Download Parallel Lines and Planes

9/24/2010
Parallel Lines
and Planes
3-1
Definitions
Parallel Lines coplanar lines that do not
intersect
Parallel Planes Parallel planes that do not
intersect
Skew Lines noncoplanar lines
THEOREM 3-1
If two parallel planes are
cut by a third plane, then
the lines of intersection
are parallel.
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Transversal is a line that intersects
each of two other
coplanar lines in different
points to produce interior
and exterior angles
Interior Angles
2
1
4
Interior Angles The angles formed
between the two coplanar
lines and the transversal
Exterior Angles The angles formed
outside the two coplanar
lines and the transversal
3
Exterior Angles
8
6
5
7
ALTERNATE INTERIOR
ANGLES two nonadjacent interior
angles on opposite sides
of a transversal
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Alternate Interior
Angles
2
1
4
3
Alternate Exterior
Angles
8
6
5
7
Same-Side Interior
Angles
2
1
4
3
ALTERNATE
EXTERIOR ANGLES two nonadjacent exterior
angles on opposite sides
of the transversal
Same-Side Interior
Angles two interior angles on the
same side of the
transversal
Corresponding Angles two angles in
corresponding positions
relative to two lines cut by
a transversal
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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 3-3
If two parallel lines are cut
by a transversal, then
same-side interior angles
are supplementary.
THEOREM 3-2
If two parallel lines are cut
by a transversal, then
alternate interior angles
are congruent.
THEOREM 3-4
If a transversal is
perpendicular to one of
two parallel lines, then it is
perpendicular to the other
one also.
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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.
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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
Triangle – is a figure
formed by the
segments that join
three noncollinear
points
3-4
Angles of a Triangle
Vertex – point of a
triangle
Side – segment of a
triangle
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Scalene triangle – is a
Isosceles Triangle – is
triangle with all three
sides of different
length.
a triangle with at least
two legs of equal
length and a third side
called the base and
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
Acute triangle – is a
Obtuse triangle – is a
triangle with three
acute angle (<90°)
triangle with one
obtuse angle (>90°)
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Right triangle – is a
triangle with one right
angle (90°)
Auxillary line – is a
line (ray or segment)
added to a diagram to
help in a proof.
Corollary
Equiangular triangle –
is a triangle with three
angles of equal
measure.
THEOREM 3-11
The sum of the measures
of the angles of a triangle is
180°
Corollary 1
A statement that can
If two angles of one
easily be proved by applying
a theorem
triangle are congruent to two
angles of another triangle,
then the third angles are
congruent.
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Corollary 2
Corollary 3
Each angle of an
In a triangle, there can be
equiangular triangle has
measure 60°.
at most one right angle or
obtuse angle.
Corollary 4
THEOREM 3-12
The acute angles of a right
The measure of an exterior
triangle are complementary.
angle of a triangle equals the
sum of the measures of the
two remote interior angles.
Polygon – is a closed
3-5
Angles of a Polygon
plane figure that is
formed by joining three
or more coplanar
segments at their
endpoints, and
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Each segment of the
polygon is called a
side, and the point
where two sides meet
is called a vertex, and
Convex polygon - is a
polygon such that no line
containing a side of the
polygon contains a point
in the interior of the
polygon.
The angles
determined by the
sides are called
interior angles.
Diagonal - a segment
of a polygon that joins
two nonconsecutive
vertices.
THEOREM 3-14
THEOREM 3-13
The sum of the
measures of the angles of
a convex polygon with n
sides is (n-2)180°
The sum of the
measures of the exterior
angles of a convex
polygon, one angle at
each vertex, is 360°
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Inductive Reasoning
3-6
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
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