PARALLEL LINES, PERPENDICULAR LINES, AND TRANSVERSAL
Transcript of PARALLEL LINES, PERPENDICULAR LINES, AND TRANSVERSAL
PARALLEL LINES,
PERPENDICULAR LINES,
AND TRANSVERSAL Shirlee Remoto Ocampo
De La Salle University-Manila
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Are the lines parallel?
Parallel Lines
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Parallel lines - coplanar lines
that do not intersect
Perpendicular lines –
intersecting lines that form right
angles
Parallel Lines and Planes
In geometry, two lines in a plane that are always the same
distance apart are ____________.
parallel lines
No two parallel lines intersect, no matter how far you extend them.
Postulate 3.1. Parallel Postulate
There is exactly one line
parallel to a line at a given point
not contained on the line.
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Transversal
a line that intersects two
coplanar lines (not necessarily
parallel) at two distinct points
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Parallel Lines and Transversals
transversal
l
m
B
A
AB is an example of a transversal. It intercepts lines l and m.
Note all of the different angles formed at the points of intersection.
1 2
3 4
5
7
6 8
Parallel Lines and Transversals
Definition of
Transversal
In a plane, a line is a transversal iff it intersects two or more
Lines, each at a different point.
The lines cut by a transversal may or may not be parallel.
l
m
1 2
3 4
5
7
6
8
ml
Parallel Lines
t is a transversal for l and m.
t
1 2
3 4
5
7
6
8
b
c
cb ||
Nonparallel Lines
r is a transversal for b and c.
r
Parallel Lines and Transversals
Two lines divide the plane into three regions.
The region between the lines is referred to as the interior.
The two regions not between the lines is referred to as the exterior.
Exterior
Exterior
Interior
l
m
1 2
3 4
5
7
6
8
Parallel Lines and Transversals
When a transversal intersects two lines, _____ angles are formed. eight
These angles are given special names.
t
Interior angles lie between the two lines.
Exterior angles lie outside the two lines.
Alternate Interior angles
Corresponding angles
Alternate Exterior angles
Same side interior angles
Same side exterior angles
Mathematical Investigation
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1) Draw two parallel lines using
the edges of a ruler.
2) Draw a transversal that
intersects the two parallel lines.
3) Measure each of the 8 angles
formed using a protractor.
4) What conjectures can you
form?
Postulate 3.2. CAP
Corresponding Angles Postulate
If two parallel lines are cut by
a transversal, then each pair of
corresponding angles are
congruent.
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Parallel Lines and Transversals
Theorem 3.1
Alternate
Interior
Angles
(AIT)
If two parallel lines are cut by a transversal, then each pair of
Alternate interior angles is _________.
1 2
3 4
5
7
6
8
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congruent
Another Approach : Indirect Proof
Given: m // n cut be transversal t
Prove: /_3 ≅ /_6
Proof:
Suppose that alternate interior
angles 3 and 6 are not congruent.
It follows that there are two lines
through point A parallel to the same
line. This contradicts the Parallel
Postulate. Thus, /_3 ≅ /_6.
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Parallel Lines and Transversals
1 2
3 4
5
7
6
8
Theorem 3.2
Alternate
Exterior
Angles
(AET)
If two parallel lines are cut by a transversal, then each pair of
alternate exterior angles is _________. congruent
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Parallel Lines and Transversals
1 2
3 4
5
7
6
8
Theorem 3.3
Consecutive
/Same Side
Interior
Angles
(CIT)
If two parallel lines are cut by a transversal, then each pair of
consecutive interior angles is _____________. supplementary
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Parallel Lines and Transversals
1 2
3 4
5
7
6
8
Theorem 3.4
Consecutive
/Same Side
Exterior
Angles
(CET)
If two parallel lines are cut by a transversal, then each pair of
Consecutive/same side exterior angles is _____________. supplementary
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Example:
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a b
y
x
Proving Parallel Lines
Postulate 3.3: Converse of CAP
If two lines are cut by a
transversal so that a pair of
corresponding angles are
congruent, then the lines are
parallel.
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Theorem 3.5: Converse of AIT
If two lines are cut by a
transversal so that a pair of
alternate interior angles are
congruent, then the lines are
parallel.
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Indirect Proof:
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?
Theorem 3.7: Consecutive/Same
Side Interior (CET)Theorem
If two lines are cut by a
transversal so that a pair of
same side interior angles are
supplementary, then the lines
are parallel.
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Theorem 3.8: Consecutive/Same
Side Exterior (CIT)Theorem
If two lines are cut by a
transversal so that a pair of
same side exterior angles are
supplementary, then the lines
are parallel.
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Theorem 3.9: Triangle Angle
Sum Theorem (TAST)
The sum of the measures of all
angles in a triangle is 180o.
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Theorem 3.10 Uniqueness of
Perpendicular Line
Only one line can be drawn
through a point perpendicular to
a given line.
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Perpendicular bisector
Perpendicular bisector of a
segment – segment, ray, line
that is perpendicular to the
segment at its midpoint.
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Problem Set 1:
On a given plane, every
segment has exactly one
perpendicular bisector through a
given point.
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Theorem 3.11
On a plane, if a line is
perpendicular to one of two
parallel lines, then it is
perpendicular to the other.
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Theorem 3.12
On a plane, two lines are
parallel if and only if they are
both perpendicular to the same
line.
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REFERENCE
Remoto-Ocampo, Shirlee.(2010)
Mathematics Ideas and Life
Applications (MILA) III:
Geometry, Philippines: ABIVA
Publishing, Inc.