Question

given: j || k
prove: ∠1 and ∠7 are supp ∠s
statements reasons
j || k
∠1 + ∠5 = 180
∠5 = ∠7
∠1 + ∠7 = 180
∠1 and ∠7 are supp ∠s

Explanation:

Step1: Given information

Given

Step2: Use same - side interior angles theorem

Same - side interior angles of parallel lines are supplementary. Since \(j\parallel k\), \(\angle1\) and \(\angle5\) are same - side interior angles, so \(\angle1+\angle5 = 180^{\circ}\)

Step3: Use vertical angles property

Vertical angles are equal. \(\angle5\) and \(\angle7\) are vertical angles, so \(\angle5=\angle7\)

Step4: Substitute \(\angle5\) with \(\angle7\)

By substitution property of equality, if \(\angle1+\angle5 = 180^{\circ}\) and \(\angle5=\angle7\), then \(\angle1+\angle7 = 180^{\circ}\)

Step5: Definition of supplementary angles

If the sum of two angles is \(180^{\circ}\), they are supplementary. Since \(\angle1+\angle7 = 180^{\circ}\), \(\angle1\) and \(\angle7\) are supplementary angles

Answer:

Statements	Reasons
\(\angle1+\angle5 = 180\)	Same - side interior angles of parallel lines are supplementary
\(\angle5=\angle7\)	Vertical angles are equal
\(\angle1+\angle7 = 180\)	Substitution property of equality
\(\angle1\) and \(\angle7\) are supp \(\angle s\)	Definition of supplementary angles