https://file.aiquickdraw.com/mathbot/file/1729645343048gql6u9le.jpg

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

### Given:
\[
m \angle 4 + m \angle 7 = 180^\circ
\]
This states that the measures of angles \( \angle 4 \) and \( \angle 7 \) add up to 180°, suggesting they are supplementary.

### Prove:
\[
c \parallel d
\]

### Proof Outline:

| **Statements**                        | **Reasons**                                |
|---------------------------------------|-------------------------------------------|
| \( \angle 4 + \angle 7 = 180^\circ \) | Given                                      |
| \( \angle 4 \cong \angle 6 \)         | Vertical Angles Theorem                    |
| \( \angle 4 = \angle 6 \)             | Definition of Congruence                   |
| \( \angle 6 + \angle 7 = 180^\circ \) | Substitution (from previous steps)         |
| Angle 6 and angle 7 are supplementary | Definition of Supplementary                |
| \( c \parallel d \)                   | Consecutive Interior Angles Converse       |

### Explanation:
1. **\( \angle 4 + \angle 7 = 180^\circ \)**: This is the given information stating that these two angles are supplementary.
   
2. **\( \angle 4 \cong \angle 6 \)**: Angles 4 and 6 are congruent by the **Vertical Angles Theorem**, which states that opposite angles formed by two intersecting lines are congruent.
   
3. **\( \angle 4 = \angle 6 \)**: By the **Definition of Congruence**, congruent angles have equal measures, so \( \angle 4 = \angle 6 \).

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

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

---

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 \( c \parallel d \)?
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.

Given: m∠4 + m∠7 = 180°. Prove: c ∥ d

This problem demonstrates how to prove that lines c and d are parallel by using angle relationships and theorems such as the Vertical Angles Theorem and Consecutive Interior Angles Converse. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation