We are given a with several blanks and are asked to fill in those blank spaces. Then, we will also write a .
Let's begin by looking at the given information and the desired outcome of the proof.
Given:& ∠1 and ∠2 are complementary.
& ∠1 and∠3 are complementary.
Prove:& ∠2 ≅ ∠3
Now, let's take a look at the statements that need to be completed one at a time.
First and Second Missing Information
Our first sentence containing blanks asks us what is the name of a pair of angles whose measures add up to 90^(∘). Since we know that the sum of the measures of is 90^(∘), we can fill the first blank with the word complementary.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of } \underline{\text{complementary}}\text{ angles,}\\
m\angle1+m \angle2= 90^\circ \text{ and } \underline{\phantom{m\angle1 +m\angle3}}= 90^\circ.
\end{gathered}
In the second blank we are supposed to write the sum of two angle measures which add to 90^(∘). From the given information we know that ∠1 and ∠3 are complementary angles. Therefore, the sum of their measures is 90^(∘). Therefore, we can fill the blank with m∠ 1+m∠ 3.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of } \underline{\text{complementary}}\text{ angles,}\\
m\angle1+m \angle2= 90^\circ \text{ and } \underline{m\angle1 +m\angle3}= 90^\circ.
\end{gathered}
Third Missing Information
The next statement asks us about the name of the property used to write the equation m∠ 1+m∠ 2 = m∠ 1+m∠ 3. Since we have two expressions that are equal to 90^(∘), these expressions must be equal to each other. This property is called .
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the } \underline{ \text{Transitive Property of Equality,} }\\
m\angle 1+m\angle 2=m\angle 1+m\angle 3.
\end{gathered}
Fourth Missing Information
The next statement asks us about the result of using the . Let's perform a subtraction on both sides of the equation.
m ∠ 1+m ∠2=m ∠ 1 +m ∠ 3
- m ∠ 1 - m ∠ 1
m ∠ 2 = 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,}\\
\underline{ m\angle 2=m\angle 3}.
\end{gathered}
Fifth Missing Information
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 . 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{So, }\angle 2 \cong \angle 3\\
\text{by the definition of } \underline{\text{congruent angles.}}
\end{gathered}
Completed Proof
∠ 1 and ∠ 2 are complementary, and ∠ 1 and ∠ 3 are complementary. By the definition of complementary angles, m∠ 1+m∠ 2=90^(∘) and m∠ 1+m∠ 3=90^(∘). By the Transitive Property of Equality, m∠ 1+m∠ 2=m∠ 1+m∠ 3. By the Subtraction Property of Equality, m∠ 2=m∠ 3. So, ∠ 2 ≅ ∠ 3 by the definition of congruent angles.
Two-column proof
Finally, we can summarize all steps in a two-column proof.
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2. Definition of complementary angles
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3. Transitive Property of Equality
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4. Subtraction Property of Equality
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5. Definition of congruent angles
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