To fill in the blank spaces in the proof, let's begin by looking at the given information and the desired outcome of the proof.
Given:& ∠ 1≅ ∠ 3
Prove:& ∠ 2≅ ∠ 4
Note that the proof corresponds to the given diagram.
We will take a look at the statements that need to be completed one at a time.
Missing Information (a.)
To prove that ∠ 2≅ ∠ 3, we will combine the results from the previous steps. From step two we know that ∠ 1 ≅∠ 2. Also, we are given that ∠ 1≅ ∠ 3. By the Transitive Property of Congruence, we obtain that ∠ 2 ≅ ∠ 3.
∠ 1≅ ∠ 2 ∠ 1≅ ∠ 3 ⇒ ∠ 2 ≅ ∠ 3
Therefore, in the flowchart, we can justify that ∠ 2 ≅ ∠ 3 with the Transitive Property of Equality.
\begin{gathered}
\underline\textbf{Statement}\\
\angle 2 \cong \angle 3 \\
\textbf{a. }\underline{\text{Transitive Property of Congruence}}
\end{gathered}
Missing Information (b.)
To complete the last information, we will again use the Transitive Property of Congruence. From the third step, we know that ∠ 2≅ ∠ 3. In the second step, we also obtained that ∠ 3≅∠ 4. Therefore, by the Transitive Property of Congruence, we conclude that ∠ 2≅∠ 4.
∠ 2≅ ∠ 3 ∠ 3≅ ∠ 4 ⇒ ∠ 2 ≅ ∠ 4
Again, the Transitive Property of Congruence justifies the statement.
\begin{gathered}
\underline\textbf{Statement}\\
\angle 2 \cong \angle 4 \\
\textbf{b. }\underline{\text{Transitive Property of Congruence}}
\end{gathered}
Flowchart Proof
Let's consider one more time the given information and the desired outcome of the proof.
Given:& ∠ 1≅ ∠ 3
Prove:& ∠ 2≅ ∠ 4
We will now write the complete flowchart proof.
Two-column Proof
Finally, we will use the flowchart proof to write a two-column proof. Recall that a two-column proof has numbered statements and corresponding reasons organized in a two-column table.
