Question

1. complete the two - column proof. given: x || y prove: ∠3 ≅ ∠5 8. in the figure, eh || bi and bi || cj. a. what is m∠1? explain. b. what is m∠3? explain. 1) x || y 2) m∠3 + m∠8 = 180° 3) m∠5 + m∠8 = 180° 4) transitive property of equality 5) subtraction property of equality 6) definition of congruence

Explanation:

Step1: Identify vertical - angles

In the figure, $\angle1$ and the angle marked $53^{\circ}$ are vertical - angles. Vertical angles are congruent. So, $m\angle1 = 53^{\circ}$.

Step2: Identify corresponding angles

Since $x\parallel y$, $\angle3$ and $\angle5$ are corresponding angles. Corresponding angles formed by parallel lines are congruent.
We know that $m\angle3 + m\angle8=180^{\circ}$ and $m\angle5 + m\angle8 = 180^{\circ}$.
By the transitive property of equality, $m\angle3=m\angle5$.
From the vertical - angle relationship in the left - hand figure, we first found $m\angle1 = 53^{\circ}$. But to find $m\angle3$, we use the parallel - line properties.
Since $\angle3$ and $\angle5$ are corresponding angles for $x\parallel y$, and we know from the linear - pair equations $m\angle3 + m\angle8=180^{\circ}$ and $m\angle5 + m\angle8 = 180^{\circ}$, we can conclude that $m\angle3$ has the same measure as the corresponding angle $\angle5$.

Answer:

a. $m\angle1 = 53^{\circ}$ because vertical angles are congruent.
b. Since $x\parallel y$, $\angle3$ and $\angle5$ are corresponding angles. Given $m\angle3 + m\angle8=180^{\circ}$ and $m\angle5 + m\angle8 = 180^{\circ}$, by the transitive property of equality, $m\angle3=m\angle5$. Without more information about the measure of $\angle5$ or other related angles, we can only say that $m\angle3$ is equal to the measure of its corresponding angle $\angle5$ due to the parallel lines $x$ and $y$. If we assume we want to find the measure in terms of the given $53^{\circ}$ angle, we need more information about the relationships between the two sub - figures. But if we just consider the parallel - line property for $\angle3$ and $\angle5$, we know they are congruent.