Math Problem Statement

Prove that \( \overline{ED} \parallel \overline{FH} \) given that \( \overline{EF} \parallel \overline{JG} \) and \( \angle 1 \cong \angle 4 \).

Solution

The problem involves proving that EDFH\overline{ED} \parallel \overline{FH} using the given information:

Given:

  1. EFJG\overline{EF} \parallel \overline{JG}
  2. 14\angle 1 \cong \angle 4

To Prove:

EDFH\overline{ED} \parallel \overline{FH}

Proof Explanation:

  1. Corresponding Angles Theorem:

    • Since EFJG\overline{EF} \parallel \overline{JG}, and FJ\overline{FJ} is a transversal, we can conclude that 23\angle 2 \cong \angle 3 by the Corresponding Angles Postulate.
  2. Given Information:

    • 14\angle 1 \cong \angle 4 is given in the problem.
  3. Angle Relationships and Parallel Lines:

    • ED\overline{ED} and FH\overline{FH} will be parallel if we can show that the corresponding angles formed by transversal EF\overline{EF} and JG\overline{JG} with these lines are congruent. From the given dataThe problem involves proving that EDFH\overline{ED} \parallel \overline{FH} using the given information:

Given:

  1. EFJG\overline{EF} \parallel \overline{JG}
  2. 14\angle 1 \cong \angle 4

To Prove:

EDFH\overline{ED} \parallel \overline{FH}

Proof Explanation:

  1. Corresponding Angles Theorem:

    • Since EFJG\overline{EF} \parallel \overline{JG}, and FJ\overline{FJ} is a transversal, we can conclude that 23\angle 2 \cong \angle 3 by the Corresponding Angles Postulate.
  2. Given Information:

    • 14\angle 1 \cong \angle 4 is given in the problem.
  3. Angle Relationships and Parallel Lines:

    • ED\overline{ED} and FH\overline{FH} will be parallel if we can show that the corresponding angles formed by transversal EF\overline{EF} and JG\overline{JG} with these lines are congruent. From the given
      • Since EFJG\overline{EF} \parallel \overline{JG}, the transversal relationships allow us to conclude that 14\angle 1 \cong \angle 4, implying EDFH\overline{ED} \parallel \overline{FH}.

Thus, the two lines EDFH\overline{ED} \parallel \overline{FH} are parallel because corresponding angles are congruent.

Would you like further details on any part of this explanation or theorems used?

Related Questions:

  1. What is the Corresponding Angles Postulate, and how does it apply to parallel lines?
  2. How does transversal FJ\overline{FJ} help in this proof?
  3. Why are corresponding angles important for proving lines parallel?
  4. Can alternate interior angles also be used to prove lines parallel in this case?
  5. How can we use congruent angles to prove more general properties about lines?

Tip:

Always check for transversals and corresponding angles when attempting to prove lines are parallel.

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Math Problem Analysis

Mathematical Concepts

Parallel Lines
Transversals
Corresponding Angles

Formulas

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Theorems

Corresponding Angles Theorem

Suitable Grade Level

Grades 8-10