Consecutive Interior Angles Theorem
Consecutive Interior Angles
When two lines are cut by a transversal, the pair of angles on one side of the transversal and inside the two lines are called the consecutive interior angles.
In the figure, the angles $3$ and $5$ are consecutive interior angles.
Also the angles $4$ and $6$ are consecutive interior angles.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary.
Proof:
Given: $k\parallel l$ , $t$ is a transversal
Prove: $\angle 3$ and $\angle 5$ are supplementary and $\angle 4$ and $\angle 6$ are supplementary.
Statement

Reason


1

$k\parallel l$
,
$t$
is a traversal.

Given

2

$\angle 1$
and
$\angle 3$
form a linear pair and
$\angle 2$
and
$\angle 4$
form a linear pair.

Definition of
linear pair

3

$\angle 1$ and $\angle 3$ are supplementary $m\angle 1+m\angle 3=180\xb0$ $\angle 2$ and $\angle 4$ are supplementary $m\angle 2+m\angle 4=180\xb0$ 

4

$\angle 1\cong \angle 5$
and
$\angle 2\cong \angle 6$

Corresponding Angles Theorem

5

$\angle 3$
and
$\angle 5$
are supplementary
$\angle 4$
and
$\angle 6$
are supplementary.

Substitution Property
