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Given: [tex] \angle 3 \cong \angle 4 [/tex]

Prove: [tex] \angle 1 \cong \angle 2 [/tex]

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] in for [tex] m \angle 3 [/tex] to rewrite the previous equation as [tex] m \angle 2 + m \angle 4 = 180^{\circ} [/tex]. Thus, [tex] \angle 2 [/tex] is supplementary to [tex] \angle 4 [/tex] by the definition of supplementary angles. By the linear pair theorem, [tex] \angle 1 [/tex] is supplementary to [tex] \angle 4 [/tex]. Since [tex] \angle 2 [/tex] and [tex] \angle 1 [/tex] are supplementary to [tex] \angle 4 [/tex], then by the congruent supplements theorem, [tex] \angle 1 \cong \angle 2 [/tex].

Use the paragraph proof to complete the two-column proof.

What statement and reason belong in line 5?

\begin{tabular}{|l|l|}
\hline
\textbf{Statements} & \textbf{Reasons} \\
\hline
1. [tex] \angle 2 [/tex] is supplementary to [tex] \angle 3 [/tex] & 1. Linear pair theorem \\
\hline
2. [tex] m \angle 2 + m \angle 3 = 180^{\circ} [/tex] & 2. Definition of supplementary angles \\
\hline
3. [tex] \angle 3 \cong \angle 4 [/tex] & 3. Given \\
\hline
4. [tex] m \angle 3 = m \angle 4 [/tex] & 4. Definition of congruence \\
\hline
5. [tex] m \angle 2 + m \angle 4 = 180^{\circ} [/tex] & 5. Substitution property of equality \\
\hline
6. [tex] \angle 2 [/tex] is supplementary to [tex] \angle 4 [/tex] & 6. Definition of supplementary angles \\
\hline
7. [tex] \angle 1 [/tex] is supplementary to [tex] \angle 4 [/tex] & 7. Linear pair theorem \\
\hline
8. [tex] \angle 1 \cong \angle 2 [/tex] & 8. Congruent supplements theorem \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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.

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