*Postulate 11* can be used to derive additional theorems regarding parallel lines cut by a transversal. Because *m* ∠1 + *m* ∠2 = 180 ° and *m* ∠5 + *m* ∠6 = 180° (because adjacent angles whose noncommon sides lie on a line are supplementary), and because *m* ∠1 = *m* ∠3, *m*∠2 = *m* ∠4, *m* ∠5 = *m* ∠7, and *m* ∠6 = *m* ∠8 (because vertical angles are equal), all of the following theorems can be proven as a consequence of *Postulate 11.*

*Theorem 13:* If two parallel lines are cut by a transversal, then alternate interior angles are equal.

*Theorem 14:* If two parallel lines are cut by a transversal, then alternate exterior angles are equal.

*Theorem 15:* If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

*Theorem 16:* If two parallel lines are cut by a transversal, then consecutive exterior angles are supplementary.

The above postulate and theorems can be condensed to the following theorems:

*Theorem 17:* If two parallel lines are cut by a transversal, then every pair of angles formed are either equal or supplementary.

*Theorem 18:* If a transversal is perpendicular to one of two parallel lines, then it is also perpendicular to the other line.

Based on *Postulate 11* and the theorems that follow it, all of the following conditions would be true if *l* // *m* (Figure 1

**Figure 1 **Two parallel lines cut by a transversal.

Based on *Postulate 11:*

*m*∠1 =*m*∠5

*m*∠4 =*m*∠8

*m*∠2 =*m*∠6

*m*∠3 =*m*∠7

Based on *Theorem 13:*

*m*∠3 =*m*∠5

*m*∠4 =*m*∠6

Based on *Theorem 14:*

*m*∠1 =*m*∠7

*m*∠2 =*m*∠8

Based on *Theorem 15:*

- ∠3 and ∠6 are supplementary

- ∠4 and ∠5 are supplementary

Based on *Theorem 16:*

- ∠1 and ∠8 are supplementary

- ∠2 and ∠7 are supplementary

Based on *Theorem 18:*

If *t* ⊥ *l,* then *t* ⊥ *m*