parallel lines, perpendicular lines, and transversal

PARALLEL LINES,

PERPENDICULAR LINES,

AND TRANSVERSAL Shirlee Remoto Ocampo

De La Salle University-Manila

Shirlee Remoto Ocampo

Are the lines parallel?

Parallel Lines


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.


Transversal

a line that intersects two

coplanar lines (not necessarily

parallel) at two distinct points


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


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


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


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


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.



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

64 53

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.



1 2

3 4

5

7

6

8

Theorem 3.2

Alternate

Exterior

Angles

(AET)


alternate exterior angles is _________. congruent

71 82


1 2

3 4

5

7

6

8

Theorem 3.3

Consecutive

/Same Side

Interior

Angles

(CIT)


consecutive interior angles is _____________. supplementary

18054 18063


1 2

3 4

5

7

6

8

Theorem 3.4

Consecutive

/Same Side

Exterior

Angles

(CET)


Consecutive/same side exterior angles is _____________. supplementary

18081 18072

Example:


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.


Theorem 3.5: Converse of AIT



alternate interior angles are

congruent, then the lines are

parallel.


Indirect Proof:



?

Theorem 3.7: Consecutive/Same

Side Interior (CET)Theorem



same side interior angles are

supplementary, then the lines

are parallel.


Theorem 3.8: Consecutive/Same

Side Exterior (CIT)Theorem



same side exterior angles are

supplementary, then the lines

are parallel.


Theorem 3.9: Triangle Angle

Sum Theorem (TAST)

The sum of the measures of all

angles in a triangle is 180o.


Theorem 3.10 Uniqueness of

Perpendicular Line

Only one line can be drawn

through a point perpendicular to

a given line.


Perpendicular bisector

Perpendicular bisector of a

segment – segment, ray, line

that is perpendicular to the

segment at its midpoint.


Problem Set 1:

On a given plane, every

segment has exactly one

perpendicular bisector through a

given point.


Theorem 3.11

On a plane, if a line is

perpendicular to one of two

parallel lines, then it is

perpendicular to the other.


Theorem 3.12

On a plane, two lines are

parallel if and only if they are

both perpendicular to the same

line.



REFERENCE

Remoto-Ocampo, Shirlee.(2010)

Mathematics Ideas and Life

Applications (MILA) III:

Geometry, Philippines: ABIVA

Publishing, Inc.

parallel lines, perpendicular lines, and transversal

Documents