**Theorem 6.1**: If two
parallel
lines are transected by a
third, the alternate interior
angles are the same
size.

**Proof**: Let* **A *and* **B *denote the points where the transverse
line meets the
parallel
lines. Let* **C *be the midpoint of the
line segment between* **A *and* **B*.* *Let* **DE *be the line
through* **C *which is perpendicular to one,
and hence, by Theorem
1.14, both of the
parallel
lines.

Let* **D *be the point where
the perpendicular
line meets the
line which contains the
point * **A *and let* **E *be the point where the
perpendicular
line meets the
line that contains the
point * B*.* *If we rotate the figure* *180^{o }about point * **C*,* *then point * A *is moved to point * **B *and vice versa. Let* D*'* *be the point to which* D *is moved. Then* **D*'* *and* B *are points on the line to which the line which is determined by* AD *is moved, So the line determined by* **D*'* *and* B* is the line to which the line determined by* AD *is moved . Since rotations preserve the size of angles, by Theorem 5.6, * *__/ __*CD*'*B* is a right angle. Thus, the line determined by* **D*'* *and* B *is a line through* B *which is parallel to* AD *by Theorem 1.15. But, there is only one line through* **B*,* *which is parallel to* **AD*,* *so* BE* = *BD*'.* *So* *__/ __*CAD *is moved to* *__/ __*CBE*.* *Since a rotation is an invertible
isometry by
Theorem 5.12, it preserves the
size of the angles by Theorem 5.6. We can conclude
that the alternate interior
angles are the same
size.