Question

1. use the given information to prove that ∠2≅∠7. given: x∥y prove: ∠2≅∠7 4. use the given information to prove that ∠3≅∠6. given: a∥b prove: ∠3≅∠6

Explanation:

Step1: Recall corresponding - angles property

If two parallel lines are cut by a transversal, corresponding angles are congruent.

Step2: Identify corresponding - angles for $\angle2$ and $\angle7$

For parallel lines $x\parallel y$ cut by a transversal, $\angle2$ and $\angle7$ are corresponding angles.

Step3: Prove $\angle2\cong\angle7$

Since $x\parallel y$ (given) and $\angle2$ and $\angle7$ are corresponding angles, by the corresponding - angles postulate, $\angle2\cong\angle7$.

Step4: Recall alternate - interior angles property

If two parallel lines are cut by a transversal, alternate - interior angles are congruent.

Step5: Identify alternate - interior angles for $\angle3$ and $\angle6$

For parallel lines $a\parallel b$ cut by a transversal, $\angle3$ and $\angle6$ are alternate - interior angles.

Step6: Prove $\angle3\cong\angle6$

Since $a\parallel b$ (given) and $\angle3$ and $\angle6$ are alternate - interior angles, by the alternate - interior angles theorem, $\angle3\cong\angle6$.

Answer:

For the first proof:

Statement	Reason
$\angle2\cong\angle7$	Corresponding angles postulate (since $x\parallel y$ and $\angle2$ and $\angle7$ are corresponding angles)

For the second proof:

Statement	Reason
$\angle3\cong\angle6$	Alternate - interior angles theorem (since $a\parallel b$ and $\angle3$ and $\angle6$ are alternate - interior angles)