for complementary angles to be equal. The requirement for two angles to be complementary is that their measures add up to 90 degrees.

Step 3: Correct the Statement if needed

The statement 'Two complementary angles can be equal in measure' is already true, therefore no corrections are necessary.

Key Concepts

Angle Measurement

Angles are our way of describing a turn or rotation. When we measure an angle, we're looking at the amount of turn between two straight lines that have a common end point called the vertex. The standard unit for measuring angles is degrees.

• Full rotation makes 360 degrees.
• A right angle, which is a quarter of a full rotation, is 90 degrees.
• Complementary angles together make 90 degrees.

The size of an angle is determined by comparing it to a right angle. When we talk about complementary angles, each pair will combine to make a right angle. This is a key idea in the problem we're examining. Understanding how these measurements work helps us see why two equal angles that each measure 45 degrees can indeed be complementary.

True or False Statements

True or false statements are questions where you need to decide if something matches reality. To evaluate a statement, we compare it to what we know from definitions or scientific facts. For our problem, let's look at the statement: "Two complementary angles can be equal in measure." This involves applying what we know about angles:

• If we know complementary angles sum to 90 degrees, what happens?
• Well, if two angles are equal, they can both be 45 degrees, for instance, which indeed sums up to 90 degrees.

Thus, the statement is true. It's crucial to carefully break down the statement against the rules or definitions you already understand. In this case, understanding that 45+45 equals 90 quickly validates the statement as true.

Definition of Angles

The definition of angles is fundamental for understanding their properties and relationships, like those in complementary angles. An angle is formed where two lines meet at a point, called the vertex. Here's some important vocabulary surrounding angles:

• Vertex: The point where the two lines intersect.
• Arms: The two lines making up the angle.
• Measurement: Usually given in degrees, describing how open the angle is.

When considering complementary angles, the focus is particularly on their relationship where they add up to a 90-degree angle. This helps us in solving problems where we're asked to determine conditions like equality or complementary status.