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Lesson 10 Parallel Lines and their Angles

Page history last edited by Math in a Box - Susan Johnsey gm 2 years, 2 months ago

Parallel Lines and their Angles by Susan Johnsey

www.mathinabox.com

To return to the HOME page of this classroom click the WIKI tab on the left above.

This is a long Lesson. You need to plan to study it for 2 days. You must write and draw and watch videos and play games!

What does parallel mean? If you have a pad of lined paper for your school class then the lines that are drawn for you to write along are usually parallel. The columns that hold up the roof of a porch or building are usually parallel. The rails laid for a train are usually parallel ( of course, some of those are not straight lines, but have a bit of curve to them). What are some other examples of parallel?

There are 3 letters in the word parallel that are parallel to each other. Do you see them? parallel .

Lines that are equal distant apart from each other are parallel lines.

Parallel lines are in the same plane (co-planar) and they do not intersect each other.

Have you heard of SKEW lines?

Lines that are NOT in the same plane and that do NOT intersect are SKEW lines.

Draw a line on your floor and then walk over to the wall or go up a ladder to the ceiling.

You can draw a line on the wall or ceiling that is SKEW to the line on the floor.

To be sure that the lines are SKEW you must NOT make the one on the ceiling parallel to the one on the floor.

To be sure that the lines are SKEW you must NOT make the one on the wall be parallel nor intersect the one on the floor so watch where the lines go if you "extend" them forever!

Can you draw 2 or 3 lines in your room now that are SKEW?

If you use the wall and the floor you can actually show all three. Skew, parallel or intersecting. Can you do that?

Here is a very good video about parallel lines and skew lines.

You just need to watch the first 3 minutes!!!

Copy and paste this into your address bar.

http://www.youtube.com/watch?v=io_fiNgxAXg

There are SKEW LINES also. Which lines will not intersect and are not parallel?

LINES are not the only thing that can be parallel!!

Planes can be parallel also. If a building has several floors or stories then those floors represent parallel planes.

The walls of a building are sometimes parallel. Most buildings are rectangular and have a front wall and back wall. These walls are usually parallel to each other.

The designs on floors created by the tiles or wood planks illustrate parallel lines too. Look around in your room. You will see many parallel lines and planes!

Now let's look more at parallel lines.

Any time you are GIVEN a line that intersects parallel lines you need to think of the 4 Theorems below.

The 4 converse theorems for these are also true!

The line or segment that crosses the parallel lines is given a special name, YOU will have to work hard to spell and say it!! TRANSVERSAL.

Diagram for Alternate Interior Angles.

The TRANSVERSAL goes across (intersects) the 2 parallel lines.

Converse: If the alternate interior angles are congruent then the lines are parallel.

________________________________________________________________

Diagram for Corresponding Angles (of parallel lines)

There are 8 angles.

Do you see the 2 groups of 4 angles. Can you tell which ones are corresponding?

I colored some of them for an example. On the first diagram I colored 2 of them orange.

They are both on the right side of the and above the parallel line that they are on.

Do you see that? Same position you could say!

Look at the 2 pinkish ones. What position do they both have?

Both are "On left side of TRANSVERSAL and below the parallel line".

Converse: If the corresponding angles are congruent then the lines are parallel.

____What does congruent mean?

It is the word we use in GEOMETRY for equal length or same exact shape or equal measurement. _____

If the angles are congruent then their measurements are equal ( we use degrees most of the time to measure angles) .

Diagrams for Same-Side Interior Angles (also called co-interior).

Look at the 2 green angles. These are Same-Side Interior Angles; they are both on the same side of the TRANSVERSAL, but they are between the 2 parallel lines. Do you see they are not the same as corresponding? They do not have the same position.

The Same-Side Interior Angles are ALWAYS between the parallel lines and on the same side of the TRANSVERSAL.

NOTE: THEY ARE NOT usually congruent!!

What? they are congruent only occasionally! but they are always supplementary!

When we have parallel lines and then a third line is drawn PERPENDICULAR

to one of the parallel lines then it will be PERPENDICULAR to the other parallel line too.

Can you write the 2 remaining converses?

Look back at mine above for the alternate interior and corresponding angles if you need help with the CONVERSE.

Watch this video. Remember you can pause it and rewind it.

Please do so that you understand it well.

We will be combining algebra and geometry.

Corresponding angles on Parallel lines

and

Same-side interior angles (also called co-interior) on parallel lines.

http://www.youtube.com/v/lTKDOkwtVX8 or go there and watch,

State the REASON from this lesson that justifies each of the statements below,

but warning 2 answers require you to remember Lesson 7 Special Pairs of Angles.

Note: WE are GIVEN that the lines j and k are parallel.

LOOK at the angles along lines j and k. What type angles do you see?

My LONG HINT for ANGLES CREATED BY PARALLEL LINES: TRACE the sides of two angles. LOOK at the 3 lines that you just traced.

If the angles are CONGRUENT corresponding or CONGRUENT alternate interior angles then which of the 2 lines are parallel? ___ and ___

IF you have 4 lines traced then you are not going to know!

These theorems will use only 3 lines at a time to create the two angles. COUNT 1-2-3 first.

If the angles are same-side interior then remember they too will be created by 3 lines, but the angles will be SUPPLEMENTARY.

Answers in the brown boxes. Highlight them so you can see the answers. Always look for the parallel lines; see the 2 arrows.

Do not assume lines are parallel. You must be given that information in the diagram or with words!

angle 3 congruent to angle 7. corresponding angles are congruent so lines J and K are parallel
measure angle 6 + measure angle 7 = 180. same-side interior angles are supplementary so lines J and K are parallel
angle 4 congruent angle 6 vertical angles are congruent ( does not tell us about parallel lines)
measure angle 5 = measure angle 7 alternate interior angles are congruent so lines J and K are parallel
line a is perpendicular to line k. line a is perpendicular to line J so it is also perpendicular to line K
angle 6 congruent angle 8 alternate interior angles are congruent so lines J and K are parallel
angle 1 congruent angle 2 corresponding angles are congruent so lines J and K are parallel
angle 5 is supplementary to angle 8 same-side interior angles are supplementary so lines J and K are parallel
angle 3 is congruent to angle 5 vertical angles are congruent ( does not tell us about parallel lines)

If measure of angle 8 is 72 degrees then what is measure of angle 3? That is NOT in one of the diagrams (theorems) above! To find angle 3 from angle 8 I must use two types of angles!

1. Do you know angle 6? It is alternate interior to angle 8 so angle 6 is also 72 degrees.

Now what do you know about angle 6 and angle 3? Look at the diagram.

They are a linear pair. That means angle 3= 180-72 or 108 degrees.

WOW, we are using several Lessons here.

You can find angle 3 also by using same-side interior angles and vertical angles. Can you do that?

2. LOOK at angle 8 and tell me the angle that is same-side interior to it. Do you see it is angle 5?

Thus angle 8 and angle 5 are supplementary so angle 5= 180-72 degrees or 108 degrees.

NOW look at angle 3 again. Do you see that angle 5 and 3 are VERTICAL angles.

Vertical angles are congruent thus angle 3 is 108 degrees too.

Here is a jazzy video to watch. It will help you with all the NEW words to know about angles around parallel lines.

I hope you enjoy it and learn a LOT.

Here is the link: http://www.youtube.com/watch?v=vmFH5ajoxKk It is at youtube.

Here is a fun game only 10 questions but you should play it 2 or 3 times. You need to recall vertical angles and linear pairs too. Take a screen shot of your scores. Save it to your Desktop.

copy this http://www.thatquiz.org/tq-C/?-j3-l8-p0 into your address bar.

In your LESSON Notes 10 page drag a screen shot of your scores.

CHALLENGING VIDEO for the BRAVE!

Here is another video. It uses two rather complicated diagrams. You will need to be quite good a finding angles when there are several line segments. The problems above used at most 3 line segments at one time. The diagrams in this video use 4 or more segments.

http://www.youtube.com/watch?v=BZyQYMw6ZVI

Let me know your questions any time, but always be specific as I have many students in different courses on various Lessons.