Unit 3 Parallel Lines

1
Unit 3Parallel Lines

• Learn about parallel line relationships
• Prove lines parallel
• Describe angle relationship in polygons

2
Lecture 1 (3-1)

• Objectives
• State the definition of parallel lines
• Describe a transverse

3
Parallel Lines

• Coplanar lines that do not intersect.

m
m n
n
Skew lines are non-coplanar, non-intersecting
lines.
4
Parallel Planes

• Planes that do not intersect.

Can a plane and a line be parallel?
5
Theorem 3-1

• If two parallel planes are cut by a third plane,
  then the lines of intersection are parallel.

6
The Transversal

• Any line that intersects two or more coplanar
  lines.

7
Special Angle Pairs
t

• Corresponding
• ?1 and ? 5
• Alternate Interior
• ? 4 and ? 5
• Same Side Interior
• ? 4 and ? 6

1
2
3
4
r
5
6
s
7
8
8
Homework Set 3.1

• 3-1 1-39 odd

9
Lecture 2 (3-2)

• Objectives
• Learn the special angle relationships
• 
  when lines
• 
  are parallel

10
When parallel lines are cut by a transversal

• Corresponding ?s ?
• ?1 ? ? 5
• Alternate Interior ?s ?
• ? 4 ? ? 5
• Same Side Interior ?s Suppl.
• ? 4 suppl. ? 6

11
Postulate 10

• If two parallel lines are cut by a
  transversal, then corresponding angles are
  congruent.

12
Theorem 3-2

• If two parallel lines are cut by a
  transversal, then alternate interior angles are
  congruent.

13
Theorem 3-3

• If two parallel lines are cut by a
  transversal, then same side interior angles are
  supplementary.

14
Theorem 3-4

• A line perpendicular to one of two parallel
  lines is perpendicular to the other.

t
r
s
15
Homework Set 3-2

• 3-2 1-21 odd
• WS PM 12

16
Lecture 3 (3-3)

• Objectives
• Learn about ways to prove lines are parallel
• Use Theorems about parallel lines
• Define an auxiliary line

17
Postulate 11

• If two lines are cut by a transversal so that
  corresponding angles are congruent, then the
  lines are parallel.

If ?1? ? 2, then m n.
1
m
2
n
18
Theorem 3-5

• If two lines are cut by a transversal so that
  alternate interior angles are congruent, then the
  lines are parallel.

If ?1? ? 2, then m n.
m
1
2
n
19
Theorem 3-6

• If two lines are cut by a transversal so that
  same side interior angles are supplementary, then
  the lines are parallel.

If ?1 suppl ? 2, then m n.
m
1
2
n
20
Theorem 3-7

• In a plane, two lines perpendicular to the same
  line are parallel.

If t ? m and t ? n , then m n.
t
m
n
21
Auxiliary Lines

• Lines added to a figure to make a proof easier.
  Postulate 6 is the first justification for adding
  a line.

Post 6 Through any two points there is exactly
one line.
A
B
22
Theorem 3-8

• Through a point outside a line, there is exactly
  one line parallel to the given line.

P
m
23
Theorem 3-9

• Through a point outside a line, there is exactly
  one line perpendicular to a given line.

P
m
24
Theorem 3-10

• Two lines parallel to the same line are parallel
  to each other

If p ?? m and m ?? n, then p ?? n
p
m
n
25
Ways to Prove Lines are Parallel

• Corresponding angles are congruent
• Alternate interior angles are congruent
• Same side interior angles are supplementary
• In a plane, that two lines are perpendicular to
  the same line
• Both lines are parallel to a third line

26
Homework Set 3.3

• 3-3 1-21, 25
• WS PM 13
• Quiz next class day

27
Lecture 4 (3-4)

• Objectives
• Classify Triangles
• State the Triangle Sum Theorem
• Apply the Exterior Angle Theorem

28
Types of Triangles
Isosceles
Equilateral
Equiangular
Scalene
Right
Acute
Obtuse
29
The Triangle Sum Theorem

• The three interior angles of a triangle add to
  180.

B
m?A m?B m?C 180
C
A
See It!
30
The Corollaries of Th 3-11

• 1. If two angles of one triangle are congruent to
  two angles of another triangle, then the third
  angles are congruent.
• 2. Each angle of an equiangular triangle has
  measure 60.
• 3. In a triangle, there can be at most one right
  or obtuse angle.
• 4. The acute angles of a right triangle are
  complimentary.

31
The Exterior Angle

• The exterior angle of a triangle is formed
  between a side and a side extended.

3
4
1
2
32
The Exterior Angle Theorem

• The measure of an exterior angle is equal to the
  sum of the measures of the two remote interior
  angles

m?1 m?3 m?4
3
4
1
2
See It!
33
Homework Set 3-4

• 3-4 1-23 odd

34
Lecture 5 (3-5)

• Objectives
• Define a regular and convex polygon
• Calculate the interior and exterior angles of a
  polygon

35
The Polygon

• 1. Each segment intersects exactly two other
  segments, one at each endpoint.
• 2. No two segments with a common endpoint are
  collinear

36
Convex Polygon

• No line containing a side of the polygon contains
  a point in the interior of the polygon.

37
Names of Polygons

• Number of sides Name
• 3
  Triangle
• 4
  Quadrilateral
• 5
  Pentagon
• 6
  Hexagon
• 8
  Octagon
• 10
  Decagon
• n
  n-gon

38
The Diagonal

• A segment that joins non-consecutive vertices.

39
Theorem 3-13

• The sum of the measures of the interior angles of
  a convex polygon with n sides is
• (n-2)180.

40
Theorem 3-14

• The sum of the measures of the exterior angles,
  one at each vertex, of a convex polygon is 360.

41
Regular Polygon

• Is equilateral and equiangular.
• Measure of each interior angle would be
  (n-2)180/n
• Measure of each exterior angle would be 360/n

42
Homework Set Lecture 4

• WS PM 14
• 3-5 1-21 odd

43
Lecture 6 (3-6)

• Objectives
• Discover the uses and hidden dangers of inductive
  logic
• Compare inductive and deductive logic

44
Inductive Logic

• The conclusion is based on seeing a pattern from
  observations.
• Is probably true, but need not be.
• Can be dangerous because of its tendency to allow
  generalization from specific information.

45
Deductive Logic

• The conclusion is based on eliminating all other
  possibilities.
• Two-column proofs are deductive.
• In doing these proofs, we work from specific
  information to a general conclusion.

46
Homework Set 3.6

• WS PM 15
• 3-6 1-25 odd
• Quiz next class day