We are given a paragraph proof with several blanks and asked to fill in those blank spaces. Let's begin by looking at the given information and the desired outcome of the proof.
Given:& ∠ X and ∠ Y are right angles.
Prove:& ∠ X ≅ ∠ Y
Now, let's take a look at the statements that need to be completed one at a time.
Missing Information a
Our first blank is looking at the given information. We know that both ∠ X and ∠ Y are right angles. This means that we can fill in the first blank with ∠ Y.
\begin{gathered}
\underline\textbf{Statement}\\
\angle X \text{ and }\textbf{a. }\underline{\,\angle Y\,}\text{ are right angles} \\
\text{because it is given.}
\end{gathered}
Missing Information b
Our next blank is looking at the definition that applies to the angles. Since we know that the angles are right angles, we also know that they have a measure of 90^(∘). We can therefore fill in the blank with right angles.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of }\textbf{b. } \underline{\text{right angles}}\text{, } \\
m\angle X = 90 \text{ and }m\angle Y = 90.
\end{gathered}
Missing Information c
This statement is looking at the transitive property. The transitive property states:
If a=b and b=c then a=c
If we apply the information from our problem we get the following:
If m∠ X =90 ^(∘) and m∠ Y =90 ^(∘)
then m∠ X = m∠ Y
We can fill in the blank with ∠ Y.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Transitive Property of Equality, } \\
m\angle X=\textbf{c. } \underline{\,\angle Y \,}.
\end{gathered}
Missing Information d
The next statement is looking at properties of congruency. If both angles have equal measures, then the angles are congruent.
If m∠ X = m∠ Y then ∠ X ≅ ∠ Y
Now we can fill in the last blank.
\begin{gathered}
\underline\textbf{Statement}\\
\text{Because angles of equal measure are congruent, }\\
\textbf{d. }\underline{\,\angle X \cong \angle Y \,}.
\end{gathered}
Final Proof
Given:& ∠ X and ∠ Y are right angles.
Prove:& ∠ X ≅ ∠ Y
Proof:∠ X and a. ∠ Y are right angles because it is given. By the definition of b. right angles, m∠ X = 90 and m∠ Y = 90. By the Transitive Property of Equality, m∠ X=c. ∠ Y. Because angles of equal measure are congruent, d.∠ X ≅ ∠ Y.
