complete the missing pieces of the proof. given: ( ell parallel m ) prove: ( angle 2 ) and ( angle 5 ) are supplementary. statements reasons 1) ( ell parallel m ) 1) given 2) ( angle 1 cong angle 5 ) 2) 3) 3) linear pair postulate 4) ( mangle 1 + mangle 2 = 180^{circ} ) 4) 5) 5) substitution property 6) 6)

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# Explanation:
## Step1: Justify ∠1 ≅ ∠5
When two parallel lines $ \ell\parallel m$ are cut by a transversal $t$, corresponding angles are congruent. So, the reason for $ \angle1\cong\angle5$ is "Corresponding Angles Postulate".
## Step2: State the linear - pair relationship
A linear pair of angles is formed when two adjacent angles form a straight line. $ \angle1$ and $ \angle2$ form a linear pair. So, the statement for reason "Linear Pair Postulate" is "$ \angle1$ and $ \angle2$ are a linear pair".
## Step3: Justify $m\angle1 + m\angle2=180^{\circ}$
By the definition of supplementary angles (if two angles form a linear pair, then they are supplementary), the reason for $m\angle1 + m\angle2 = 180^{\circ}$ is "Definition of supplementary angles".
## Step4: Substitute $m\angle1$ with $m\angle5$
Since $ \angle1\cong\angle5$, then $m\angle1=m\angle5$. Substituting $m\angle1$ in $m\angle1 + m\angle2=180^{\circ}$, we get the statement $m\angle5 + m\angle2=180^{\circ}$.
## Step5: Conclude $ \angle2$ and $ \angle5$ are supplementary
By the definition of supplementary angles (if $m\angle A+m\angle B = 180^{\circ}$, then $ \angle A$ and $ \angle B$ are supplementary), the reason for $ \angle2$ and $ \angle5$ being supplementary is "Definition of supplementary angles".

# Answer:
| Statements | Reasons |
|---|---|
| $ \ell\parallel m$ | Given |
| $ \angle1\cong\angle5$ | Corresponding Angles Postulate |
| $ \angle1$ and $ \angle2$ are a linear pair | Linear Pair Postulate |
| $m\angle1 + m\angle2=180^{\circ}$ | Definition of supplementary angles |
| $m\angle5 + m\angle2=180^{\circ}$ | Substitution Property ($m\angle1 = m\angle5$ as $ \angle1\cong\angle5$) |
| $ \angle2$ and $ \angle5$ are supplementary | Definition of supplementary angles |

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Question

Explanation:

Step1: Justify ∠1 ≅ ∠5

Step2: State the linear - pair relationship

Step3: Justify \(m\angle1 + m\angle2=180^{\circ}\)

Step4: Substitute \(m\angle1\) with \(m\angle5\)

Step5: Conclude \( \angle2\) and \( \angle5\) are supplementary

Answer:

Statements	Reasons
\( \angle1\cong\angle5\)	Corresponding Angles Postulate
\( \angle1\) and \( \angle2\) are a linear pair	Linear Pair Postulate
\(m\angle1 + m\angle2=180^{\circ}\)	Definition of supplementary angles
\(m\angle5 + m\angle2=180^{\circ}\)	Substitution Property (\(m\angle1 = m\angle5\) as \( \angle1\cong\angle5\))
\( \angle2\) and \( \angle5\) are supplementary	Definition of supplementary angles