Problem 6 - Precalculus

Question

Find \((a)\) the complement and \((b)\) the supplement of the given angles. $$75^{\circ}$$

Step-by-Step Solution

Answer

The complement of 75° is 15°, and the supplement is 105°.

Step 1: Understanding Complements and Supplements

The complement of an angle is what, when added to it, results in a right angle (90°). The supplement of an angle is what, when added to it, results in a straight angle (180°). Our task is to find both for a given angle of 75°.

Step 1: Calculate the Complement

To find the complement of 75°, subtract it from 90°: \[ 90^{ ext{°}} - 75^{ ext{°}} = 15^{ ext{°}} \] So, the complement of 75° is 15°.

Step 2: Calculate the Supplement

To find the supplement of 75°, subtract it from 180°: \[ 180^{ ext{°}} - 75^{ ext{°}} = 105^{ ext{°}} \] Thus, the supplement of 75° is 105°.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This means that if you know one of the angles, you can easily find the other by subtracting the known angle from 90 degrees. This property is helpful for solving various math problems, especially when dealing with right triangles or certain geometric configurations.

Here are some important points about complementary angles:

• If one angle is 30°, the other must be 60° to be complementary.
• Complementary angles do not have to be adjacent; they just need to sum to 90°.
• Knowing that an angle's complement is 15° tells you immediately that the original angle is 75° because 75° + 15° = 90°.

Understanding complements is key in trigonometry and geometry as it helps to determine missing angles.

Supplementary Angles

Supplementary angles refer to two angles whose measures add up to 180 degrees. They are an essential part of understanding angle relationships, especially when dealing with straight lines and polygons. Just like with complementary angles, if you know one angle, you can determine its supplement by subtracting the angle from 180 degrees.

Consider these features of supplementary angles:

• If you have a 75° angle, its supplement must be 105° because together they equal 180° (75° + 105° = 180°).
• Supplementary angles can be non-adjacent; they simply need to sum to 180°.
• They are commonly used in problems involving linear pairs or angles on a straight line.

Recognizing supplementary relationships assists in easily finding unknown angles in various geometric settings.

Angle Measurement

Understanding angle measurements is fundamental to solving problems related to geometry. Angles are measured in degrees, with the total circle measuring 360 degrees. This system aids in understanding how angles relate to one another within various shapes and figures.

Here are some basics about angle measurement:

• A right angle measures exactly 90°.
• A straight angle, often seen as a line, measures exactly 180°.
• An acute angle is any angle less than 90°, a right angle is precisely 90°, and an obtuse angle is any angle more than 90° but less than 180°.

Practicing angle measurements involves not only calculating angles when given partial information but also using complementary and supplementary relationships to determine unknown values. This skill is widely applicable, making it crucial in both academic and practical situations.