Math Problem Statement
Solution
We are given two parallel lines, , and asked to find the values of and using the angles provided in the image. Let’s break down the problem:
Step 1: Identify the relationships between angles
Since and the transversal intersects these lines, the following angle relationships apply:
- Corresponding angles: Angles on the same side of the transversal and in corresponding positions are equal.
- Alternate interior angles: Angles on opposite sides of the transversal but inside the parallel lines are equal.
Step 2: Set up equations
From the image:
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The angles and are corresponding angles, so:
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The angles and are alternate interior angles, so:
Step 3: Solve for
We first solve the equation involving : Subtract from both sides: Subtract 2 from both sides: Divide by 4:
Step 4: Solve for
Substitute into the first equation: Simplify: Subtract 20 from both sides: Divide by 2:
Final answer:
Would you like further details on any part of this solution?
Here are 5 questions to deepen your understanding:
- How do corresponding angles help in solving parallel line problems?
- What other angle properties are commonly used in problems involving parallel lines?
- Could we have used another pair of angles to solve for and ?
- How would the solution change if the lines were not parallel?
- Why are alternate interior angles always equal when lines are parallel?
Tip:
Always double-check the relationships between angles in diagrams involving parallel lines, as they frequently provide the key to solving such problems.
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Math Problem Analysis
Mathematical Concepts
Algebra
Angle Relationships
Parallel Lines
Transversal
Formulas
Corresponding angles are equal when two parallel lines are cut by a transversal.
Alternate interior angles are equal when two parallel lines are cut by a transversal.
Theorems
Corresponding Angle Theorem
Alternate Interior Angle Theorem
Suitable Grade Level
Grades 9-10
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