Problem 33 - Intermediate Algebra

Question

Geometry. In the illustration, lines \(r\) and \(s\) are cut by a third line \(l\) to form \(\angle 1\) (angle 1 ) and \(\angle 2\). When lines \(r\) and \(s\) are parallel, \(\angle 1\) and \(\angle 2\) are parallel, \(\angle 1\) and \(\angle 2\) are supplementary. If \(\angle 1=x+50^{\circ}\) \(\angle 2=2 x-20^{\circ},\) and lines \(r\) and \(s\) are parallel, find \(x\)

Step-by-Step Solution

Answer

The value of \(x\) is 50 degrees.

Step 1: Set up the equation

Since lines \(r\) and \(s\) are parallel, we are given that \(\angle 1\) and \(\angle 2\) are supplementary. That means the sum of the two angles is 180 degrees. So, we can write the equation: \[(x + 50^{\circ}) + (2x - 20^{\circ}) = 180^{\circ}.\]

Step 2: Simplify the equation

Combine like terms in the equation: \[x + 2x + 50^{\circ} - 20^{\circ} = 180^{\circ}.\] This simplifies to \[3x + 30^{\circ} = 180^{\circ}.\]

Step 3: Solve for \(x\)

Subtract \(30^{\circ}\) from both sides of the equation to isolate the terms with \(x\): \[3x = 150^{\circ}.\] Then divide both sides by 3 to solve for \(x\): \[x = 50^{\circ}.\]

Key Concepts

Supplementary Angles

When two angles are supplementary, their measures add up to 180 degrees. This concept is especially useful when dealing with parallel lines and transversals, as often, supplementary angles appear in this setup.
In the context of this exercise, angles 1 and 2 are supplementary because they are created by a transversal cutting across two parallel lines. This relationship,

• helps determine unknown angle measures,
• is crucial in various geometric proofs,
• provides a foundation for solving complex problems.

Recognizing supplementary angles involves:

• identifying pairs of angles that form a straight line,
• checking if the angles are adjacent or directly opposite when parallel lines are involved.

Geometry Problem-Solving

Problem-solving in geometry often requires connecting different concepts together. Understanding how angles interact in relation to parallel lines is key.
When working with problems involving parallel lines:

• Identify the relationships present (such as supplementary, complementary, vertical angles).
• Use known properties of these relationships to set up equations.
• Keep track of which angles you know the measures of and which you need to find.

In this exercise, understanding the relationship between the parallel lines and the angles helped us formulate an equation, which was pivotal to finding the value of \(x\). This approach can be generalized to solve various types of geometric problems.

Equation Solving

Solving equations is a fundamental aspect of mathematics, essential for deriving values from given conditions. When equations stem from geometric setups, like the one in this problem, you translate physical aspects into algebraic expressions.
Here's a quick guide to solving such equations:

• Start by establishing the equation using known geometric properties; here, leveraging the supplementary nature of angles 1 and 2.
• Simplify by combining like terms; this reduces complexity and helps in isolating the variable.
• Perform algebraic operations to solve for the unknown; this involved accurate arithmetic steps like addition, subtraction, and division.

Understanding each step in equation solving ensures a precise and reliable approach to tackling problems, especially those encountered in geometry. It's crucial to maintain accuracy throughout the process to correctly determine solutions, such as finding that \(x = 50^{\circ}\) in this problem.