Question

statements | reasons

1. $overleftrightarrow{ab} parallel overleftrightarrow{cd}$ and $overleftrightarrow{eh}$ is a transversal. | 1. given
2. $angle afe cong angle cgf$ | 2. corresponding angles theorem
3. $mangle afe = mangle cgf$ | 3. definition of congruent angles
4. $angle afe$ and $angle afg$ are a linear pair. | 4. definition of a linear pair
5. | 5.

Explanation:

Step1: Recall Linear Pair Postulate

A linear pair of angles is supplementary, so their measures sum to \(180^\circ\). So, for \(\angle AFE\) and \(\angle AFG\) (linear pair), \(m\angle AFE + m\angle AFG = 180^\circ\).

Step2: Substitute \(m\angle AFE\)

From step 3, \(m\angle AFE = m\angle CGF\). Substitute into the equation from step 1: \(m\angle CGF + m\angle AFG = 180^\circ\).

Step3: Define Supplementary Angles

If the sum of two angles is \(180^\circ\), they are supplementary. So, \(\angle CGF\) and \(\angle AFG\) are supplementary. Also, by the Linear Pair Postulate (step 1 reasoning) and substitution, we can state the relationship. But for the table, the statement could be \(m\angle AFE + m\angle AFG = 180^\circ\) (using Linear Pair Postulate) or the supplementary relationship. Let's fill the table:

Statement 5: \(m\angle AFE + m\angle AFG = 180^\circ\)
Reason 5: Linear Pair Postulate (or Definition of Linear Pair, which states linear pairs are supplementary)

Alternatively, using substitution:
Statement 5: \(m\angle CGF + m\angle AFG = 180^\circ\)
Reason 5: Substitution Property (substituting \(m\angle AFE\) with \(m\angle CGF\) from step 3 into \(m\angle AFE + m\angle AFG = 180^\circ\) from Linear Pair Postulate)

Let's go with the first approach for the table:

Step1: Identify Linear Pair Property

Linear pair angles sum to \(180^\circ\). So, for \(\angle AFE\) and \(\angle AFG\) (linear pair), \(m\angle AFE + m\angle AFG = 180^\circ\).

Step2: Write Statement and Reason

Statement 5: \(m\angle AFE + m\angle AFG = 180^\circ\)
Reason 5: Linear Pair Postulate (or Definition of a linear pair, which says linear pairs are supplementary)

Answer:

Statement 5: \( \boldsymbol{m\angle AFE + m\angle AFG = 180^\circ} \)
Reason 5: Linear Pair Postulate (or Definition of a linear pair)

(Alternatively, if substituting:
Statement 5: \( \boldsymbol{m\angle CGF + m\angle AFG = 180^\circ} \)
Reason 5: Substitution Property)