Question

1. $overleftrightarrow{cm}$, $overleftrightarrow{hi}$, $overleftrightarrow{en}$, and $overleftrightarrow{ab}$ are shown where $angle lki = 94^{circ}$. determine if the given conditions could be used to justify that $overleftrightarrow{en}paralleloverleftrightarrow{ab}$ or $overleftrightarrow{cm}paralleloverleftrightarrow{hi}$.

condition	is $overleftrightarrow{en}paralleloverleftrightarrow{ab}$ or $overleftrightarrow{cm}paralleloverleftrightarrow{hi}$?	justification
$angle cdecongangle mla$	o yes <br> o no
$mangle edl + mangle njk = 180^{circ}$	o yes <br> o no
$angle klmcongangle hjn$	o yes <br> o no
$angle jklcongangle bki$	o yes <br> o no
if $angle bki = 86^{circ}$, then $mangle bki + mangle lki = 180^{circ}$	o yes <br> o no

Explanation:

Step1: Recall parallel - line postulates

If corresponding angles are congruent, or alternate - interior angles are congruent, or same - side interior angles are supplementary, then two lines are parallel.

Step2: Analyze $\angle CDJ\cong\angle KLM$

$\angle CDJ$ and $\angle KLM$ are not corresponding, alternate - interior, or same - side interior angles for either $\overleftrightarrow{EN}\parallel\overleftrightarrow{AB}$ or $\overleftrightarrow{CM}\parallel\overleftrightarrow{HI}$. So the answer is No. Justification: Non - relevant angle pair.

Step3: Analyze $\angle CDE\cong\angle MLA$

$\angle CDE$ and $\angle MLA$ are corresponding angles for $\overleftrightarrow{CM}\parallel\overleftrightarrow{HI}$. By the corresponding - angles postulate, if corresponding angles are congruent, then the lines are parallel. So the answer is Yes. Justification: Corresponding angles are congruent.

Step4: Analyze $m\angle EDL + m\angle NJK=180^{\circ}$

$\angle EDL$ and $\angle NJK$ are not corresponding, alternate - interior, or same - side interior angles for either $\overleftrightarrow{EN}\parallel\overleftrightarrow{AB}$ or $\overleftrightarrow{CM}\parallel\overleftrightarrow{HI}$. So the answer is No. Justification: Non - relevant angle pair.

Step5: Analyze $\angle KLM\cong\angle HJN$

$\angle KLM$ and $\angle HJN$ are not corresponding, alternate - interior, or same - side interior angles for either $\overleftrightarrow{EN}\parallel\overleftrightarrow{AB}$ or $\overleftrightarrow{CM}\parallel\overleftrightarrow{HI}$. So the answer is No. Justification: Non - relevant angle pair.

Step6: Analyze $\angle JKL\cong\angle BKI$

$\angle JKL$ and $\angle BKI$ are vertical angles and not relevant to the parallelism of the given lines. So the answer is No. Justification: Vertical angles, non - relevant.

Step7: Analyze if $\angle BKI = 86^{\circ}$ and $m\angle BKI+m\angle LKI = 180^{\circ}$

$\angle BKI$ and $\angle LKI$ are not corresponding, alternate - interior, or same - side interior angles for either $\overleftrightarrow{EN}\parallel\overleftrightarrow{AB}$ or $\overleftrightarrow{CM}\parallel\overleftrightarrow{HI}$. So the answer is No. Justification: Non - relevant angle pair.

Answer:

$\angle CDJ\cong\angle KLM$: No, Non - relevant angle pair.
$\angle CDE\cong\angle MLA$: Yes, Corresponding angles are congruent.
$m\angle EDL + m\angle NJK = 180^{\circ}$: No, Non - relevant angle pair.
$\angle KLM\cong\angle HJN$: No, Non - relevant angle pair.
$\angle JKL\cong\angle BKI$: No, Vertical angles, non - relevant.
If $\angle BKI = 86^{\circ}$, then $m\angle BKI+m\angle LKI = 180^{\circ}$: No, Non - relevant angle pair.