Welcome to a lesson on the properties of parallel lines.
The goals are to define the different types of angles
formed by a transversal and parallel lines, as well as how they relate.
And then we'll state the corresponding angle postulate,
which will be the basis for proving many of the properties of parallel lines.
In the previous video we defined the different types of
angles formed when a transversal cuts two lines.
We'll go ahead and review that as well as state how they relate to one another.
Vertical angles are nonadjacent angles formed by two intersecting lines.
And we know from before that these angles are congruent.
So angle one and four are vertical angles and congruent,
so are two, and three, five and eight,
and six and seven. These are vertical angles
and they are also congruent angles. Corresponding angles are two angles in
the same position
with respect to the transversal, but on a different line.
And these angles are also congruent. So if we view angle one as being in the upper
left hand corner of the intersection of the transversal, and the first black line.
Angle five is also on the upper left hand corner
of the intersection of the transversal in red and the second black line.
These are corresponding angles and they are also congruent angles,
as well as angle three and angle seven,
angle two and angle six, as well as angle four and angle eight.
These are corresponding angles and they're also congruent angles.
Alternate interior angles are two angles on the interior,
but opposite side of the transversal. Interior angles are the angles
between the two parallel lines, therefore, the alternate interior angles will be
angle three and angle six,
as well as angle four and angle five.
And again these angles are congruent. Next we have the alternate exterior angles.
Two angles on the exterior but on the opposite side of the transversal.
So angle one and angle eight
are alternate exterior angles, which are congruent,
as well as angle two and angle seven.
And for the last two we have consecutive
interior angles, or same side interior angles.
These are two angles on the interior and on the same side of the transversal.
And these angles are supplementary. Remember supplementary angles
have a sum of one hundred eighty degrees. So angle four and angle six
are consecutive interior angles, as well as angle three angle five,
and these angles are supplementary. And then lastly we have the
consecutive exterior angles, or same side exterior angles.
Two angles on the exterior, but on the same side of the transversal,
and once again these angles are supplementary. So angle two
and angle eight are same side
exterior angles, which are supplementary, as well as angle one and angle seven.
Remember the relationship stated in red are only true if the two lines
intersected by the transversal are parallel lines.
Let's take a look at a wolfram demonstration just to reinforce this idea.
So again, here we see vertical angles in blue, red,
orange and purple. Notice regardless of
how the transversal cuts the two parallel
lines, the angles remain
congruent, or equal in measure.
Now we see the corresponding angles in red,
blue, purple and orange.
And notice that the corresponding angles are congruent
regardless of how the transversal intersects
the two lines.
Consecutive, or same side interior angles in red
and in blue. Notice these are supplementary
not congruent. Again, regardless of
how the transversal cuts the two parallel lines.
Next we have the same side exterior,
or consecutive exterior angles in red and in blue,
again these are supplementary.
The alternate interior angles,
again in blue and red, notice that these are congruent.
And then lastly we have the alternate exterior angles,
again notice they are congruent.
Now we're going to prove many of these in the next several videos.
But in order to prove these we have to first state the corresponding angle
postulate. A postulate is not proved,
but it is assumed to be true because it is self-evident.
So we're going to use this postulate in the next several videos to prove
many of the relationships that we previously stated. And here's the corresponding angle postulate.
This state that if two parallel lines are cut by a transversal,
the corresponding angles are congruent. So we accept this to be true
without a proof. So that means angle one is congruent to angle
five, angle two is congruent to angle six,
angle three is congruent to angle seven,
and angle four is congruent to angle
eight. And hopefully this does make sense. If these two parallel lines
are slanted in the same direction, then if a line was to cut those two,
the corresponding angles that would be formed would be congruent.
We'll go ahead and stop here. I hope you found this helpful.