Question

the diagram below illustrates the construction of $overrightarrow{ps}$ parallel to $overrightarrow{rq}$ through point $p$. what statement justifies the construction? a $overline{pr}congoverline{rq}$ b $mangle2 + mangle3 = 180^{circ}$ c $overline{ps}congoverline{rq}$ d $mangle1=mangle2$

Explanation:

Step1: Recall parallel - line postulates

When two lines are parallel, corresponding angles are equal. In the construction of $\overleftrightarrow{PS}$ parallel to $\overleftrightarrow{RQ}$ through point $P$, $\angle1$ and $\angle2$ are corresponding angles.

Step2: Analyze each option

• Option A: $\overline{PR}\cong\overline{RQ}$ has nothing to do with the parallel - line construction.
• Option B: $m\angle2 + m\angle3=180^{\circ}$ is the property of supplementary angles on the same side of a transversal for two parallel lines, but it is not the key property used in this construction.
• Option C: $\overline{PS}\cong\overline{RQ}$ is not relevant to the parallel - line construction.
• Option D: If $m\angle1 = m\angle2$, by the corresponding - angles postulate, $\overleftrightarrow{PS}\parallel\overleftrightarrow{RQ}$.

Answer:

D. $m\angle1 = m\angle2$