Question

complete the two - column proof. given: x||y prove: ∠3≅∠5 statements: 1) x||y 2) m∠3 + m∠8 = 180 3) m∠5 + m∠8 = 180

Explanation:

Step1: Given parallel lines

Given \(x\parallel y\).

Step2: Apply same - side interior angles postulate

Since \(x\parallel y\), by the Same - Side Interior Angle Postulate, \(\angle3\) and \(\angle8\) are same - side interior angles, so \(m\angle3 + m\angle8=180\).

Step3: Another pair of same - side interior angles

\(\angle5\) and \(\angle8\) are also same - side interior angles. So \(m\angle5 + m\angle8 = 180\) by the Same - Side Interior Angle Postulate.

Step4: Use subtraction property of equality

From \(m\angle3 + m\angle8=180\) and \(m\angle5 + m\angle8 = 180\), we can subtract \(m\angle8\) from both equations.
If \(m\angle3 + m\angle8=180\) and \(m\angle5 + m\angle8 = 180\), then \(m\angle3=180 - m\angle8\) and \(m\angle5=180 - m\angle8\).
By the Subtraction Property of Equality, \(m\angle3=m\angle5\).

Step5: Use definition of congruence

By the Definition of Congruence (if two angles have the same measure, they are congruent), \(\angle3\cong\angle5\).

Answer:

1. Given
2. Same - Side Interior Angle Postulate
3. Same - Side Interior Angle Postulate
4. Subtraction Property of Equality
5. Definition of congruence; \(\angle3\cong\angle5\)