complete the table to prove congruency between angles
statement
1 ( a parallel b ) and ( c parallel f )
2 ( m angle 5 = m angle 13 ) and ( m angle 9 = m angle 13 )
3 ( m angle 5 + m angle 6 = 180^{circ} ) and ( m angle 9 + m angle 11 = 180^{circ} )
4 ( m angle 13 + m angle 6 = 180^{circ} ) and ( m angle 13 + m angle 11 = 180^{circ} )
5 ( m angle 13 + m angle 6 = m angle 13 + m angle 11 )
6 ( m angle 6 = m angle 11 )
reason
given
corresponding angles theorem
definition of linear pair
substitution property of equality
transitive property of equality
subtraction property of equality

Question

Accepted Answer

# Explanation:
## Step1: Use Corresponding Angles Theorem
When two parallel lines are cut by a transversal, corresponding angles are equal. Since $a\parallel b$, $\angle5$ and $\angle13$ are corresponding angles, so $m\angle5 = m\angle13$. Since $c\parallel f$, $\angle9$ and $\angle13$ are corresponding angles, so $m\angle9 = m\angle13$.
## Step2: Use Substitution Property of Equality
We know that $m\angle5 + m\angle6=180^{\circ}$ (linear - pair) and $m\angle5 = m\angle13$. Substitute $m\angle5$ with $m\angle13$ in $m\angle5 + m\angle6 = 180^{\circ}$, we get $m\angle13 + m\angle6=180^{\circ}$. Also, since $m\angle9 + m\angle11 = 180^{\circ}$ (linear - pair) and $m\angle9 = m\angle13$, substitute $m\angle9$ with $m\angle13$ in $m\angle9 + m\angle11 = 180^{\circ}$, we get $m\angle13 + m\angle11 = 180^{\circ}$.
## Step3: Use Transitive Property of Equality
If $m\angle13 + m\angle6 = 180^{\circ}$ and $m\angle13 + m\angle11 = 180^{\circ}$, then by the transitive property of equality ($a = c$ and $b = c$ implies $a = b$), we have $m\angle13 + m\angle6=m\angle13 + m\angle11$.
## Step4: Use Subtraction Property of Equality
Subtract $m\angle13$ from both sides of the equation $m\angle13 + m\angle6=m\angle13 + m\angle11$. Using the subtraction property of equality ($a + c=b + c$ implies $a = b$), we get $m\angle6=m\angle11$.

# Answer:
The table is completed as follows:
| Statement | Reason |
|---|---|
| 1. $a\parallel b$ and $c\parallel f$ | given |
| 2. $m\angle5 = m\angle13$ and $m\angle9 = m\angle13$ | Corresponding Angles Theorem |
| 3. $m\angle5 + m\angle6 = 180^{\circ}$ and $m\angle9 + m\angle11 = 180^{\circ}$ | definition of linear pair |
| 4. $m\angle13 + m\angle6 = 180^{\circ}$ and $m\angle13 + m\angle11 = 180^{\circ}$ | Substitution Property of Equality |
| 5. $m\angle13 + m\angle6=m\angle13 + m\angle11$ | Transitive Property of Equality |
| 6. $m\angle6=m\angle11$ | Subtraction Property of Equality |

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QUESTION IMAGE

Question

Explanation:

Step1: Use Corresponding Angles Theorem

Step2: Use Substitution Property of Equality

Step3: Use Transitive Property of Equality

Step4: Use Subtraction Property of Equality

Answer:

Statement	Reason
2. \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\)	Corresponding Angles Theorem
3. \(m\angle5 + m\angle6 = 180^{\circ}\) and \(m\angle9 + m\angle11 = 180^{\circ}\)	definition of linear pair
4. \(m\angle13 + m\angle6 = 180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\)	Substitution Property of Equality
5. \(m\angle13 + m\angle6=m\angle13 + m\angle11\)	Transitive Property of Equality
6. \(m\angle6=m\angle11\)	Subtraction Property of Equality