Math Problem Statement

Prove that lines c and d are parallel, given that ∠4 + ∠7 = 180°.

Solution

It looks like you've uploaded a proof involving parallel lines. The given information is:

  • 4+7=180\angle 4 + \angle 7 = 180^\circ (which suggests these angles are supplementary).
  • The goal is to prove that line cdc \parallel d (that is, lines cc and dd are parallel).

Based on the diagram and the given information, here's how you can approach the proof:

Proof outline:

  1. Given: 4+7=180\angle 4 + \angle 7 = 180^\circ.

    • This shows that 4\angle 4 and 7\angle 7 are supplementary.
  2. Reason: Supplementary angles.

    • Since the sum of the angles is 180°, these angles must be supplementary.
  3. Reason: Consecutive interior angles.

    • If two consecutive interior angles formed by a transversal are supplementary, then the lines cut by the transversal are parallel.
  4. Conclusion: Since 4\angle 4 and 7\angle 7 are supplementary, we can conclude that cdc \parallel d.

Does this explanation help? Let me know if you would like further clarification or additional details on any steps!

Additional questions to explore:

  1. How do you define consecutive interior angles in geometry?
  2. What other theorems can be applied to prove parallel lines?
  3. How does the concept of supplementary angles help in proofs involving parallel lines?
  4. Can you identify other pairs of supplementary angles in this diagram?
  5. What happens if the angles are not supplementary?

Tip: A good habit is to always check if you can apply the Consecutive Interior Angles Theorem when proving parallel lines with angles involving a transversal!

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Math Problem Analysis

Mathematical Concepts

Geometry
Parallel Lines
Supplementary Angles
Transversal

Formulas

Supplementary Angles Property: ∠4 + ∠7 = 180°

Theorems

Consecutive Interior Angles Theorem

Suitable Grade Level

Grades 8-10