Transversals are fun to work with, - stratlab.wikispaces.com - Geom F - 7...system of logic....
Transcript of Transversals are fun to work with, - stratlab.wikispaces.com - Geom F - 7...system of logic....
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.