1. given $overleftrightarrow{ab}$, $overleftrightarrow{cd}$, and $overleftrightarrow{er}$, $overleftrightarrow{er}$ intersects $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$. determine if the given condition could be used to justify that $overleftrightarrow{ab}paralleloverleftrightarrow{cd}$. then, justify answer.
| condition | is $overleftrightarrow{ab}paralleloverleftrightarrow{cd}$? | justification |
|--|--|--|
| if $angle elacongangle dmr$ | $\begin{array}{l}\bullet ext{ yes}\circ ext{ no}end{array}$ | alternate |
| if $angle almcongangle ble$ | $\begin{array}{l}circ ext{ yes}\\bullet ext{ no}end{array}$ | vertical |
| if $mangle cmr + mangle rmd=180^{circ}$ | $\begin{array}{l}circ ext{ yes}\circ ext{ no}end{array}$ | |
| if $angle elbcongangle lmd$ | $\begin{array}{l}circ ext{ yes}\circ ext{ no}end{array}$ | |
| if $angle elbcongangle rmd$ | $\begin{array}{l}circ ext{ yes}\circ ext{ no}end{array}$ | |
| if $mangle elb = 102^{circ}$, then $mangle ela=78^{circ}$ | $\begin{array}{l}circ ext{ yes}\circ ext{ no}end{array}$ | |
| if $mangle rmc = 91^{circ}$, then $mangle alm=89^{circ}$ | $\begin{array}{l}circ ext{ yes}\circ ext{ no}end{array}$ | |

Question

1. given $overleftrightarrow{ab}$, $overleftrightarrow{cd}$, and $overleftrightarrow{er}$, $overleftrightarrow{er}$ intersects $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$. determine if the given condition could be used to justify that $overleftrightarrow{ab}paralleloverleftrightarrow{cd}$. then, justify answer.
| condition | is $overleftrightarrow{ab}paralleloverleftrightarrow{cd}$? | justification |
|--|--|--|
| if $angle elacongangle dmr$ | $\begin{array}{l}\bullet	ext{ yes}\circ	ext{ no}end{array}$ | alternate |
| if $angle almcongangle ble$ | $\begin{array}{l}circ	ext{ yes}\\bullet	ext{ no}end{array}$ | vertical |
| if $mangle cmr + mangle rmd=180^{circ}$ | $\begin{array}{l}circ	ext{ yes}\circ	ext{ no}end{array}$ |  |
| if $angle elbcongangle lmd$ | $\begin{array}{l}circ	ext{ yes}\circ	ext{ no}end{array}$ |  |
| if $angle elbcongangle rmd$ | $\begin{array}{l}circ	ext{ yes}\circ	ext{ no}end{array}$ |  |
| if $mangle elb = 102^{circ}$, then $mangle ela=78^{circ}$ | $\begin{array}{l}circ	ext{ yes}\circ	ext{ no}end{array}$ |  |
| if $mangle rmc = 91^{circ}$, then $mangle alm=89^{circ}$ | $\begin{array}{l}circ	ext{ yes}\circ	ext{ no}end{array}$ |  |

Accepted Answer

# Explanation:
## Step1: Recall parallel - line postulates
Parallel lines have equal corresponding, alternate - interior, and alternate - exterior angles, and same - side interior angles are supplementary.
## Step2: Analyze $\angle ELA\cong\angle DMR$
$\angle ELA$ and $\angle DMR$ are alternate - exterior angles. If alternate - exterior angles are congruent, then $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$. So the answer is Yes.
## Step3: Analyze $\angle ALM\cong\angle BLE$
$\angle ALM$ and $\angle BLE$ are vertical angles. Vertical - angle congruence does not imply parallel lines. So the answer is No.
## Step4: Analyze $m\angle CMR + m\angle RMD=180^{\circ}$
$\angle CMR$ and $\angle RMD$ are a linear pair. The sum of angles in a linear pair is $180^{\circ}$ regardless of whether $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel. So the answer is No.
## Step5: Analyze $\angle ELB\cong\angle LMD$
$\angle ELB$ and $\angle LMD$ are corresponding angles. If corresponding angles are congruent, then $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$. So the answer is Yes.
## Step6: Analyze $\angle ELB\cong\angle RMD$
$\angle ELB$ and $\angle RMD$ are not corresponding, alternate - interior, or alternate - exterior angles. Their congruence does not imply parallel lines. So the answer is No.
## Step7: Analyze $m\angle ELB = 102^{\circ},m\angle ELA = 78^{\circ}$
$\angle ELB$ and $\angle ELA$ are a linear pair. This information does not relate to the parallelism of $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$. So the answer is No.
## Step8: Analyze $m\angle RMC = 91^{\circ},m\angle ALM = 89^{\circ}$
$\angle RMC$ and $\angle ALM$ are not corresponding, alternate - interior, or alternate - exterior angles. Their angle measures do not imply parallel lines. So the answer is No.

# Answer:
1. Yes
2. No
3. No
4. Yes
5. No
6. No
7. No

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall parallel - line postulates

Step2: Analyze $\angle ELA\cong\angle DMR$

Step3: Analyze $\angle ALM\cong\angle BLE$

Step4: Analyze $m\angle CMR + m\angle RMD=180^{\circ}$

Step5: Analyze $\angle ELB\cong\angle LMD$

Step6: Analyze $\angle ELB\cong\angle RMD$

Step7: Analyze $m\angle ELB = 102^{\circ},m\angle ELA = 78^{\circ}$

Step8: Analyze $m\angle RMC = 91^{\circ},m\angle ALM = 89^{\circ}$

Answer:

condition	is $overleftrightarrow{ab}paralleloverleftrightarrow{cd}$?	justification
if $angle almcongangle ble$	$\begin{array}{l}circ\text{ yes}\\bullet\text{ no}end{array}$	vertical
if $mangle cmr + mangle rmd=180^{circ}$	$\begin{array}{l}circ\text{ yes}\circ\text{ no}end{array}$
if $angle elbcongangle lmd$	$\begin{array}{l}circ\text{ yes}\circ\text{ no}end{array}$
if $angle elbcongangle rmd$	$\begin{array}{l}circ\text{ yes}\circ\text{ no}end{array}$
if $mangle elb = 102^{circ}$, then $mangle ela=78^{circ}$	$\begin{array}{l}circ\text{ yes}\circ\text{ no}end{array}$
if $mangle rmc = 91^{circ}$, then $mangle alm=89^{circ}$	$\begin{array}{l}circ\text{ yes}\circ\text{ no}end{array}$