Geometry
Chapter 4.1 Parallel Lines and Planes
Objectives:
• Expand on our definition of parallel lines
• Introduce the idea of parallel planes.
What do we recall about parallel lines?
In geometry, we have to be concerned about the different planes lines can be drawn.
Definition of Parallel Lines: Two lines are parallel if and only if they are in the same plane
and do not intersect.
Since segments and rays are parts of lines, segments and rays can be parallel also.
To designate that two lines, segments, or rays are parallel, arrowheads are used to
show this.
A
B
𝐴𝐵 ∥ 𝐶𝐷
𝐴𝐵 ∥ 𝐶𝐷
C
D
𝐴𝐵 ∥ 𝐶𝐷
𝐵𝐴 ∥ 𝐷𝐶
Geometry
Chapter 4.1 Parallel Lines and Planes
Objectives:
• Expand on our definition of parallel lines
• Introduce the idea of parallel planes.
When we look at the shelves of a bookshelf, we see planes, do we not?
These planes are parallel planes.
In geometry, two or more planes that do not intersect are called parallel planes.
The top and bottom planes are parallel. The front and back planes are parallel. The two side
planes are parallel.
Geometry
Chapter 4.1 Parallel Lines and Planes
Objectives:
• Expand on our definition of parallel lines
• Introduce the idea of parallel planes.
What about lines that do not intersect but are on different planes?
B
Does BC intersect AX?
C
Is BC ∥ AX then?
A
X
Definition of Skew Lines: Two lines that are not in the same plane are skew if and only if
they do not intersect.
Are AC and BC skew lines?
No, because points A, B, and C can form their own plane.
Geometry
Chapter 4.1 Parallel Lines and Planes
Objectives:
• Expand on our definition of parallel lines
• Introduce the idea of parallel planes.
Points of Focus
A
B
C
D
Notice that when we have more
than one set of parallel lines,
additional arrowheads are used for
each set of parallel lines.
C
A
AB and CD lie in the same plane and do
not intersect. Are they parallel?
No, these are line segments, not lines.
B
D
Geometry
Chapter 4.1 Parallel Lines and Planes
Objectives:
• Expand on our definition of parallel lines
• Introduce the idea of parallel planes.
Points of Focus
B
A
C
D
Name all the segments that are
skew to WX.
AD, BC, DZ, CY
CD is not skew to WX because a
plane could be drawn through the
four points.
X
W
Y
Z
Bookwork: page 145; problems 12-45
Geometry
Chapter 4.2 Parallel Lines and Transversals
Objectives:
• Define what a transversal line is.
• Define what interior and exterior angles are.
In geometry, a line, line segment, or ray that intersects two or more lines at different points
is called a transversal. The lines intersected do not have to be parallel.
1
l
2
4
1
2
4
3
5 6
8 7
b
m
t
t is a transversal for l
and m, and l ∥ m.
r is a transversal for b
and c, and b ∦ c.
When a transversal intersects two lines, eight angles are formed.
These angles are given special names.
3
5 6
8 7
c
r
Geometry
Chapter 4.2 Parallel Lines and Transversals
Objectives:
• Define what a transversal line is.
• Define what interior and exterior angles are.
1
l
2
4
1
2
4
3
5 6
8 7
b
m
t
3
5 6
8 7
c
r
Interior Angles lie between the two lines. Exterior Angles lie outside the two lines.
∠1, ∠2, ∠7, ∠8
∠3, ∠4, ∠5, ∠6
Alternate Interior Angles lie opposite of
the transversal
∠3 𝑎𝑛𝑑 ∠5, ∠4 𝑎𝑛𝑑 ∠6
Consecutive Interior Angles lie on the
same side of the transversal
∠3 𝑎𝑛𝑑 ∠6, ∠4 𝑎𝑛𝑑 ∠5
Alternate Exterior Angles lie opposite of
the transversal
∠1 𝑎𝑛𝑑 ∠7, ∠2 𝑎𝑛𝑑 ∠8
Geometry
Chapter 4.2 Parallel Lines and Transversals
Objectives:
• Define what a transversal line is.
• Define what interior and exterior angles are.
1
l
2
4
3
m
5 6
8 7
t
1. Compare the measures of the alternate interior angles?
They are equal.
2. What is the sum of the measures of the consecutive interior angles? 180 degrees
3. Compare the measures of the alternate exterior angles?
If l ∦ m, would the same conclusions be made?
No
They are equal.
Geometry
Chapter 4.2 Parallel Lines and Transversals
Objectives:
• Define what a transversal line is.
• Define what interior and exterior angles are.
The following theorems can now be stated:
Theorem 4-1 Alternate Interior Angles: Given two parallel lines and a transversal, alternate
interior angles are congruent.
Theorem 4-2 Consecutive Interior Angles: Given two parallel lines and a transversal, each
pair of consecutive interior angles is supplementary.
Theorem 4-3 Alternate Exterior Angles: Given two parallel lines and a transversal, each pair
of alternate exterior angles is congruent.
Bookwork: page 152; problems 13-42
Geometry
Chapter 4.3 Transversals and Corresponding
Angles
Objectives:
• Identify the relationship of corresponding angles.
When two lines are cut by a transversal, an additional set of angles is created.
1
l
2
4
3
5 6
8 7
m
Angles 2 and 6 are also called corresponding angles. There are three other pairs of
corresponding angles.
As with alternate interior angles, if the lines cut by the transversal are parallel, then
corresponding angles are congruent.
Geometry
Chapter 4.3 Transversals and Corresponding
Angles
Objectives:
• Identify the relationship of corresponding angles.
Postulate 4-1 Corresponding Angles: If two parallel lines are cut by a transversal, then each
pair of corresponding angles is congruent.
Look at Preparing for Proof on page 157.
This proof leads to the following theorem.
Theorem 4-4 Perpendicular Transversal: If a transversal is perpendicular to one of two
lines, then is perpendicular to the other line.
Bookwork: page 159; problems 13-28, and 31
Geometry
Chapter 4.4 Proving Lines Parallel
Objectives:
• Learn to construct parallel lines
• Identify conditions that lead to parallel lines.
When using alternate interior angles, alternate exterior angles, corresponding angles,
what must be true to determine the measure of congruent angles?
Lines must be parallel.
Hands-On Geometry page 162 – Lets Construct Parallel Lines
Postulate 4-2: If two lines are cut by a transversal so that a pair of corresponding angles
is congruent, then the lines are parallel.
Theorem 4-5: If two lines are cut by a transversal so that a pair of alternate interior
angles are congruent, then the two lines are parallel.
Theorem 4-6: If two lines are cut by a transversal so that a pair of alternate exterior
angles are congruent, then the two lines are parallel.
Geometry
Chapter 4.4 Proving Lines Parallel
Objectives:
• Learn to construct parallel lines
• Identify conditions that lead to parallel lines.
Theorem 4-7: If two lines are cut by a transversal so that a pair of consecutive interior
angles is supplementary, then the two lines are parallel.
Theorem 4-8: If two lines are perpendicular to the same line, then the two lines are
parallel.
In Conclusion, the five ways to show two lines are parallel:
1.
2.
3.
4.
5.
Pair of corresponding angles are congruent.
Pair of alternate interior angles are congruent.
Pair of alternate exterior angles are congruent.
Pair of consecutive interior angles is supplementary.
Two lines perpendicular to the same third line.
Bookwork: page 166; problems 9-23
Geometry
Chapter 4.5 and 4.6 Slope and Equations of
Lines
Objectives:
• Find the slopes of line and identify parallel and perpendicular lines.
• Write and graph equations of lines.
Recall from Algebra I, the steepness of a line is called slope.
𝑟𝑖𝑠𝑒
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
Δ𝑦
Slope can be defined as 𝑟𝑢𝑛 , which is equal to 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 or Δ𝑥.Type equation here.
Slope can be calculated by 𝑚 =
Why can 𝑥2 ≠ 𝑥1 ?
𝑦2 − 𝑦1
,
𝑥2 − 𝑥1
where 𝑥2 ≠ 𝑥1 .
The denominator would equal zero, which is undefined.
An undefined slope is what kind of line?
What is the slope of a horizontal line?
A vertical line.
Zero.
What does a line with a positive slope look like?
Up and to the right.
What does a line with a negative slope look like?
Down and to the right.
Geometry
Chapter 4.5 and 4.6 Slope and Equations of
Lines
Objectives:
• Find the slopes of line and identify parallel and perpendicular lines.
• Write and graph equations of lines.
Postulate 4-3: Two non-vertical lines are parallel if and only if they have the same slopes.
All vertical lines are considered parallel.
Postulate 4-4: Two non-vertical lines are perpendicular if and only if the product of their
slopes is -1.
Or their slopes are negative reciprocals of each other.
What are the three forms of an equation of a line?
Point-slope:
𝑦2 − 𝑦1 = 𝑚(𝑥2 − 𝑥1 )
Slope-intercept: 𝑦 = 𝑚𝑥 + 𝑏, where m is the slope and b is the y-intercept.
Standard Form or Intercept Form: 𝐴𝑥 + 𝐵𝑦 = 𝐶
Look at examples 1-6.
Bookwork: page 172; problems 10-29, and page 178; problems 13-34