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Lesson 3. Parallel Lines. Definition. Two lines are parallel if they lie in the same plane and do not intersect. If lines m and n are parallel we write. p. q. p and q are not parallel. m. n.

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lesson 3

Lesson 3

Parallel Lines

definition
Definition
  • Two lines are parallel if they lie in the same plane and do not intersect.
  • If lines m and n are parallel we write

p

q

p and q are not parallel

m

n

slide4
We also say that two line segments, two rays, a line segment and a ray, etc. are parallel if they are parts of parallel lines.

D

C

A

B

P

m

Q

the parallel postulate
The Parallel Postulate
  • Given a line m and a point P not on m, there is one and no more than one line that passes through P and is parallel to m.

P

m

transversals
Transversals
  • A transversal for lines m and n is a line t that intersects lines m and n at distinct points. We say that tcutsm and n.
  • A transversal may also be a line segment and it may cut other line segments.

E

m

A

B

n

D

C

t

slide7
We will be most concerned with transversals that cut parallel lines.
  • When a transversal cuts parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary.
corresponding angles
Corresponding Angles
  • A transversal creates two groups of four angles in each group. Corresponding angles are two angles, one in each group, in the same relative position.

2

1

m

3

4

6

5

n

8

7

alternate interior angles
Alternate Interior Angles
  • When a transversal cuts two lines, alternate interior angles are angles within the two lines on alternate sides of the transversal.

m

1

3

4

2

n

alternate exterior angles
Alternate Exterior Angles
  • When a transversal cuts two lines, alternate exterior angles are angles outside of the two lines on alternate sides of the transversal.

1

3

m

n

4

2

interior angles on the same side of the transversal
Interior Angles on the Same Sideof the Transversal
  • When a transversal cuts two lines, interior angles on the same side of the transversal are angles within the two lines on the same side of the transversal.

m

1

3

2

4

n

exterior angles on the same side of the transversal
Exterior Angles on the Same Sideof the Transversal
  • When a transversal cuts two lines, exterior angles on the same side of the transversal are angles outside of the two lines on the same side of the transversal.

1

3

m

n

4

2

example
Example
  • In the figure and
  • Find
  • Since angles 1 and 2 are vertical, they are congruent. So,
  • Since angles 1 and 3 are

corresponding angles,

they are congruent.

So,

1

3

2

n

m

example14
Example
  • In the figure,

and

  • Find
  • Consider as a transversal for the

parallel line segments.

  • Then angles B and D are

alternate interior angles

and so they are congruent.

So,

A

B

C

E

D

example15
Example
  • In the figure, and
  • If then find
  • Considering as a transversal, we see that angles A and B are interior angles on the same side of the transversal and so they are supplementary.
  • So,
  • Considering as a transversal, we see that angles B and D are interior angles on the same side of the transversal and so they are supplementary.
  • So,

A

B

?

C

D

example16
Example
  • In the figure, bisects

and

  • Find
  • Note that is twice
  • So,
  • Considering as a transversal for the parallel line segments, we see that

are corresponding angles and so they are congruent.

  • So,

A

D

E

C

B

example17
Example
  • In the figure is more than and is less than twice
  • Also, Find
  • Let denote Then
  • Note that angles 2 and 4 are alternate interior angles and so they are congruent.
  • So, Adding 44 and subtracting from both sides gives
  • So, Note that angles 1 and 5 are alternate interior angles, and so

m

n

2

4

3

5

1

proving lines parallel
Proving Lines Parallel
  • So, far we have discussed that if we have a pair of parallel lines, then certain pairs of angles created by a transversal are congruent or supplementary.
  • Now we consider the converse.
  • If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
  • If the alternate interior or exterior angles are congruent, then the lines are parallel.
  • If the interior or exterior angles on the same side of the transversal are supplementary, then the lines are parallel.
example19
Example
  • In the figure, angles A and B are right angles and
  • What is
  • Since these angles are supplementary. Note that they are interior angles on the same side of the transversal This means that
  • Now, since angles C and D are interior angles on the same side of the transversal they are supplementary.
  • So,

A

D

B

C

slide20
In the previous example, there were two lines each perpendicular to a third, and we concluded that the two lines are parallel.
  • This is a nice fact to remember.
  • Given a line m,

if p is perpendicular to m,

and q is perpendicular to m,

then

p

q

m

three parallel lines
Three Parallel Lines
  • In the diagram,

if

and

then

m

n

p

example22
Example
  • In the figure, ,

and Find

  • According to the parallel postulate, there is a line through E parallel to Draw this line and notice that this line is also parallel to
  • Note that and are alternate interior angles and so they are congruent. So,
  • Similarly, and so
  • Therefore,

D

C

2

1

E

A

B