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:& ∠1 and ∠2 are complementary.
& ∠2 and∠3 are complementary.
Prove:& ∠1 ≅ ∠3
Now, let's take a look at the statements that need to be completed one at a time.
Missing Information a and b
Our first sentence containing blanks asks us to look at what it means for angles to be complementary. We know that the sum of complementary angles is 90^(∘). Therefore, we can fill in both blanks, a and b, with 90.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of complementary angles,}\\
m\angle1+m \angle2= \textbf{a. }\underline{\,90\,} \text{ and } m\angle3 +m\angle2= \textbf{b. }\underline{\,90\,}.
\end{gathered}
Missing Information c
Our next statement creates an equation using the Transitive Property of Equality and then asks us to subtract m∠2 from either side.
m ∠ 1+m ∠2=m ∠ 3 +m ∠ 2
-m ∠ 2 -m ∠ 2
m ∠ 1 = m ∠ 3
Now, we can use this calculation to fill in the appropriate blank.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the Subtraction Property of Equality,}\\
\text{you get }m\angle1 = \textbf{ c.}\underline{\,m\angle3\,}.
\end{gathered}
Missing Information d
Our last statement is asking us to think about the relationship between angles which have the same measure. Angles that have the same measure are considered congruent. This is also the meaning of the symbol ( ≅) used in the conclusion. Let's fill in the last blank!
\begin{gathered}
\underline\textbf{Statement}\\
\text{Angles with the same measure are }\\
\textbf{d. } \underline{\text{congruent}}, \text{ so }\angle1 \cong \angle3.
\end{gathered}
Completed Proof
Considering the given information, we can summarize all the steps in a paragraph proof.
Given:& ∠1 and ∠2 are complementary.
& ∠2 and∠3 are complementary.
Prove:& ∠1 ≅ ∠3
Proof: ∠1 and ∠2 are complementary and ∠3 and ∠2 are complementary because it is given. By the definition of complementary angles, m∠1+m∠2= a. 90 and m∠3 +m∠2= b. 90. Then m∠1+m∠2=m∠3 +m∠2 by the Transitive Property of Equality. Subtract m∠2 from each side. By the Subtraction Property of Equality, you get m∠1 = c.m∠3. Angles with the same measure are d.congruent, so ∠1 ≅ ∠3.
