Lines and Angles

1
Lines and Angles

• You will be able to identify relationships
  between lines and angles formed by transversals.

2
Relationships between lines

• Parallel Lines? two lines that are coplanar and
  do not intersect.
• Skew Lines? lines that do not intersect and are
  not coplanar.
• Parallel Planes? two planes that do not
  intersect.
• Segments and rays can also be parallel if they
  lie on parallel lines.
• The notation for parallel lines
• line m and line n are parallel ? m n

3
Parallel and Perpendicular Postulates

• Parallel Postulate? If there is a line and a
  point not on the line, then there is exactly one
  line through the point parallel to the given
  line.
• Perpendicular Postulate? If there is a line and a
  point not on the line, then there is exactly one
  line through the point perpendicular to the given
  line.

4
Try It Out!

• Draw a line (a straight one!) and a point
  anywhere in relation to that line (above or
  below).
• How many lines can you draw through that one
  point that are parallel to your line?
• How many lines can you draw through that one
  point hat are perpendicular to your line?

5
Identifying Angles formed by transversals.

• Transversal? a line that intersects two or more
  coplanar lines at different points.

Transversal
Coplanar Lines
6
Corresponding Angles? two angles that are in
corresponding (the same) positions. Angles 1
and 5 are corresponding angles.
2
1
4
3
5
6
8
7
7
Alternate Exterior Angles? two angles that lie
outside the parallel lines and on opposite sides
of the transversal. Angles 1 and 8 are alternate
exterior angles.
2
1
4
3
5
6
8
7
8
Alternate Interior Angles? two angles that lie
between the parallel lines and on opposite sides
of the transversal. Angles 3 and 6 are alternate
interior angles.
2
1
4
3
5
6
8
7
9
Consecutive Interior Angles? two angles that lie
between the parallel lines and are on the same
side of the transversal. Angles 3 and 5 are
consecutive interior angles. These angles are
sometimes called same side interior angles.
2
1
4
3
5
6
8
7
10
Using the diagram, list all pairs of angles that
fit the description.

1. Corresponding
2. Alternate exterior
3. Alternate interior
4. Consecutive interior

8
5
7
6
3
1
4
2
11
Homework Assignment

• Worksheet 3.1 all of it

12
Parallel Lines and Transversals

• Geometry
• Section 3.2
• Objective To identify relationships of angles
  formed by parallel lines cut by a transversal.

13
Angles and Parallel Lines Activity

• Using a ruler, trace over two of the parallel
  lines on your paper that are near the middle of
  the your half piece of paper and about an inch
  apart.
• Draw a transversal that makes clearly acute and
  clearly obtuse angles near the center of the
  paper
• Label the angles with numbers from 1 to 8
• Sketch the parallel lines, transversal, and
  number labels in your notes. We will use this to
  record observations.

14
Angles and Parallel Lines Activity

• Cut the paper carefully along the lines you first
  drew to make six pieces.
• Try stacking different numbered angles onto each
  other and see what you observe.
• Try placing different numbered angles next to
  each other and see what you Observe
• Mark your observations on the sketch in your
  notes

15
Angles and Parallel Lines Activity

• Answer the following questions
• How many different sizes of angles where formed?
• 2
• What special relationships exist between the
  angles
• Congruent and supplementary
• Indicate the two different sizes of angles in
  your sketch.

16
Angles and Parallel Lines Activity

• How can we use the vocabulary learned yesterday,
  to describe these relationships?
• IF parallel lines are cut by a transversal, THEN
  
• corresponding angles are congruent (Postulate in
  Text)
• alternate interior angles are congruent (Theorem
  in Text)
• alternate exterior angles are congruent (Theorem
  in Text)
• Consecutive Interior angles are Supplementary
  (Theorem in Text)

17
Perpendicular Transversal

• In your notes, trace over two of the parallel
  lines about one inch apart.
• Using a protractor, draw a line perpendicular to
  one of the parallel lines.
• Extend this perpendicular so that it crosses the
  other parallel line.
• Based on your observations in the previous
  exercise, what should be true about the new
  angles formed?
• Verify this with your protractor.
• If a line is perpendicular to one of two parallel
  lines, then it is perpendicular to the other.
  (Theorem in Text)

18
3.3 Proving Lines are Parallel
19
Standard/Objectives

• Standard 3 Students will learn and apply
  geometric concepts
• Objectives
• Prove that two lines are parallel.
• Use properties of parallel lines to solve
  real-life problems, such as proving that
  prehistoric mounds are parallel.
• Properties of parallel lines help you predict.

20
Properties

• Reflexive Property -
• General a a
• Angles
• Segments AB AB
• Symmetric Property
• General If a b then b a.
• Angles
• Segments If AB CD then CD AB
• Transitive Property-
• General If a b and b c then a c
• Angles
• Segments If AB CD and CD EF then AB EF

21
Postulate Corresponding Angles Converse

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

22
Theorem Alternate Interior Angles Converse

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

23
Theorem Consecutive Interior Angles Converse

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

24
Theorem Alternate Exterior Angles Converse

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

25
Prove the Alternate Interior Angles Converse

• Given ?1 ? ?2
• Prove m n

3
m
2
1
n
26
Example 1 Proof of Alternate Interior Converse

• Statements
• ?1 ? ?2
• ?2 ? ?3
• ?1 ? ?3
• m n

• Reasons
• Given
• Vertical Angles
• Transitive prop.
• Corresponding angles converse

27
Proof of the Consecutive Interior Angles Converse

• Given ?4 and ?5 are supplementary
• Prove g h

g
6
5
4
h
28
Paragraph Proof

• You are given that ?4 and ?5 are supplementary.
  By the Linear Pair Postulate, ?5 and ?6 are also
  supplementary because they form a linear pair.
  By the Congruent Supplements Theorem, it follows
  that ?4 ? ?6. Therefore, by the Alternate
  Interior Angles Converse, g and h are parallel.

29
Find the value of x that makes j k.

• Solution
• Lines j and k will be parallel if the marked
  angles are supplementary.
• x? 4x? 180 ?
• 5x 180 ?
• X 36 ?
• 4(36) 144 ?
• So, if x 36, then
• j k.

4x?
x?
30
Using Parallel ConversesUsing Corresponding
Angles Converse

• SAILING. If two boats sail at a 45? angle to the
  wind as shown, and the wind is constant, will
  their paths ever cross? Explain

31
Solution

• Because corresponding angles are congruent, the
  boats paths are parallel. Parallel lines do not
  intersect, so the boats paths will not cross.

32
Example 5 Identifying parallel lines

• Decide which rays are parallel.

H
E
G
61?
58?
62?
59?
C
A
B
D
A. Is EB parallel to HD? B. Is EA parallel to
HC?
33
Example 5 Identifying parallel lines

• Decide which rays are parallel.

H
E
G
61?
58?
B
D

• Is EB parallel to HD?
• m?BEH 58?
• m ?DHG 61? The angles are corresponding, but
  not congruent, so EB and HD are not parallel.

34
Example 5 Identifying parallel lines

• Decide which rays are parallel.

H
E
G
120?
120?
C
A

• B. Is EA parallel to HC?
• m ?AEH 62? 58?
• m ?CHG 59? 61?
• ?AEH and ?CHG are congruent corresponding angles,
  so EA HC.

35
Conclusion

• Two lines are cut by a transversal. How can you
  prove the lines are parallel?
• Show that either a pair of alternate interior
  angles, or a pair of corresponding angles, or a
  pair of alternate exterior angles is congruent,
  or show that a pair of consecutive interior
  angles is supplementary.