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Parallel Lines
and Planes
Section 3 - 1
Definitions
Parallel Lines coplanar lines that do not
intersect
Skew Lines noncoplanar lines
that do not intersect
Parallel Planes Parallel planes do not
intersect
THEOREM 3-1
If two parallel planes are
cut by a third plane, then
the lines of intersection
are parallel.
Transversal is a line that intersects
each of two other
coplanar lines in different
points to produce interior
and exterior angles
ALTERNATE INTERIOR
ANGLES two nonadjacent interior
angles on opposite sides
of a transversal
Alternate Interior
Angles
2
1
4
3
ALTERNATE
EXTERIOR ANGLES two nonadjacent exterior
angles on opposite sides
of the transversal
Alternate Exterior
Angles
6
5
7
8
Same-Side Interior
Angles two interior angles on the
same side of the
transversal
Same-Side Interior
Angles
2
1
4
3
Corresponding Angles two angles in
corresponding positions
relative to two lines cut by
a transversal
Corresponding
Angles
6 2
5 1
4 8
3 7
3-2
Properties of Parallel Lines
Postulate 10
If two parallel lines are cut
by a transversal, then
corresponding angles are
congruent.
THEOREM 3-2
If two parallel lines are cut
by a transversal, then
alternate interior angles
are congruent.
THEOREM 3-3
If two parallel lines are cut
by a transversal, then
same-side interior angles
are supplementary.
THEOREM 3-4
If a transversal is
perpendicular to one of
two parallel lines, then it is
perpendicular to the other
one also.
Section 3 - 3
Proving Lines Parallel
Postulate 11
If two lines are cut by a
transversal and
corresponding angles are
congruent, then the lines are
parallel
THEOREM 3-5
If two lines are cut by a
transversal and alternate
interior angles are
congruent, then the lines
are parallel.
THEOREM 3-6
If two lines are cut by a
transversal and sameside interior angles are
supplementary, then the
lines are parallel.
THEOREM 3-7
In a plane two lines
perpendicular to the same
line are parallel.
THEOREM 3-8
Through a point outside a
line, there is exactly one
line parallel to the given
line.
THEOREM 3-9
Through a point outside a
line, there is exactly one
line perpendicular to the
given line.
THEOREM 3-10
Two lines parallel to a
third line are parallel to
each other.
Ways to Prove Two Lines
Parallel
1. Show that a pair of corresponding angles
2.
3.
4.
5.
are congruent.
Show that a pair of alternate interior angles
are congruent
Show that a pair of same-side interior angles
are supplementary.
In a plane show that both lines are  to a
third line.
Show that both lines are  to a third line
Section 3 - 4
Angles of a Triangle
Triangle – is a figure
formed by the
segments that join
three noncollinear
points
Scalene triangle – is a
triangle with all three
sides of different
length.
Isosceles Triangle – is a
triangle with at least two
legs of equal length and a
third side called the base
Angles at the base are
called base angles
and the third angle is
the vertex angle
Equilateral triangle –
is a triangle with three
sides of equal length
Obtuse triangle – is a
triangle with one
obtuse angle (>90°)
Acute triangle – is a
triangle with three
acute angles (<90°)
Right triangle – is a
triangle with one right
angle (90°)
Equiangular triangle –
is a triangle with three
angles of equal
measure.
Auxillary line – is a
line (ray or segment)
added to a diagram to
help in a proof.
THEOREM 3-11
The sum of the measures
of the angles of a triangle is
180
Corollary
A statement that can
easily be proved by applying
a theorem
Corollary 1
If two angles of one
triangle are congruent to two
angles of another triangle,
then the third angles are
congruent.
Corollary 2
Each angle of an
equiangular triangle has
measure 60°.
Corollary 3
In a triangle, there can be
at most one right angle or
obtuse angle.
Corollary 4
The acute angles of a right
triangle are complementary.
THEOREM 3-12
The measure of an exterior
angle of a triangle equals the
sum of the measures of the
two remote interior angles.
Section 3 - 5
Angles of a Polygon
Polygon – is a closed
plane figure that is
formed by joining three
or more coplanar
segments at their
endpoints, and
Each segment of the
polygon is called a
side, and the point
where two sides meet
is called a vertex, and
The angles
determined by the
sides are called
interior angles.
Convex polygon - is a
polygon such that no line
containing a side of the
polygon contains a point
in the interior of the
polygon.
Diagonal - a segment
of a polygon that joins
two nonconsecutive
vertices.
THEOREM 3-13
The sum of the
measures of the angles of
a convex polygon with n
sides is (n-2)180°
THEOREM 3-14
The sum of the
measures of the exterior
angles of a convex
polygon, one angle at
each vertex, is 360°
Regular Polygon
A polygon that is
both equiangular
and equilateral.
To find the measure
of each interior angle
of a regular polygon
3-6
Inductive Reasoning
Inductive Reasoning
Conclusion based on
several past observations
Conclusion is probably
true, but not necessarily
true.
Deductive Reasoning
Conclusion based on
accepted statements
Conclusion must be true if
hypotheses are true.
THE END