Problem 84 - 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. When the measure of a given angle is added to three times the measure of its complement, the sum equals the sum of the measures of the complement and supplement of the angle.

Step-by-Step Solution

Answer

The original statement is false. The corrected statement will read: 'When the measure of a given angle is added to two times the measure of its complement, the sum equals the sum of the measures of the complement and supplement of the angle.'

Step 1: Understand the statement

From the statement, you have an equation which will look like the following expression: \( Angle + 3 \cdot Complement = Complement + Supplement \). Therefore, mark the angle as 'x', then the complement will be \(90-x\), and the supplement will be \(180-x\).

Step 2: Substitution

Substitute these values into the equation: \( x + 3 \cdot (90 - x) = (90 - x) + (180 - x) \). This translates into \( x + 270 - 3x = 270 - x \).

Step 3: Simplify the equation

Simplifying the equation, you get: \( 270 - 2x = 270 - x \). There is a discrepancy in the coefficients of 'x' on both sides, which implies that the original statement is not correct.

Step 4: Amend the statement

The correct statement would be: 'When the measure of a given angle is added to two times the measure of its complement, the sum equals the sum of the measures of the complement and supplement of the angle.'

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This is a fundamental concept in geometry. For example, if one angle measures 30 degrees, the angle complement to it would measure 60 degrees, because 30 + 60 equals 90.
Think of complementary angles like two pieces of a puzzle that fit together perfectly to form a right angle. They are often used in problems to help find unknown angle measures.

• Remember: The sum of complementary angles is always 90 degrees.
• Complementary angles do not need to be adjacent or connected in any specific way; they just need to add up to 90 degrees when combined.
• This concept is useful in various fields such as architecture, engineering, and design for determining the necessary angles in structures or objects.

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees, forming a straight line when placed adjacent to each other. For instance, if one angle measures 110 degrees, its supplementary angle would measure 70 degrees, because when added, 110 + 70 equals 180.
Supplementary angles are crucial when dealing with linear pairs and other geometric configurations.

• Always remember: The sum of supplementary angles is 180 degrees.
• Like complementary angles, supplementary angles do not need to be connected or adjacent as long as their measures total 180 degrees.
• They are often used in geometric proofs and to solve problems involving parallel lines cut by a transversal where alternate interior angles or corresponding angles may be supplementary.

Equation Solving

Solving equations is a critical mathematical skill used to find unknown values. In this context, we use equation solving to verify truths about angle relationships. In the example exercise, the initial task was to evaluate an equation involving complementary and supplementary angles.
Here's a breakdown of that process:

• Identifying the components: "angle + 3 times its complement = complement + supplement" translates to a mathematical expression based on angle relationships.
• Substitution: Replace unknowns and expressions with expressions based on known quantities, like using \(x\) for an angle, \(90-x\) for its complement, and \(180-x\) for its supplement.
• Simplification: Combine like terms and simplify the equation to see if the initial statement holds true.
• Correction: Sometimes solving the equation leads to revealing a false statement, requiring a correction to make it true as seen in the exercise.

Mastering equation solving involves practice and careful consideration of each step to ensure mathematical correctness. This skill is foundational not just in geometry but in all branches of mathematics.