To prove that lines $c$ and $d$ are parallel given the conditions that:

1. $a \parallel b$ (lines $a$ and $b$ are parallel),
2. $\angle 2 \parallel \angle 3$ (angles $2$ and $3$ are corresponding angles),

we will use properties of parallel lines and corresponding angles.

Proof:

1. Given: $a \parallel b$ (lines $a$ and $b$ are parallel).

   • From this, we know that corresponding angles formed by a transversal with parallel lines are congruent.
   • Thus, $\angle 2 \cong \angle 3$ because they are corresponding angles formed by line $c$ (acting as a transversal) intersecting parallel lines $a$ and $b$.

2. Given: $\angle 2 \parallel \angle 3$, meaning that $\angle 2$ and $\angle 3$ are equal by definition.

3. To Prove: Lines $c \parallel d$.

4. Consider the transversal formed by line $a$ or $b$ cutting across lines $c$ and $d$:

   • Since $\angle 1$ and $\angle 2$ form a pair of corresponding angles with lines $c$ and $d$, if $\angle 1 = \angle 2$, then $c \parallel d$ by the corresponding angles postulate.

5. Conclusion:

   • Since $\angle 2 \parallel \angle 3$, and $\angle 1$ is also equal to $\angle 2$, the lines $c$ and $d$ must be parallel by the converse of the corresponding angles postulate.

Thus, we have proven that $c \parallel d$.

Would you like any more details, or do you have any further questions?

Related Questions:

1. What is the significance of corresponding angles in proving parallelism?
2. How can the converse of the corresponding angles postulate be used in other problems?
3. Can we use alternate interior angles in a similar proof?
4. What happens if one of the given lines is not parallel? How would that affect the proof?
5. How would you prove parallelism if the angles were not explicitly given as corresponding?

Tip:

Always remember, corresponding angles are key in determining whether lines are parallel when a transversal is present.