Question

1. 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° (4) transitive property of equality (5) subtraction property of equality (6) definition of congruence reasons:

Explanation:

Step1: State the given

$x\parallel y$ (Given)

Step2: Use same - side interior angles theorem

Since $x\parallel y$, $\angle3$ and $\angle8$ are same - side interior angles, so $m\angle3 + m\angle8=180^{\circ}$ (Same - side interior angles are supplementary when lines are parallel)

Step3: State another angle - sum equation

$\angle5$ and $\angle8$ are also related in a way that $m\angle5 + m\angle8 = 180^{\circ}$ (Linear pair of angles are supplementary)

Step4: Apply transitive property of equality

Since $m\angle3 + m\angle8=180^{\circ}$ and $m\angle5 + m\angle8 = 180^{\circ}$, then $m\angle3=m\angle5$ (Transitive Property of Equality)

Step5: Use definition of congruence

If $m\angle3=m\angle5$, then $\angle3\cong\angle5$ (Definition of congruence: Two angles are congruent if their measures are equal)

Answer:

The two - column proof is completed as follows:

Statements	Reasons
2. $m\angle3 + m\angle8=180^{\circ}$	Same - side interior angles are supplementary when lines are parallel
3. $m\angle5 + m\angle8 = 180^{\circ}$	Linear pair of angles are supplementary
4. $m\angle3=m\angle5$	Transitive Property of Equality
5. $\angle3\cong\angle5$	Definition of congruence