Wize High School Geometry Textbook (Common Core) > Lines

Pairs of Lines

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Pairs of Lines
  

When we have a pair of lines in the same plane, they either intersect or they don't.  If we step out of the plane, then we can have other types of lines as well.

Types of lines
A pair of lines are parallel if they are in the same plane, and do not intersect.
A pair of lines are skew if they are not in the same plane, and they do not intersect.

Pairs of angles from lines
A transversal is a line that cuts through two or more lines at different points in the same plane.  

  

When we have a transversals it can create many different types of angles.
Corresponding angles -  These are located in corresponding, or similar positions.  For example one angle inside the pair of lines, and one outside the pair of lines, but both on the same side of the transversal. 
Alternate interior angles - These are located between the pair of lines, and either side of the transversal. 
Alternate exterior angles -  These are located outside the pair of lines, and either side of the transversal.
Consecutive interior angles -  These are located inside the pair of lines, and the same side of the transversal.
Important Postulates
Parallel Lines Postulate - If given and line and a point, not on the line, then there is exactly one line through the point parallel to the given line. 

Perpendicular Lines Postulate - If given a line and a point not on the line, there there is exactly one line through the point perpendicular to the given line.  

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Constructing perpendicular lines
  
Our goal in this construction is to create a new line that is perpendicular to a given line and goes through a given point.

Using Technology
  

Begin with a line AB↔\overleftrightarrow{AB}AB , and a point CCC off the line.
Draw a circle with the center at CCC.  Make sure it is large enough to intersect the given line at two points.  Label these new points as DDD and EEE.
Use the midpoint tool to place a point in the middle of segment DE‾\overline{DE}DE.  Label this as point FFF.
Draw a line through points CCC and FFF.
From this we now have that AB↔\overleftrightarrow{AB}AB is perpendicular to CF↔\overleftrightarrow{CF}CF.

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Construct a perpendicular bisector
  
Our goal in this construction is to create a new line that bisects, and is perpendicular to a given line segment.

Using Technology
  

Begin with a segment AB‾\overline{AB}AB
Draw a circle whose radius is slightly larger than the midpoint of AB‾\overline{AB}AB, centered at AAA.  Label the intersection of the segment and the circle as point CCC.
Use the compass tool to draw a circle whose radius is the length of AC‾\overline{AC}AC , with the center at BBB.  Label the intersection of the circles as point DDD and EEE.
Draw a line through points DDD and EEE.  Label the intersection of the line and AB‾\overline{AB}AB as point FFF.
From this we now that point FFF is the midpoint of AB‾\overline{AB}AB, and that DE↔\overleftrightarrow{DE}DE is perpendicular to AB‾\overline{AB}AB

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Lines and Rails

The Canadian National railway is the largest railroad in Canada.  It has about 32,000 km of rails that cross the country and help people get where they need to go.  

Use the diagram of two railroad tracks crossing to answer the questions below.  

1.  In the diagram it appears that lines AC↔\overleftrightarrow{AC}AC and DF↔\overleftrightarrow{DF}DF are parallel.  What is a good argument that they are indeed parallel?

ANSWER:  Railroad tracks should stay the same distance apart so that the train can ride over them smoothly.  This means they will not intersect.

2.  Line GB↔\overleftrightarrow{GB}GB  is incident to lines AC↔\overleftrightarrow{AC}AC, and DF↔\overleftrightarrow{DF}DF.  What angles are considered alternate interior angles?  What are these angles interior ?

ANSWER:  Alternate Interior Angles:  ∠DEB and ∠EBC\angle{DEB} \text{ and } \angle{EBC}∠DEB and ∠EBC
These are interior the pair of lines AC↔\overleftrightarrow{AC}AC and DF↔\overleftrightarrow{DF}DF.

3.  If given a line such as DF↔\overleftrightarrow{DF}DFand a point off the line likeBBB, then there exist a unique line through BBB.  According to the diagram we have two lines that go through BBB.  Namely AB↔\overleftrightarrow{AB}AB and GB↔\overleftrightarrow{GB}GB.  How can this be?

ANSWER:  We have to be careful with the wording of the postulate.  If given a line and a point off of the line, then there exist a unique line that is parallel to the original line.  The diagram does not break this postulate since line AB↔\overleftrightarrow{AB}AB would be the unique parallel line to DF↔\overleftrightarrow{DF}DF that goes through point BBB.

Identifying Angle Pairs

Use the diagram to math the pairs of angles with their best description.

A.

Corresponding angles

B.

Consecutive interior angles

C.

Alternate interior angles

D.

Alternate exterior angles

∠4 and ∠8\angle{4} \text{ and } \angle{8}∠4 and ∠8

∠4 and ∠5\angle{4} \text{ and } \angle{5}∠4 and ∠5

∠3 and ∠6\angle{3} \text{ and } \angle{6}∠3 and ∠6

∠2 and ∠5\angle{2} \text{ and } \angle{5}∠2 and ∠5

I don't know

Thinking about lines

Everyday objects can be modeled using geometry in a variety of ways.  The following diagram shows a simple box that his ready for shipping.  Assume that each segment is part of a line and that all angles are right angles.

Fill in the blanks with the best description of the lines.  

1.  Lines AD↔\overleftrightarrow{AD}AD and BE↔\overleftrightarrow{BE}BE are considered ________ .

2.  Lines AB↔\overleftrightarrow{AB}AB and EF↔\overleftrightarrow{EF}EF are considered ________ .

3.  Lines BE↔\overleftrightarrow{BE}BE and EF↔\overleftrightarrow{EF}EF are considered ________ .

1.  Lines AD↔\overleftrightarrow{AD}AD and BE↔\overleftrightarrow{BE}BE are considered 
.

I don't know

The game of Chess

In the game of Chess, the piece known as the Rook can only move in a strait line.
One way we can model the game using geometry is to use points for the location of the pieces and lines for the their possible movement of the rook.  For example moving the rook either up or down would give us a line.  This is labeled in the diagram as ℓ\ellℓ.

1.  How many unique lines go through the white pieces and are perpendicular to line ℓ\ellℓ ?

2.  How many unique lines go through the white pieces and are parallel to line ℓ\ellℓ ?

1.  How many unique lines go through the white pieces and are perpendicular to line ℓ\ellℓ ?

Answer

I don't know