Math Problem Statement

Given: m∠4 + m∠7 = 180°. Prove: c ∥ d

Solution

The proof shown in the image aims to demonstrate that lines cdc \parallel d using properties of angles. Here is a breakdown of the proof's steps based on the image:

Given:

m4+m7=180m \angle 4 + m \angle 7 = 180^\circ This states that the measures of angles 4\angle 4 and 7\angle 7 add up to 180°, suggesting they are supplementary.

Prove:

cdc \parallel d

Proof Outline:

StatementsReasons
4+7=180\angle 4 + \angle 7 = 180^\circGiven
46\angle 4 \cong \angle 6Vertical Angles Theorem
4=6\angle 4 = \angle 6Definition of Congruence
6+7=180\angle 6 + \angle 7 = 180^\circSubstitution (from previous steps)
Angle 6 and angle 7 are supplementaryDefinition of Supplementary
cdc \parallel dConsecutive Interior Angles Converse

Explanation:

  1. 4+7=180\angle 4 + \angle 7 = 180^\circ: This is the given information stating that these two angles are supplementary.

  2. 46\angle 4 \cong \angle 6: Angles 4 and 6 are congruent by the Vertical Angles Theorem, which states that opposite angles formed by two intersecting lines are congruent.

  3. 4=6\angle 4 = \angle 6: By the Definition of Congruence, congruent angles have equal measures, so 4=6\angle 4 = \angle 6.

  4. Substitution: Since 4=6\angle 4 = \angle 6, we substitute 6\angle 6 for 4\angle 4 in the given equation, resulting in 6+7=180\angle 6 + \angle 7 = 180^\circ.

  5. Supplementary: By the Definition of Supplementary, if two angles add up to 180°, they are supplementary. Thus, 6\angle 6 and 7\angle 7 are supplementary.

  6. Consecutive Interior Angles Converse: The final step uses the Consecutive Interior Angles Converse theorem. When two lines are cut by a transversal and the consecutive interior angles are supplementary, the lines are parallel. Hence, cdc \parallel d.


Would you like further details or clarifications? Here are some follow-up questions you might consider:

  1. How is the Vertical Angles Theorem applied in this proof?
  2. What is the significance of the Consecutive Interior Angles Converse theorem?
  3. Could a different pair of angles be used to prove cdc \parallel d?
  4. Why is the Substitution step important in this proof?
  5. How do you know when to use the Definition of Supplementary?

Tip: Always start a geometry proof by identifying given information and known theorems that can help establish the required result.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Parallel Lines

Formulas

Supplementary Angles: ∠4 + ∠7 = 180°

Theorems

Vertical Angles Theorem
Definition of Congruence
Definition of Supplementary
Consecutive Interior Angles Converse

Suitable Grade Level

Grades 8-10