geometry
unit 3
lesson 5 homework
name_____________________
date_____________________
period_____________________
1. given $\overleftrightarrow{ab}$, $\overleftrightarrow{cd}$, and $\overleftrightarrow{er}$. $\overleftrightarrow{er}$ intersects $\overleftrightarrow{ab}$ and $\overleftrightarrow{cd}$.
image of lines and angles
determine if the given condition could be used to justify that $\overleftrightarrow{ab} \parallel \overleftrightarrow{cd}$. then, justify your answer.
| condition | is $\overleftrightarrow{ab} \parallel \overleftrightarrow{cd}$? | justification |
| --- | --- | --- |
| $\angle ela \cong \angle dmr$ | $\circ$ yes $\circ$ no | |
| $\angle alm \cong \angle ble$ | $\circ$ yes $\circ$ no | |
| $m\angle cmr + m\angle rmd = 180^\circ$ | $\circ$ yes $\circ$ no | |
| $\angle elb \cong \angle lmd$ | $\circ$ yes $\circ$ no | |
| $\angle elb \cong \angle rmd$ | $\circ$ yes $\circ$ no | |
| if $m\angle elb = 102^\circ$, then $m\angle ela = 78^\circ$ | $\circ$ yes $\circ$ no | |
| if $m\angle rmc = 91^\circ$, then $m\angle alm = 89^\circ$ | $\circ$ yes $\circ$ no | |

Question

geometry
unit 3
lesson 5 homework
name_____________________
date_____________________
period_____________________
1. given $\overleftrightarrow{ab}$, $\overleftrightarrow{cd}$, and $\overleftrightarrow{er}$. $\overleftrightarrow{er}$ intersects $\overleftrightarrow{ab}$ and $\overleftrightarrow{cd}$.
image of lines and angles
determine if the given condition could be used to justify that $\overleftrightarrow{ab} \parallel \overleftrightarrow{cd}$. then, justify your answer.
| condition | is $\overleftrightarrow{ab} \parallel \overleftrightarrow{cd}$? | justification |
| --- | --- | --- |
| $\angle ela \cong \angle dmr$ | $\circ$ yes $\circ$ no |  |
| $\angle alm \cong \angle ble$ | $\circ$ yes $\circ$ no |  |
| $m\angle cmr + m\angle rmd = 180^\circ$ | $\circ$ yes $\circ$ no |  |
| $\angle elb \cong \angle lmd$ | $\circ$ yes $\circ$ no |  |
| $\angle elb \cong \angle rmd$ | $\circ$ yes $\circ$ no |  |
| if $m\angle elb = 102^\circ$, then $m\angle ela = 78^\circ$ | $\circ$ yes $\circ$ no |  |
| if $m\angle rmc = 91^\circ$, then $m\angle alm = 89^\circ$ | $\circ$ yes $\circ$ no |  |

Accepted Answer

# Explanation:
## Step1: Analyze $\angle ELA\cong\angle DMR$
These angles are not corresponding, alternate - interior, or alternate - exterior angles. So, $\overleftrightarrow{AB}$ is not parallel to $\overleftrightarrow{CD}$ based on this condition.
## Step2: Analyze $\angle ALM\cong\angle BLE$
These are vertical angles. Vertical angles are equal for any two intersecting lines, not just parallel lines. So, $\overleftrightarrow{AB}$ is not parallel to $\overleftrightarrow{CD}$ based on this condition.
## Step3: Analyze $m\angle CMR + m\angle RMD=180^{\circ}$
These angles are adjacent and form a linear pair. A linear pair of angles summing to $180^{\circ}$ is a property of adjacent angles on a straight line (not related to parallel lines). So, $\overleftrightarrow{AB}$ is not parallel to $\overleftrightarrow{CD}$ based on this condition.
## Step4: Analyze $\angle ELB\cong\angle LMD$
These are not corresponding, alternate - interior, or alternate - exterior angles. So, $\overleftrightarrow{AB}$ is not parallel to $\overleftrightarrow{CD}$ based on this condition.
## Step5: Analyze $\angle ELB\cong\angle RMD$
These are corresponding angles. If corresponding angles are congruent, then the lines are parallel (Corresponding Angles Postulate). So, $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$ based on this condition.
## Step6: Analyze $m\angle ELB = 102^{\circ}$, then $m\angle ELA=78^{\circ}$
$\angle ELB$ and $\angle ELA$ are a linear pair ($m\angle ELB+m\angle ELA = 180^{\circ}$). This is a property of adjacent angles on a straight line (not related to parallel lines). So, $\overleftrightarrow{AB}$ is not parallel to $\overleftrightarrow{CD}$ based on this condition.
## Step7: Analyze $m\angle RMC = 91^{\circ}$, then $m\angle ALM=89^{\circ}$
$\angle RMC$ and $\angle ALM$ are not related by a parallel - line theorem. $\angle RMC$ and $\angle ALM$  being supplementary ($m\angle RMC+m\angle ALM = 180^{\circ}$) is not a sufficient condition for $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$

# Answer:
| Condition | Is $\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$? | Justification |
|---|---|---|
|$\angle ELA\cong\angle DMR$| No | Not corresponding/alternate - interior/alternate - exterior angles |
|$\angle ALM\cong\angle BLE$| No | Vertical angles (not parallel - line related) |
| $m\angle CMR + m\angle RMD = 180^{\circ}$| No | Linear pair (not parallel - line related) |
|$\angle ELB\cong\angle LMD$| No | Not corresponding/alternate - interior/alternate - exterior angles |
|$\angle ELB\cong\angle RMD$| Yes | Corresponding Angles Postulate |
|If $m\angle ELB = 102^{\circ}$, then $m\angle ELA=78^{\circ}$| No | Linear pair (not parallel - line related) |
|If $m\angle RMC = 91^{\circ}$, then $m\angle ALM = 89^{\circ}$| No | Not a parallel - line theorem |

condition	is \\(\overleftrightarrow{ab} \parallel \overleftrightarrow{cd}\\)?	justification
\\(\angle alm \cong \angle ble\\)	\\(\circ\\) yes \\(\circ\\) no
\\(m\angle cmr + m\angle rmd = 180^\circ\\)	\\(\circ\\) yes \\(\circ\\) no
\\(\angle elb \cong \angle lmd\\)	\\(\circ\\) yes \\(\circ\\) no
\\(\angle elb \cong \angle rmd\\)	\\(\circ\\) yes \\(\circ\\) no
if \\(m\angle elb = 102^\circ\\), then \\(m\angle ela = 78^\circ\\)	\\(\circ\\) yes \\(\circ\\) no
if \\(m\angle rmc = 91^\circ\\), then \\(m\angle alm = 89^\circ\\)	\\(\circ\\) yes \\(\circ\\) no

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze \(\angle ELA\cong\angle DMR\)

Step2: Analyze \(\angle ALM\cong\angle BLE\)

Step3: Analyze \(m\angle CMR + m\angle RMD=180^{\circ}\)

Step4: Analyze \(\angle ELB\cong\angle LMD\)

Step5: Analyze \(\angle ELB\cong\angle RMD\)

Step6: Analyze \(m\angle ELB = 102^{\circ}\), then \(m\angle ELA=78^{\circ}\)

Step7: Analyze \(m\angle RMC = 91^{\circ}\), then \(m\angle ALM=89^{\circ}\)

Answer:

Condition	Is \(\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}\)?	Justification
\(\angle ALM\cong\angle BLE\)	No	Vertical angles (not parallel - line related)
\(m\angle CMR + m\angle RMD = 180^{\circ}\)	No	Linear pair (not parallel - line related)
\(\angle ELB\cong\angle LMD\)	No	Not corresponding/alternate - interior/alternate - exterior angles
\(\angle ELB\cong\angle RMD\)	Yes	Corresponding Angles Postulate
If \(m\angle ELB = 102^{\circ}\), then \(m\angle ELA=78^{\circ}\)	No	Linear pair (not parallel - line related)
If \(m\angle RMC = 91^{\circ}\), then \(m\angle ALM = 89^{\circ}\)	No	Not a parallel - line theorem