\[\overleftrightarrow{AB}\] is parallel to

\[\overleftrightarrow{CD}\].
Parallel lines C D and A B. There is a transversal line through points B and C. Angle A B C is labeled one. Angle B C D is labeled two..
\[A\]
\[B\]
\[C\]
\[D\]
\[1\]
\[2\]
Parallel lines C D and A B. There is a transversal line through points B and C. Angle A B C is labeled one. Angle B C D is labeled two..
Complete the proof that the alternate interior angles of transversals of parallel lines are congruent.
Note: this proof is for the case where
\[m\angle1\] is less than
\[90\degree\].
This proof uses the following theorem: Any point on one parallel line is the same distance from the other line on a perpendicular transversal.
Statement or construction Reason
1


\[\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}\] Given
2 Construct

\[\overline{BE}\] perpendicular to

\[\overleftrightarrow{CD}\] such that point
\[E\] is on

\[\overleftrightarrow{CD}\].
3 Construct

\[\overline{CF}\] perpendicular to

\[\overleftrightarrow{AB}\] such that point
\[F\] is on

\[\overleftrightarrow{AB}\].
4
\[m\angle CFB = m\angle BEC = 90\degree\] All perpendicular angles measure
\[90\degree\] (2, 3).
5
\[CF=\]

Any point on one parallel line is the same distance from the other line on a perpendicular transversal (1, 2, 3).
6
\[BC=BC\] They are measures of the same segment.
7
\[\triangle BCF \cong \triangle CBE\]

congruence (4, 6, 5)
8
\[\angle FBC \cong \angle ECB\] Corresponding parts of congruent figures are congruent (7).