Question

given: lines \\( \ell \\) and \\( m \\) cut by transversal, \\( t ; \angle 2 \\) is supplementary to \\( \angle 8 \\).
prove: \\( \ell \\| m \\)

statements

1. lines \\( \ell \\) and \\( m \\) cut by transversal, \\( t \\);

\\( \angle 2 \\) is supplementary to \\( \angle 8 \\)

1. \\( m \angle 2 + m \angle 8 = 180 ^ { \circ } \\)
2. \\( \angle 6 \\) and \\( \angle 8 \\) are a linear pair of angles

4.

1. \\( m \angle 6 + m \angle 8 = \\)

1. \\( m \angle 2 + m \angle 8 = m \angle 6 + m \angle 8 \\)

7.

1. \\( \angle 2 \cong \angle 6 \\)
2. \\( \ell \\| m \\)

reasons

1. given

2.

1. def. linear pair of angles
2. linear pair theorem
3. def. supplementary angles

6.

1. subtraction prop. of equality

8.

1. converse of the corresponding angles

postulate

Explanation:

Step1: Definition of supplementary angles

Since \(\angle2\) is supplementary to \(\angle8\), by the definition of supplementary angles (if two angles are supplementary, the sum of their measures is \(180^{\circ}\)), we have \(m\angle2 + m\angle8=180^{\circ}\). So, the reason for statement 2 is "Def. supplementary angles".

Step2: Apply Linear Pair Theorem

By the Linear Pair Theorem (if two angles form a linear pair, then they are supplementary), for the linear pair \(\angle6\) and \(\angle8\), we get \(m\angle6 + m\angle8 = 180^{\circ}\). So, statement 4 is \(m\angle6+m\angle8 = 180^{\circ}\)

Step3: Transitive property of equality

Since \(m\angle2 + m\angle8=180^{\circ}\) (from step 1) and \(m\angle6 + m\angle8 = 180^{\circ}\) (from step 2), by the transitive property of equality (if \(a = c\) and \(b=c\), then \(a = b\)), we have \(m\angle2 + m\angle8=m\angle6 + m\angle8\). So, the reason for statement 6 is "Transitive Property of Equality"

Step4: Subtract \(m\angle8\) from both sides

Subtracting \(m\angle8\) from both sides of the equation \(m\angle2 + m\angle8=m\angle6 + m\angle8\) (using the Subtraction Property of Equality: if \(a + b=c + b\), then \(a=c\)), we get \(m\angle2=m\angle6\). So, statement 7 is \(m\angle2=m\angle6\)

Step5: Definition of congruent angles

Since \(m\angle2=m\angle6\), by the definition of congruent angles (two angles are congruent if their measures are equal), we have \(\angle2\cong\angle6\). So, the reason for statement 8 is "Def. congruent angles"

Answer:

1. Def. supplementary angles; 4. \(m\angle6 + m\angle8=180^{\circ}\); 5. \(180^{\circ}\); 6. Transitive Property of Equality; 7. \(m\angle2=m\angle6\); 8. Def. congruent angles