Transversals, and their special angle pairs

 
 
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There are lots of special angle pairs created when a transversal crosses two lines

In this lesson we’ll look at the angles formed when a pair of parallel lines is crossed by another line, called a “transversal.”

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Transversals

transversal is a line that crosses at least two other lines.

 
transversals cross at least two lines
 


Special angle pairs

When transversals cross parallel lines, they form angles with special angle relationships. The angle pair relationships form two types of special angles.

Congruent angles have the same measure.

Supplementary angles have measures that add up to ???180{}^\circ???.


Types of special angle pairs

Vertical angles share a vertex, but lie on opposite sides of both the parallel lines and the transversal. Vertical angle pairs are congruent.

 
a transversal creates vertical angles
 

???m\angle a=m\angle b???

???m\angle c=m\angle d???

???m\angle e=m\angle f???

???m\angle g=m\angle h???

Corresponding angles are matching angles that lie, one on each parallel line, on the same side of the parallel lines and the same side of the transversal. Corresponding angle pairs are congruent.

 
a transversal creates corresponding angles
 

???m\angle a=m\angle e???

???m\angle d=m\angle h???

???m\angle c=m\angle g???

???m\angle b=m\angle f???

Alternate interior angles are angles that lie, one on each line, on opposite sides of the parallel lines and opposite sides of the transversal, inside of the parallel lines. Alternate interior angle pairs are congruent.

 
a transversal creating alternate interior angles
 

???m\angle d=m\angle g???

???m\angle b=m\angle e???

Alternate exterior angles are angles that lie, one on each line, on opposite sides of the parallel lines and opposite sides of the transversal, outside of the parallel lines. Alternate exterior angle pairs are congruent.

 
a transversal creating alternate exterior angles
 

???m\angle a=m\angle f???

???m\angle c=m\angle h???

Consecutive interior angles are angles that lie, one on each parallel line, on opposite sides of the parallel lines, but on the same side of the transversal. Consecutive interior angle pairs are supplementary, which means they sum to ???180^\circ???.

 
a transversal creating consecutive interior angles
 

???m\angle d+m\angle e=180{}^\circ???

???m\angle b+m\angle g=180{}^\circ???

Adjacent angles are angles with the same vertex that lie on opposite sides of the parallel line, but on the same side of the transversal. Adjacent angle pairs are supplementary, which means they sum to ???180^\circ???.

 
a transversal creating adjacent angles
 

???m\angle a+m\angle d=180{}^\circ???

???m\angle a+m\angle c=180{}^\circ???

???m\angle b+m\angle c=180{}^\circ???

???m\angle d+m\angle b=180{}^\circ???

???m\angle e+m\angle g=180{}^\circ???

???m\angle e+m\angle h=180{}^\circ???

???m\angle g+m\angle f=180{}^\circ???

???m\angle f+m\angle h=180{}^\circ???

We often use these angle pair relationships to solve problems.

 
 

Working with the angles created by a transversal


 
 
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An example with corresponding and adjacent angle pairs

Example

Solve for the variable. Find the value of ???x??? to the nearest tenth, given ???m||n???.

a corresponding angle pair


We can think of which angle pair relationships to use.

a supplementary angle pair

Angles ???\angle 1??? and ???\angle 2??? are congruent because they are a corresponding angle pair. Angles ???\angle 2??? and ???\angle 3??? are supplementary because they are adjacent angles. This means the two angles with variables are supplementary, so

???(3x+5){}^\circ +(8x-3){}^\circ =180{}^\circ???

???3x{}^\circ +5{}^\circ +8x{}^\circ -3{}^\circ =180{}^\circ???

???11x{}^\circ +2{}^\circ =180{}^\circ???

???11x{}^\circ =178{}^\circ???

???x\approx 16.2{}^\circ???

Angles of transversals for Geometry.jpg

When transversals cross parallel lines, they form angles with special angle relationships. The angle pair relationships form two types of special angles.

Example

What is the measure of ???\angle XYZ???, given ???m||n???, ???m\angle ZYR=(5x+35){}^\circ??? and ???m\angle QRY=(15x-5){}^\circ????

finding an angle for a transversal

Looking at the diagram, we can see that ???\angle XYZ??? and ???\angle ZYR??? are adjacent angles that lie on the same parallel line, so they’re supplementary. ???\angle ZYR??? and ???\angle QYR??? are alternate interior angles, so they’re congruent.

Now we can use these facts to find the ???m\angle XYZ???. Let’s begin by solving for ???x???. ???m\angle ZYR=m\angle QYR??? because they are congruent angles. So,

???m\angle ZYR=m\angle QYR???

???(5x+35){}^\circ =(15x-5){}^\circ???

???35{}^\circ =10x{}^\circ -5{}^\circ???

???40{}^\circ =10x{}^\circ???

???x=4{}^\circ???

Now we can find ???m\angle ZYR??? when ???x=4{}^\circ???.

???m\angle ZYR=(5x+35){}^\circ???

???m\angle ZYR=(5\cdot 4+35){}^\circ???

???m\angle ZYR=(20+35){}^\circ???

???m\angle ZYR=55^\circ???

And because ???\angle XYZ??? and ???\angle ZYR??? are supplementary angles,

???m\angle XYZ+m\angle ZYR=180{}^\circ???

???m\angle XYZ+55{}^\circ =180{}^\circ???

???m\angle XYZ+55{}^\circ -55{}^\circ =180{}^\circ -55{}^\circ???

???m\angle XYZ=125{}^\circ???

 
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