Wize High School Geometry Textbook (Common Core) > Lines

Angles from Intersecting Lines

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Pairs of angles from lines
  

When we have a pair of lines, we can cross them with a third line.  This creates many new angles which have important properties associated with them. 


Wize Concept
Remember that we can call pairs of these angles by special names:
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 Theorems
Corresponding Angles Theorem - If two parallel lines are cut by a transversal, then the pair of corresponding angles are congruent.
Alternate Interior Angles Theorem- If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Alternate Exterior Angles Theorem - If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
Consecutive Interior Angles Theorem - If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

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

Using Technology
  

Begin with a line AB↔\overleftrightarrow{AB}AB, and a point off the line labeled CCC.
Make a line through AAA and CCC.
Place a point DDDalong line AC↔\overleftrightarrow{AC}AC
Create a circle with a radius the length of AD‾\overline{AD}AD centered at AAA.  Label the intersection of the circle and the segment AB‾\overline{AB}AB as point EEE.
Use the compass tool to make a circle with the radius of AD‾\overline{AD}AD centered at CCC.  Label the intersection of the circle and the line as point FFF.
Use the compass tool to make a circle with the radius of DE‾\overline{DE}DEcentered at FFF.  Label the intersection of the two circles as point GGG.
Draw the line through CCCand GGG.
From this we now have that AB↔\overleftrightarrow{AB}AB is parallel to CG↔\overleftrightarrow{CG}CG.

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Angles and Rails

Having parallel lines and a transversal creates many angles with properties associated with them.
Use the diagram below to answer questions about rails.

Given that line AC↔\overleftrightarrow{AC}AC is parallel to line DF↔\overleftrightarrow{DF}DF, and that ∠DEG=34∘\angle{DEG} = 34^\circ∠DEG=34∘

1.  What is the measure of angle ∠BEF\angle{BEF}∠BEF?

ANSWER:  Angle ∠DEG\angle{DEG}∠DEG and ∠BEF\angle{BEF}∠BEF are vertical angles, so we have 

∠BEF≅∠DEGm∠BEF=m∠DEGm∠BEF=34∘\begin{aligned}
\angle{BEF} \cong &\angle{DEG} \\
m\angle{BEF} &= m\angle{DEG} \\
m\angle{BEF} &= 34^\circ
\end{aligned}∠BEF≅m∠BEFm∠BEF​∠DEG=m∠DEG=34∘​

2.  What is the measure of angle ∠ABE\angle{ABE}∠ABE?  

ANSWER:  Angle ∠DEG\angle{DEG}∠DEG and ∠ABE\angle{ABE}∠ABE are corresponding angle, so we have

∠ABE≅∠DEGm∠ABE=m∠DEGm∠ABE=34∘\begin{aligned}
\angle{ABE} \cong &\angle{DEG} \\
m\angle{ABE} &= m\angle{DEG} \\
m\angle{ABE} &= 34^\circ
\end{aligned}∠ABE≅m∠ABEm∠ABE​∠DEG=m∠DEG=34∘​

3.  What is the total measure of ∠ABE+∠BED\angle{ABE} + \angle{BED}∠ABE+∠BED? 

ANSWER:  These angles are consecutive interior angles.  From the consecutive interior angles theorem we know that these will be supplementary.  This means the total of their measures is 180∘180^\circ180∘   .

Find the angles measure

Use the diagram, and the fact that m∠2=140∘m\angle{2} = 140^\circm∠2=140∘ ,to find the measure of the following angles or sum of angles.

1.  What is the measure of angle ∠7\angle{7}∠7?

2.  What is the measure of angle ∠5\angle{5}∠5?

3.  What is the sum of angles ∠2\angle{2}∠2 and ∠4\angle{4}∠4?

4.  What is the sum of angles ∠1\angle{1}∠1 and ∠4\angle{4}∠4?

1.  What is the measure of angle ∠7\angle{7}∠7?

Answer

I don't know

Shapes and Geometry

A trapezoid is a polygon with four sides, and a pair of parallel sides.
Use the diagram below to answer the following questions.  Note that the parallel sides have been marked with arrows.

1.  What is the value of xxx?

2.  What is the value of the obtuse angle?

3.  What is the value of yyy?

1.  What is the value of xxx?

Answer

I don't know

Power and Mathematics

Some power lines can carry as much as 345,000 volts.  Special poles are needed to keep these high off the ground safely supported when spanned over large distances.

Below is a diagram of two parallel poles that are supported together.  Use it, and the given information to answer the following questions.

1.  What type of angles have been marked in the diagram?

2.  If ∠1\angle{1}∠1 is 123∘123^\circ123∘, then what is the measure of ∠3\angle{3}∠3?

3.  Suppose that the span between the two poles was changed so that the measure of ∠1\angle1∠1 was no smaller.  What effect would this have on the measure of ∠3\angle{3}∠3?

1.  What type of angles have been marked in the diagram?

Answer

I don't know