Question

1. given: ∠1 and ∠2 form a linear pair; ( mangle2 + mangle3 = 180^circ ) prove: ( angle1congangle3 ) statements 1. ∠1 and ∠2 form a linear pair 2. 3. given 4. definition of supplementary angles 5. ( angle1congangle3 ) reasons 1. 2. the supplement theorem 3. 4. 5.

Explanation:

Step1: <Linear pair property>

Since \(\angle1\) and \(\angle2\) form a linear pair, by the definition of a linear pair, \(m\angle1 + m\angle2=180^{\circ}\) (Statement 2: \(m\angle1 + m\angle2 = 180^{\circ}\), Reason: Definition of a linear pair).

Step2: <Given equation>

We are given that \(m\angle2+m\angle3 = 180^{\circ}\) (Statement 3: \(m\angle2 + m\angle3=180^{\circ}\), Reason: Given).

Step3: <Transitive property of equality>

From \(m\angle1 + m\angle2=180^{\circ}\) and \(m\angle2 + m\angle3=180^{\circ}\), we can say \(m\angle1+m\angle2=m\angle2 + m\angle3\) (Statement 4: \(m\angle1=m\angle3\), Reason: Subtraction property of equality (subtract \(m\angle2\) from both sides)).

Step4: <Definition of congruent angles>

If \(m\angle1=m\angle3\), then \(\angle1\cong\angle3\) (Statement 5: \(\angle1\cong\angle3\), Reason: Definition of congruent angles (angles with equal measures are congruent)).

Answer:

The completed proof table has the following statements and reasons:

Statements	Reasons
2. \(m\angle1 + m\angle2 = 180^{\circ}\)	Definition of a linear pair
3. \(m\angle2 + m\angle3=180^{\circ}\)	Given
4. \(m\angle1=m\angle3\)	Subtraction property of equality
5. \(\angle1\cong\angle3\)	Definition of congruent angles