Question

Given: $\angle 1\cong \angle 2;\angle 4\cong \angle 5$

What can you prove about $\overline{PQ}$and $\overline{RS}$? Be prepared to give your reasons in class, if asked.

Short answer

The provided segments are having a parallel connection between them.

Step-by-step solution

Step 1. Determine the theorem or postulate that can be used

The theorem 2-3 can be used which states that the relation between two angles positioned on opposite side vertically is congruent.

The theorem 3-5 can be used which states that the relation between two lines is parallel only when a separate line travels through these two lines and the angles which are positioned on the alternate interior side represents the congruent angles.

Step 2. Determine the angles that are congruent except the provided angles

The provided figure consists of two angles $\angle 2$and $\angle 5$that are positioned on vertically opposite sides. According to the theorem 2-3 these angles are congruent.

$\angle 2\cong \angle 5$

Transitive property of states that if two angles are representing them as congruent and one of the two angles is congruent to another angle so the second angle from the two angles is also congruent to that another angle.

So according to the transitive property and the provided information $\angle 1\cong \angle 4$.

Step 3. Make the conclusion

According to the theorem 3-5 determine the lines $\overline{PQ}$and $\overline{RS}$that are having a parallel relation between them because the formed by these lines are $\angle 1$and $\angle 4$. Also a separate line $\overline{QR}$is travelling through the lines $\overline{PQ}$ and $\overline{RS}$.