_____(0-10 pts) Describe parallel lines and parallel planes. Include a discussion of skew lines. Give at least 3 examples.

 

Parallel Lines and Parallel Planes

 

 

Description: 

 

They are straight planes or lines that always stay on a certain distance between each other no matter how far they extend.

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Examples:

 

You can find parallel lines in your daily life, like the railway, sometimes the floor and the roof (if they are flat) are parallel planes.

 

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In this picture you can see that the plane TUZ is parallel to the plane SRW. 

Or plane TUS and YZX are parallel too. Because they are straight to each other, have always the same distance between them and they are not touching and will never touch.

                                                                                                                  

                                                                                                                                                                                                                                                             

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In this picture you can see that there  are parallel lines and one intersecting them. The one intersecting them is called the transversal line. You can see that there are some arrows on the parallel lines, that symbol means that they are parallel.

 

 

 

 

 

 

 

 

 

This image shows how skew lines look like, they are lines that do no intersect BUT they are not parallel. 

 

 

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PARALLEL AND PERPENDICULAR POSTULATES

_____(0-10 pts) Describe the parallel and perpendicular postulates. Give an example of each.

 

 

 

 

 

Background Info.

Parallel lines: Are lines that never touch and share the same plane

 

 

Perpendicular lines: Are lines that intercept and form 90º angles.

 

 

 

 

 

 

 

Parallel Postulate

 

• Though a point, which isn't not located on the line, you may a draw a line exactly parallel to that line.

 

Ex. 

 

Through  point Q, which isn't on line X, there is only a single line that is parrallel to line X.

 

 

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To draw a parallel line you have to follow some basic steps:

 

 

1. Draw a point on a plane that isn't on the line.
2. Take a ruler and draw a line through the point.
3. Take a compass and measure it to half the distance to the point.
4. Without changing the measurement of the compass draw an arc by using the point as reference source.
5. Now move the compass to the upper part of the arc you just drew and draw another line
6. Use the ruler to draw a line were the two arcs converged. 

 

 

 

 

 

 

 

 

 

 

 

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Perpendicular Postulate

 

• If there is a line and a point not on the line, then there is just one line through that point that is perpendicular to that given line. 

 

Example:

Through point Q you can draw an exact line(perpendicular to line x)

 

 

 

 

 

 

 

 

 

 

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To draw a perpendicular line you have to follow some basic steps:

 

1. Find a point in the plane which is NOT on the line
2. Place compass on the point.
3. Extend the compass so the point of the compass is below the line
4. Draw an arc on both sides of the line.  (REMEMBER not to adjust the width of the compass or the measurement will be inacurate)
5. From the intersection of the line and compass draw a seond arc
6. Where the arcs converge, place a ruler and draw a straight line through the point.

 

1.                                                        2.                                        

 

 

 

3.

4.                                5.

 

 

 

 

 

 

I also found a video talking about how to draw perpendicualar lines which is very helpful

Link:

http://www.mathopenref.com/constperpextpoint.html

 

 

 

 

Marcela Janssen

Parallel line postulate

Given a line and a point off the line, there is exactly one line parallel given through that point.

 

This means that for every line and a point off that line there is exactly one parallel line.

 

Example 1

Example 1 parallel lines.emf

 

Example 2

Find m<1 in the figure below. Line PQ and line RS are parallel.

parallel ex 2.emf

180 degrees - 55 degrees = 125 degrees

 

Example 3

perpendicular ex 3.emf

 

Given: <1 is congruent to <3

          <3 is congruent to < 6

 

Which  lines are parallel?

 

 Answer:  F  ll G

              G ll h

 

 

Perpendicular Line Postulate

Given a line and a point off that line, there is exactly one line that is perpendicular to the give line through that point.

 

This means that for every line and a point off that line there is exactly one perpendicular line.

 

Example 1

perpendicular 1.emf

 

Example 2

 

Given: m is perpendicular to p, <1 and <2 are complementary 

Prove: p ll Q

perpendicular 2.emf

 

Proof:

It is given that M is perpendicular to P. <1 and <2 are complementary, so m<1 + m<2 = 90 degrees. Thus m is perpendicular to Q. Two lines perpendicular to the same line are parallel, so p ll q.

 

Example 3

Given: l is perpendicular to n, m is perpendicular n

Proof: <3 angd <6 are supplementary

perpendicular ex 3.emf

Stament	Reason
1) n is perpendicular to L	Given
2) L ll n	2 lines perpendicular to same line are ll
3) <5 and <6 are supplementary	LPP
4) <5 is congruent to <3	Alt. Int. <s
5) m<5+ m<6 = 180	Def. suppl.
6) m<5 = m<3	subst.
7) m<3 + m<6 =108	Subst.
8) <3 supplementary to <6	Def. Supp-

 

 

 

Constructing a Perpendicular Line.

1) Draw seg. AB. Open the compass wider than half of AB and draw an arc centered at A. Construction step 1.emf

 

2) Using the same coimpass setting draw an arc at the center at B that intersects the first arc at C and D. Construction step 2.emf

 

3) Draw line CD. Line CD is the perpendicular to the seg. AB. Construction step 3.emf