geometry chapter 4.1 parallel lines and planes

Geometry

Objectives:• Expand on our definition of parallel lines• Introduce the idea of parallel planes.

Chapter 4.1 Parallel Lines and Planes

What do we recall about parallel lines?

Definition of Parallel Lines: Two lines are parallel if and only if they are in the same plane and do not intersect.

In geometry, we have to be concerned about the different planes lines can be drawn.

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

C D

𝐴𝐵 ∥ 𝐶𝐷𝐴𝐵 ∥ 𝐶𝐷𝐴𝐵 ∥ 𝐶𝐷𝐵𝐴 ∥ 𝐷𝐶

Geometry



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



What about lines that do not intersect but are on different planes?

B

C

A X

Does BC intersect AX?

Is BC ∥ AX then?

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



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.

A

DB

C

AB and CD lie in the same plane and do not intersect. Are they parallel?

No, these are line segments, not lines.

Geometry



Points of Focus

A

B C

D

W

XY

Z

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.

Bookwork: page 145; problems 12-45

Geometry

Objectives:• Define what a transversal line is.• Define what interior and exterior angles are.

Chapter 4.2 Parallel Lines and Transversals

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.

l

m

b

c

rt

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.

1 24 3

1 24 3

5 68 7

5 68 7

Geometry



l

m

b

c

rt

1 24 3

1 24 3

5 68 7

5 68 7

Interior Angles lie between the two lines.∠3, ∠4, ∠5, ∠6

Exterior Angles lie outside the two lines.∠1, ∠2, ∠7, ∠8

Alternate Interior Angles lie opposite of the transversal

∠3 𝑎𝑛𝑑 ∠5, ∠4 𝑎𝑛𝑑 ∠6

Alternate Exterior Angles lie opposite of the transversal

∠1 𝑎𝑛𝑑 ∠7, ∠2 𝑎𝑛𝑑 ∠8

Consecutive Interior Angles lie on the same side of the transversal

∠3 𝑎𝑛𝑑 ∠6, ∠4 𝑎𝑛𝑑 ∠5

Geometry



l

m

t

1 24 3

5 68 7

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

If l ∦ m, would the same conclusions be made? No

3. Compare the measures of the alternate exterior angles? They are equal.

Geometry



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.


Geometry

Objectives:• Identify the relationship of corresponding angles.

Chapter 4.3 Transversals and Corresponding Angles

When two lines are cut by a transversal, an additional set of angles is created.

l

m

1 24 3

5 68 7

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

Objectives:• Identify the relationship of corresponding angles.

Chapter 4.3 Transversals and 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

Objectives:• Learn to construct parallel lines• Identify conditions that lead to parallel lines.

Chapter 4.4 Proving Lines Parallel

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

Objectives:• Learn to construct parallel lines• Identify conditions that lead to parallel lines.

Chapter 4.4 Proving Lines Parallel

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. Pair of corresponding angles are congruent.2. Pair of alternate interior angles are congruent.3. Pair of alternate exterior angles are congruent.4. Pair of consecutive interior angles is supplementary.5. Two lines perpendicular to the same third line.


Geometry

Objectives:• Find the slopes of line and identify parallel and perpendicular lines.• Write and graph equations of lines.

Chapter 4.5 and 4.6 Slope and Equations of Lines

Recall from Algebra I, the steepness of a line is called slope.

Slope can be defined as 𝑟𝑖𝑠𝑒

𝑟𝑢𝑛, which is equal to

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥or

Δ𝑦

Δ𝑥.Type equation here.

Slope can be calculated by 𝑚 =𝑦2 − 𝑦1

𝑥2 − 𝑥1, where 𝑥2 ≠ 𝑥1.

Why can 𝑥2 ≠ 𝑥1? The denominator would equal zero, which is undefined.

An undefined slope is what kind of line? A vertical line.

What is the slope of a horizontal 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

Objectives:• Find the slopes of line and identify parallel and perpendicular lines.• Write and graph equations of lines.

Chapter 4.5 and 4.6 Slope and Equations of Lines

Postulate 4-3: Two non-vertical lines are parallel if and only if they have the same slopes.

Postulate 4-4: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.

All vertical lines are considered parallel.

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

geometry chapter 4.1 parallel lines and planes

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