Problem 15 - A Graphical Approach to Precalculus with Limits

Question

An angle measures \(x\) degrees. (a) What is the measure of its complement? (b) What is the measure of its supplement?

Step-by-Step Solution

Answer

(a) Complement is \(90 - x\) degrees, (b) Supplement is \(180 - x\) degrees.

Step 1: Understanding Complements

The complement of an angle is found by subtracting the angle from 90 degrees. This is because complementary angles add up to 90 degrees.

Step 2: Calculate the Complement

To find the complement of an angle measuring \(x\) degrees, use the formula: \[90 - x\] This expression gives the measure of the complementary angle.

Step 3: Understanding Supplements

The supplement of an angle is found by subtracting the angle from 180 degrees. This is because supplementary angles add up to 180 degrees.

Step 4: Calculate the Supplement

To find the supplement of an angle measuring \(x\) degrees, use the formula: \[180 - x\] This expression gives the measure of the supplementary angle.

Key Concepts

complementary angles

Complementary angles are a fundamental concept in geometry. These angles are pairs that add up to 90 degrees. Think of them as two angles that together form a right angle. For instance, if you have one angle measuring 30 degrees, its complement will measure 60 degrees, because 30 + 60 = 90.
This relationship is important in various applications, such as determining unknown angles in geometric figures.

• To find a complementary angle, subtract the given angle from 90 degrees.
• If an angle is represented by \(x\) degrees, its complementary angle is \(90 - x\).
• Understanding complementary angles can help improve problem-solving skills in geometry.

supplementary angles

Supplementary angles are pairs of angles whose measures add up to 180 degrees. Imagine them as angles that form a straight line when combined. If one angle measures 110 degrees, the supplementary angle measures 70 degrees because 110 + 70 = 180.
This concept is crucial in both geometry and trigonometry, especially when dealing with linear pairs and parallel lines.

• To find a supplementary angle, subtract the given angle from 180 degrees.
• For an angle represented by \(x\), its supplementary angle is \(180 - x\).
• Grasping the idea of supplementary angles assists with understanding various geometric configurations.

angle measurement

Angle measurement refers to the size of an angle, typically expressed in degrees.
When measuring angles, it helps to understand the basic units and relationships that define them. The degree is the most common unit, where a full circle is 360 degrees. Understanding this can aid in grasping more advanced concepts like radians and even angular velocity in physics.

• A right angle is 90 degrees, a straight angle is 180 degrees, and a full circle measures 360 degrees.
• Knowing how to measure angles precisely is essential for solving geometric problems and equations.
• Proficiency in angle measurement also aids in various real-world applications, like construction and navigation.