In this problem, we need to prove that the lines $a \parallel c$, using the given information and without applying the Transitive Property of parallel lines.

Given Information:

• $b \parallel c$ (line $b$ is parallel to line $c$)
• $\angle 7 \cong \angle 5$ (angle 7 is congruent to angle 5)

Steps to prove $a \parallel c$:

1. Use the given parallel lines: Since $b \parallel c$, by the properties of parallel lines, we can infer that certain angle relationships hold true based on the transversal $b$. For example, alternate interior angles formed by a transversal cutting two parallel lines are congruent.

2. Transversal and angles: Notice that line $a$ acts as a transversal that intersects lines $b$ and $c$. The key is to use the given congruent angles and the fact that alternate interior angles are congruent when lines are parallel.

3. Given $\angle 7 \cong \angle 5$: Since $\angle 7 \cong \angle 5$ and these angles lie on the transversal that intersects lines $a$ and $b$, and line $b$ is parallel to line $c$, we can conclude that line $a$ must also be parallel to line $c$.

Conclusion:

By the Alternate Interior Angle Theorem, since the congruent angles (7 and 5) are formed by the transversal of lines $a$ and $c$, and the angles are congruent, this implies that $a \parallel c$.

Would you like me to explain any specific step in more detail?

Related Questions:

1. What is the Alternate Interior Angle Theorem?
2. How does the congruency of angles imply parallelism between lines?
3. Why can we not use the Transitive Property for parallel lines in this proof?
4. What other theorems apply to parallel lines and transversals?
5. How can you prove lines are parallel using only congruent angles?

Tip: Always look for angle relationships when transversals intersect parallel lines, as they often provide the key to proving lines are parallel.