Question

1. complete the following proof:

given: \\( \angle qsr \cong \angle wvx \\)
prove: \\( \overleftrightarrow { r t } \parallel \overleftrightarrow { u w } \\)

statements	reasons
b. \\( \angle qsr \cong \angle tsv \\)	b.
c. \\( \angle wvx \cong \angle tsv \\)	c.
d. \\( \overleftrightarrow { r t } \parallel \overleftrightarrow { u w } \\)	d.

Explanation:

Step1: Vertical angles are congruent

Vertical angles are formed when two lines intersect. $\angle QSR$ and $\angle TSV$ are vertical angles. So, $\angle QSR\cong\angle TSV$ (by the Vertical Angles Theorem).

Step2: Transitive property of congruence

If $\angle QSR\cong\angle WVX$ (given) and $\angle QSR\cong\angle TSV$ (from step 1), then by the transitive property of congruence (if $a = b$ and $a = c$, then $b = c$ for congruent angles), $\angle WVX\cong\angle TSV$.

Step3: Corresponding angles converse

$\angle WVX$ and $\angle TSV$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, $\overleftrightarrow{RT}\parallel\overleftrightarrow{UW}$ (by the Corresponding Angles Converse Theorem).

Answer:

b. Vertical Angles Theorem
c. Transitive Property of Congruence
d. Corresponding Angles Converse Theorem