Answer :
To find the correct statement and reason for line 5, let's analyze the paragraph proof:
By the linear pair theorem, [tex]$\angle 2$[/tex] is supplementary to [tex]$\angle 3$[/tex], which means [tex]$m \angle 2 + m \angle 3 = 180^{\circ}$[/tex]. It is given that [tex]$\angle 3 \cong \angle 4$[/tex], so by the definition of congruent angles, [tex]$m \angle 3 = m \angle 4$[/tex]. Using the substitution property of equality, substitute [tex]$m \angle 4$[/tex] for [tex]$m \angle 3$[/tex] to rewrite the previous equation as [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex].
This process by substitution is justified as follows:
- Statement for line 5: [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex].
- Reason for line 5: Substitution property of equality.
So, the correct entries for the two-column proof are:
[tex]\[ \begin{array}{|l|l|} \hline \text{Statements} & \text{Reasons} \\ \hline 1. \angle 2 \text{ is supplementary to } \angle 3 & 1. \text{Linear pair theorem} \\ \hline 2. m \angle 2 + m \angle 3 = 180^\circ & 2. \text{Definition of supplementary angles} \\ \hline 3. \angle 3 \cong \angle 4 & 3. \text{Given} \\ \hline 4. m \angle 3 = m \angle 4 & 4. \text{Definition of congruence} \\ \hline 5. m \angle 2 + m \angle 4 = 180^\circ & 5. \text{Substitution property of equality} \\ \hline 6. m \angle 2 \text{ is supplementary to } m \angle 4 & 6. \text{Definition of supplementary angles} \\ \hline 7. m \angle 1 \text{ is supplementary to } m \angle 4 & 7. \text{Linear pair theorem} \\ \hline 8. \angle 1 \cong \angle 2 & 8. \text{Congruent supplements theorem} \\ \hline \end{array} \][/tex]
The above table completes the two-column proof based on the step-by-step logical reasoning provided in the paragraph proof.
By the linear pair theorem, [tex]$\angle 2$[/tex] is supplementary to [tex]$\angle 3$[/tex], which means [tex]$m \angle 2 + m \angle 3 = 180^{\circ}$[/tex]. It is given that [tex]$\angle 3 \cong \angle 4$[/tex], so by the definition of congruent angles, [tex]$m \angle 3 = m \angle 4$[/tex]. Using the substitution property of equality, substitute [tex]$m \angle 4$[/tex] for [tex]$m \angle 3$[/tex] to rewrite the previous equation as [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex].
This process by substitution is justified as follows:
- Statement for line 5: [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex].
- Reason for line 5: Substitution property of equality.
So, the correct entries for the two-column proof are:
[tex]\[ \begin{array}{|l|l|} \hline \text{Statements} & \text{Reasons} \\ \hline 1. \angle 2 \text{ is supplementary to } \angle 3 & 1. \text{Linear pair theorem} \\ \hline 2. m \angle 2 + m \angle 3 = 180^\circ & 2. \text{Definition of supplementary angles} \\ \hline 3. \angle 3 \cong \angle 4 & 3. \text{Given} \\ \hline 4. m \angle 3 = m \angle 4 & 4. \text{Definition of congruence} \\ \hline 5. m \angle 2 + m \angle 4 = 180^\circ & 5. \text{Substitution property of equality} \\ \hline 6. m \angle 2 \text{ is supplementary to } m \angle 4 & 6. \text{Definition of supplementary angles} \\ \hline 7. m \angle 1 \text{ is supplementary to } m \angle 4 & 7. \text{Linear pair theorem} \\ \hline 8. \angle 1 \cong \angle 2 & 8. \text{Congruent supplements theorem} \\ \hline \end{array} \][/tex]
The above table completes the two-column proof based on the step-by-step logical reasoning provided in the paragraph proof.
