Parallel Lines Theorems

presenterawfulElectronics - Devices

Oct 10, 2013 (3 years and 8 months ago)

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Paraphrase of Euclid’s Parallel Postulate


Through a point outside a line, exactly one parallel can be drawn to the line.


Parallel Lines Theorems


If two parallel lines are cut by a transversal, then corresponding angles are
congruent.


If two parallel li
nes are cut by a transversal, then alternate interior angles are
congruent.


If two parallel lines are cut by a transversal, then same side interior (consecutive
interior) angles are supplementary.


If a transversal is perpendicular to one of two parallel
lines, then it is also
perpendicular to the other.


Converses of Parallel Lines Theorems


If two lines are cut by a transversal and corresponding angles are congruent,
then the lines are parallel.


If two lines are cut by a transversal and alternate interi
or angles are congruent,
then the lines are parallel.


If two lines are cut by a transversal and same side interior angles are
supplementary, then the lines are parallel.


If a transversal is perpendicular to each of two lines, then the two lines are
paral
lel.



Congruent Triangle Postulates


If three sides of one triangle are congruent to three sides of another triangle, then
the triangles are congruent (SSS).


If two sides of one triangle are congruent to two sides of another triangle, and the
angles betw
een these pairs of sides are congruent, then the triangles are
congruent (SAS).


If two angles of one triangle are congruent to two angles of another triangle, and
the sides between these pairs of angles are congruent, then the triangles are
congruent (ASA
).


Correspondin
g parts of congruent triangles
are congruent (CPCTC).

Theorems

(that should make perfect sense to you)


1. If two angles are complements of congruent angles (or of the same angle),


then the two angles are congruent.


2. If two ang
les are supplements of congruent angles (or of the same angle),


then the two angles are congruent.


3. Vertical angles are congruent.


4. If two lines intersect, then they intersect in exactly one point.


5. If there is a line and a point not i
n the line, then exactly one plane contains


them.


6. If two lines intersect, then exactly one plane contains them.


7. Every segment has exactly one midpoint.


8. Every angle has exactly one bisector.



9.
The angles of a linear pair are supplem
entary.


10.
If two angles are congruent and supplementary, they are right angles.



Corollaries of the
Sum of the Angles of a Triangle

Theorem



Through a point outside a line, exactly one perpendicular ca
n be drawn
to the line.


If two angles of one tr
iangle are congruent to two angles of another

triangle, then
the third angles are congruent.



Each angle of an equiangular triangle has measure 60
o
.



In a triangle, there can be at most one right angle or one obtuse angle.




The acute angles of a ri
ght triangle are complementary.



If one side of a triangle is extended, then the measure of the exterior
angle
formed is equal to the sum of
the measures of the two remote
interior (non
-
adjacent interior) angles.



The sum of the measures of the interior

ang
les of a convex polygon with
n sides
is (n


2)180
o
.



The sum of the measures of the exterio
r angles of any convex polygon
one angle
at each vertex, is 360
o
.