_____(0-10 pts) Describe the perpendicular transversal theorem. Give at least 3 examples.

 

_____(0-10 pts) Describe how the transitive property also applies to parallel and perpendicular lines. Include a discussion about theorems 3.11 and 3.12. Give at least 2 examples of each.

 

_____(0-10 pts) Describe how to find the slope of a line. How is slope related to parallel and perpendicular lines. Give at least 3 examples of each.

 

Perpendicular Transversal Theorem

 

The theorem says that if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

 

http://CoSketch.com/Saved/paTj5iMb

 

 

 

 

Examples: 

 

In these windows, the horizontal frames are parallel. So the vertical frame is perpendicualar to the top frame. Since it is perpendicular to the top one, it has to be perpendicular to the bottom one too.

 

Example #2:

Given: The transversal M is perpendicular to line J. Line J and line K are parallel

Prove: The transversal M is perpendicualr to line K.

 

http://CoSketch.com/Saved/pToUNlks

 

M is perpendicular to J. J is parallel to K = Given

M is perpendicular to K= Perpendicular transversal theorem

Q.E.D

 

Example #3:

Prove that line G is perpendicular to line D

Given: Angle A and Angle B are corresponding, Angle A measures 90 degrees.

 

http://CoSketch.com/Saved/p7z4e9ph

 

<A and <B are equal = Given

<A measures 90 degrees

Line G and F are perpendicular=definition of perpendicular

Line F and D are parallel= Corresponding Angles

Line G and D are perpendicular= Perpendicular transversal theorem

T.T.T!

  

Example #4:

 

Given:

• A is perpendicular to B
• B is parallel to C

Prove:

• A is perpendicular to C

 

A is perpendicular to B, B is parallel to C= Given

<1 is a right angle = definition of perpendicular lines

m<1= 90 degrees - definition of a right angle

<1 <2 = Corresponding Angles

m<2 = 90 degrees = Substitution Property

<2 is a right angle=definition of a right angle

Therefore, A is perpendicular to C

 

 

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Transitive, Perpendicular, and Parallel

 

The transitive property applies to parallel and perpendicular lines because if one line is parallel to another line. And a 3rd line is parallel to that same line, then the first and the third have to be parallel also.

 

It's the same for perpendicular. In a same plane, if one line is perpendicular to another one. And a 3rd line is perpendicular to that same one, then the first and last lines are parallel.

 

Theorem 3.11 states if two lines are parallel to the same line, then they are parallel to each other.

 

http://www.cosketch.com/Saved/pmZopkUm

 

Theorem 3.12 says that in a plane, it two lines are perpendicular to the same line, then they are parallel to each other.

 

http://www.cosketch.com/Saved/pFqq1IfC

 

Examples:

 

In this picture, the first horizontal line is perpendicular to the vertical line. The second horizontal line is also perpendicular to the vertical one. That means that the 2 horizontal lines must be parallel.

 

2.

 

This is a squared paper. Prove that the first line horizontal line in this paper is parallel to the last line. Given: The first line is parallel to the second one, the second line is parallel to the third and it keeps going like that.

 

1st line ll 2nd line (continuous) = Given

1st line ll last line= Transitive property of parallelism

 

Q.E.D 

 

Example 3:

http://CoSketch.com/Saved/pIoMlUBr

 

Given: Angle H and Angle T are right angles.

Prove: Line A and Line C are parallel

 

Angle H and Angle T are right angles= Given

Angle H and Angle T measure 90 degrees= Definition of right angle

Line A is perpendicular to line F= defintion of perpendicular lines

Line C is perpendicular to line F= definition of perpendicular lines

Line A and Line C are parallel= Transitive property of perpendicular lines

Q.E.D

 

Example 4:

 

 

The first line made between the books and the second line made between the books are parallel. The second line is also parallel to the third line. By the transitive property of parallelism, the 1st line and the 3rd line are parallel also.

 

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Slope of a Line

 

The postulate says that in a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

 

 

To find the slope of a line you must use this equation: m=y₂ - y₁

                                                                                       _{x2 - x1} 

 

Use this equation for slope intercept form: y=mx+b (b being y-int.)

 

Use this equation for point slope form: y-y₁=m(x-x₁)

 

SLOPE OF A LINE:

The slope of a line is the ratio of the change in y over the change in  x. Simply put, find the difference of both the y and x coordinates and place them in a ratio.  Pick two points on the line. Because you are using two sets of ordered pairs both having x and y values, a subscript must be used to difference between the two values.

 

Examples using the formula of m= y2-y1

                                                     x2-x1

 

To find the slope of this line, we can use the formula that was stated above and replace the variables with the coordinates of the points.

 

3-(-2)  = slope                          So the slope of this line would be  = 5

6-3                                                                                                   3

 

Example #2:

 

 

-3-8          =slope                            So the slope would be =   -5

5-(-4)                                                                                        9

 

SLOPES OF PARALLEL AND PERPENDICULAR LINES:

If two lines are parallel, their slopes must be equal. Each line will cut the X axis at the same angle.

if two lines are perpendicular, the slope of one is the negative reciprocal of the slope of the other, and backwwards.

 

Examples of parallel lines and their slopes:

If the slope of one line is 2, then the slope of the other line is 2 because their slopes must be equal.

Real life example: Railroad tracks

 

Examples of perpendicular lines and their slopes:

• If the slope of one line is 2, then the slope of another line is -1/2 because the slope of one line is the negative reciprocal of another.
• A real life example of perpendicular lines would be a cross because the two lines would form a right angle.