Questions: Given: y z Prove: x z

Solution Steps

Given that line $y$ is parallel to line $z$, we know that $\angle 4$ and $\angle 6$ are corresponding angles. By the corresponding angles postulate, if two parallel lines are cut by a transversal, then the corresponding angles are congruent. Therefore, $\angle 4 \cong \angle 6$.

We are given that $\angle 4 \cong \angle 8$. We also determined that $\angle 4 \cong \angle 6$. Therefore, by the transitive property of congruence (if a ≅ b and b ≅ c, then a ≅ c), $\angle 6 \cong \angle 8$.

We have established that $\angle 6 \cong \angle 8$. These angles are corresponding angles formed by lines $x$ and $z$ cut by the transversal. By the converse of the corresponding angles postulate, if two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Therefore, $x$ || $z$.

Final Answer