In order for corresponding angles to be congruent, the lines must be parallel.

If the lines are not parallel, the angles are not congruent!

Here's a new postulate:

If a transversal cuts through two parallel

lines, then the corresponding angles are

congruent.

Remember that postulates are accepted as true for the foundation of our

system of logic.

Transversals are fun to work with,

especially when the lines through which

they cut are parallel!

When the lines are parallel, all sorts of

secrets begin to unfold.

are alternate exterior

angles. We can use the same method

to prove that all pairs of alternate

exterior angles are congruent, if the

lines are parallel.

But we also know that are

vertical angles, so their measures must be

equal as well.

The fact that the corresponding angles are congruent can then be used to

reveal the relationships of other angle pairs.

If the lines are parallel:

Type an answer, then press Enter.

What angle forms corresponding

angles with

Question #2:

Type an answer, then press Enter.

Question #1:

If the lines are parallel, then

truefalse

Click on the correct answer.

Question #3:

Some people remember alternate interior

angles by the Z shape they form. Do you

see it?

Theorem:

If a transversal cuts through two parallel lines, the alternate interior angles are

congruent.

(Remember, vertical angles are congruent.)

Theorem:

If a transversal cuts through two parallel lines,

the alternate exterior angles are congruent.

Remember that a theorem is a rule that can

be proven true. We don't have to just accept it.

And that means a new theorem!

That means if angle 6 is substituted in

the place of angle 3, angles 4 and 6 are

supplementary.

are supplementary (their

measures total since they form a

linear pair.

With parallel lines as

shown, are congruent,

because they form alternate interior

angles.

The Z shape can be forward or

backward, or sideways, or crooked, as

long as it's some sort of Z.

Sounds like another theorem,

don't you think?

Same-side interior angles are supplementary.

Theorem:

If a transversal cuts through two parallel lines, the same-side interior angles

are supplementary.

Here's a typical congruent angle problem.

4x = 2x + 504x - 2x = 2x - 2x + 50 2x = 50

x = 25

Many geometry problems involving angle

measures also involve algebra.

When the problem involves parallel lines and a

transversal, remember that any two angles

must either be congruent or they are

supplementary.

Theorem:

If a transversal is perpendicular to one of two parallel lines, it is also

perpendicular to the other.

Select any two angles.

Any two angles you select will be either

congruent or supplementary.

If x = 15,

then 4x = 60,

and 2x + 30 = 60.

All the other angles in the figure

are either

If x = 20,then 5x + 15 = 115,

and 2x + 25 = 65.

All the angles in the figure are either

For either of the two previous examples, once you know the value of x, you

can use that value to find all the angle measures.

Here's a typical supplementary angle problem.

(2x + 25) + (5x + 15) = 180 7x + 40 = 180

7x + 40 - 40 = 180 - 40

7x = 140

x = 20

Each of the postulates and theorems discussed in this section are true only if the two lines crossed by the transversal are parallel.

That means, of course, that once you identify one angle in this figure, you can

identify them all.

The converse of a conditional is not necessarily true. For these theorems,

however, they just so happen to be as true as the originals.

Each theorem we have learned also works in reverse. That is, the converse of

the theorem is also true.

The converse of a conditional (if - then) statement is formed by switching the

hypothesis and the conclusion.

If "a" is true, then "b" is true.

becomes

If "b" is true, then "a" is true.

TWO LINES ARE PARALLEL IF:

Corresponding angles are congruent.

We proved that if the lines are parallel, then the angles must fit certain patterns.

The converses tell us that if the angles fit certain patterns, then the lines are

parallel.

Therefore: We can prove that two lines are parallel if we know any of the

following:

TWO LINES ARE PARALLEL IF:

Corresponding angles are congruent.

Alternate exterior angles are congruent.

Alternate interior angles are congruent.

TWO LINES ARE PARALLEL IF:

Corresponding angles are congruent.

Alternate exterior angles are congruent.

So, now you know what happens when you cross two parallel lines with a

transversal!

TWO LINES ARE PARALLEL IF:

Corresponding angles are congruent.

Alternate exterior angles are congruent.

Alternate interior angles are congruent.

Same-side interior angles are supplementary.

You have completed the lesson on Parallelism.

You may go on to take the practice test or review the study guide if you are

unsure about some of the material.