Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Fill in the blanks with the appropriate short answers. Do not use a calculator. The complement of a \(40^{\circ}\) angle is ___ and the supplement of a \(40^{\circ}\) angle is___.

Step-by-Step Solution

Answer

50°; 140°

Step 1: Understanding Complements

To find the complement of an angle, we use the fact that two angles are complementary if their measures add up to 90 degrees. Therefore, if one angle is 40 degrees, the complement can be found by subtracting 40 degrees from 90 degrees.

Step 2: Calculating the Complement

Subtract the given angle from 90 degrees. For a 40-degree angle, the calculation is: \[90^{\circ} - 40^{\circ} = 50^{\circ}\]Thus, the complement of a 40-degree angle is 50 degrees.

Step 3: Understanding Supplements

To find the supplement of an angle, we use the fact that two angles are supplementary if their measures add up to 180 degrees. Therefore, if one angle is 40 degrees, the supplement can be found by subtracting 40 degrees from 180 degrees.

Step 4: Calculating the Supplement

Subtract the given angle from 180 degrees. For a 40-degree angle, the calculation is: \[180^{\circ} - 40^{\circ} = 140^{\circ}\]Thus, the supplement of a 40-degree angle is 140 degrees.

Key Concepts

Complementary Angles

Complementary angles are pairs of angles whose degree measures sum up to exactly 90 degrees. These angles can often be found in various geometric figures, like right triangles or intersecting lines, where their adjoining measures bring about perfect right angles.
In order to identify a complementary angle, simply subtract the given angle from 90 degrees. For example, if you have a 40-degree angle, the complement is calculated as:

• Subtract the angle from 90 degrees: \[90^{\circ} - 40^{\circ} = 50^{\circ}\]
• Thus, the complement of a 40-degree angle is 50 degrees.

Understanding complementary angles is essential in solving many geometry problems, ensuring that the angles contribute to constructing right angles in figures.

Supplementary Angles

Supplementary angles are pairs of angles whose measures add up to 180 degrees. These angles are commonly looked at in context of straight lines and complex geometric shapes such as quadrilaterals and triangles.
To find the supplementary angle, subtract the given angle from 180 degrees. Take a 40-degree angle as an example to see it in action:

• Subtract the given angle from 180 degrees: \[180^{\circ} - 40^{\circ} = 140^{\circ}\]
• The supplementary angle of 40 degrees is thus 140 degrees.

Supplementary angles are crucial in geometry as they often describe relationships between angles formed by two intersecting lines or shapes that build upon parallel lines.

Degree Measurement

Degree measurement is a fundamental concept in geometry that determines the size of an angle. It provides a standardized way to express and quantify angles, enabling easier calculation and comprehension.
Angles are typically measured in degrees, where a complete circle is 360 degrees. Some important benchmarks include:

• A right angle is 90 degrees.
• A straight angle is 180 degrees.
• Circles sum up to 360 degrees.

Understanding degree measurement is key, as it allows students and practitioners to accurately describe and solve geometric problems. Whether calculating the complement or supplement of angles or working with more complex geometric configurations, degree measurement plays a pivotal role.