To fill the blank spaces in the , let's begin by looking at the given information and the desired outcome.
Given:& ∠ 1 and ∠ 3 are supplementary.
Prove:& a ∥ b
Note that we are also given a diagram that corresponds to the proof.
Let's take a look at the statements that need to be completed one at a time.
a.
First, we are asked to give the reasoning behind the first sentence , which says that ∠ 1 and ∠ 3 are . Note that this information is given. Therefore, we can fill in the blank space.
\begin{gathered}
\underline\textbf{Statement}\\
\angle 1 \text{ and } \angle 3 \text{ are supplementary.} \\
\textbf{a. }\underline{\,\text{Given}\,}
\end{gathered}
b.
The second sentence containing blanks asks us to give a statement for which the reason is the definition of a . This means that we need to fill in the blank with the names of two that form a linear pair. There is only one pair that satisfies this, which is formed by ∠ 1 and ∠ 2.
\begin{gathered}
\underline\textbf{Statement}\\
\textbf{b. }\underline{\, \angle 1 \text{ and } \angle 2 \text{ form a linear pair.}\,} \\
\text{Def. of linear pair}
\end{gathered}
c.
Now we are asked to state why ∠ 1 and ∠ 2 are supplementary angles. From part b we know that ∠ 1 and ∠ 2 form a linear pair. Therefore, by definition of a linear pair, the sum of their measures is 180^(∘). This means that they are supplementary angles.
\begin{gathered}
\underline\textbf{Statement}\\
\angle 1 \text{ and } \angle 2 \text{ are supplementary.} \\
\textbf{c. }\underline{\,\text{Angles that form a linear pair} \,} \\ \underline{\, \text{are supplementary}\,}
\end{gathered}
d.
Next, we are asked to identify angles that are because they are supplements of the same angle. It was given that ∠ 1 and ∠ 3 are supplementary angles. Furthermore, from part c. we know that ∠ 1 and ∠ 2 are also supplementary. Therefore, ∠ 2 and ∠ 3 are supplements of the same angle — ∠ 1. By the , they are congruent.
\begin{gathered}
\underline\textbf{Statement}\\
\textbf{d. }\underline{\, \angle 2 \cong \angle 3 \,} \\
\text{Supplements of the}\\ \text{same } \angle \text{ are } \cong
\end{gathered}
e.
The last sentence containing blanks asks us to state why a is to line b. We know that ∠ 2 and ∠ 3 are congruent angles. Furthermore, from the diagram we can see that they are .
Therefore, we can apply the .
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Converse of the Corresponding Angles Theorem
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If two lines and a form corresponding angles that are congruent, them the lines are parallel.
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We can use the statement of the theorem to fill in the blank.
\begin{gathered}
\underline\textbf{Statement}\\
a \parallel b \\
\textbf{e. }\underline{\,\text{If corresponding angles are congruent,} \,} \\ \underline{\, \text{then lines are parallel.}\,}
\end{gathered}
Complete Proof
Finally, we can write the complete proof.
Given:& ∠ 1 and ∠ 3 are supplementary.
Prove:& a ∥ b
Proof:
