7.02 Proving lines parallel
The converse theorems
In the previous lesson, we learned that if lines are parallel then certain relationships exist between angles formed by those lines and a transversal.
But what about the other way around? In other words, how can we use these relationships to find out whether lines are parallel?
Exploration
Let's start by considering the converse of the consecutive interior angle postulate:
"If consecutive interior angles are supplementary, then the two lines are parallel"
In the applet below, move the points around so that consecutive interior angles are supplementary. Can you make this true when $\overleftrightarrow{AB}$›‹AB is parallel to $\overleftrightarrow{CD}$›‹CD? Can you make it true when these lines are not parallel?
Notice that when the consecutive interior angles are not supplementary, the lines are not parallel (they intersect). This means that the converse of the consecutive interior angle postulate is also true.
In fact, the converses of all the results from the previous lesson are true!
- If consecutive interior angles are supplementary, then the two lines are parallel
- If consecutive exterior angles are supplementary, then the two lines are parallel
- If alternate interior angles are congruent, then the two lines are parallel
- If alternate exterior angles are congruent, then the two lines are parallel
- If corresponding angles are congruent, then the two lines are parallel
We can use these criteria to determine whether lines in a diagram are parallel.
Geometrical diagrams are always included as a rough guide only and are not to scale. If you're not sure, trust the number values written on the diagram over how the diagram looks!
Worked example
Question 1
Consider the diagram below.
Are the lines $AB$AB and $CD$CD parallel?
Think: This diagram shows the measure of consecutive interior angles. We need to find their sum to determine whether or not the lines are parallel.
Do: The sum of the measures of $\angle BFE$∠BFE and $\angle DEF$∠DEF is $114^\circ+63^\circ=177^\circ$114°+63°=177°, which is not $180^\circ$180°. This means these angles are not supplementary, so the lines are not parallel.
Reflect: How does this problem relate to the applet used at the beginning of the previous lesson?
Practice questions
Question 2
Are the lines $\overleftrightarrow{AB}$›‹AB and $\overleftrightarrow{CD}$›‹CD parallel?
It cannot be determined
AYes
BNo
C
Question 3
Consider the following image:
Two lines, $AB$AB and $CD$CD, are intersected by a transversal line $ED$ED such that lines $AB$AB and $CD$CD are on the left side of the transversal $ED$ED. Line $AB$AB is above $CD$CD. Transversal line $ED$ED intersects $AB$AB at point $B$B, forming $\angle EBA$∠EBA between $EB$EB and $AB$AB. Transversal line $ED$ED intersects $CD$CD at point $D$D, forming $\angle EDC$∠EDC between $ED$ED and $CD$CD. Both angles $\angle EBA$∠EBA and $\angle EDC$∠EDC are labeled and measures $50^\circ$50°. Both angles $\angle EBA$∠EBA and $\angle EDC$∠EDC are located on the left side of the transversal, and above their respective lines $AB$AB and $CD$CD. Another line connect points $C$C to $B$B to form a $\triangle CBD$△CBD. The two $50^\circ$50° angles, $\angle EBA$∠EBA and $\angle EDC$∠EDC, are corresponding angles.
Are the lines $\overleftrightarrow{AB}$›‹AB and $\overleftrightarrow{CD}$›‹CD parallel?
No
AIt cannot be determined
BYes
C
Question 4
$\overline{AB}$AB and $\overline{CD}$CD are not parallel because:
Three connected segments: $AB$AB, $BC$BC, and $CD$CD. Segment $AB$AB is connected to segment $BC$BC at point $B$B. Segment $CD$CD is connected to segment $BC$BC at point $C$C. Segment $AB$AB and segment $BC$BC formed $\angle ABC$∠ABC which measured $119^\circ$119°. Segment $BC$BC and segment $CD$CD formed $\angle BCD$∠BCD which measured $122^\circ$122°. $\angle ABC$∠ABC is on the left of segment $BC$BC, while $\angle BCD$∠BCD is on the right of segment $BC$BC.
$\angle ABC$∠ABC and $\angle BCD$∠BCD are vertical angles, and they're not congruent.
A$\angle ABC$∠ABC and $\angle BCD$∠BCD are alternate interior angles, and they're not congruent.
B$\angle ABC$∠ABC and $\angle BCD$∠BCD are consecutive interior angles, and they're not supplementary.
C$\angle ABC$∠ABC and $\angle BCD$∠BCD are corresponding angles, and they're not congruent.
D
Writing formal proofs of parallel lines
We are now going to consolidate everything we have learned so far about parallel lines, and use the theorems (and their converses) to construct proofs and solve problems in more difficult settings. We will use a method called a two-column proof to make sure we always move carefully and deliberately.
Two-column proofs
We begin by writing what we want to prove at the top of the table and divide the rest of the table into two columns. On the left will be each line of work, and on the right, there must always be a reason that justifies why it is true. Here's what it will look like once it's finished:
| To prove: $\left[\text{Statement to be proved}\right]$[Statement to be proved] | |
| Statement | Reason |
| Statement 1 | Reason 1 |
| Statement 2 | Reason 2 |
| Statement 3 | Reason 3 |
| $\ldots$… | $\ldots$… |
| $\left[\text{Statement to be proved}\right]$[Statement to be proved] | Reason $n$n |
We use as many lines of reasoning as we need, and make sure the last line always has the statement we want to prove in the end. Any information provided to us should have the reason Given written next to it, to indicate that no reasoning actually took place to arrive at the statement.
Worked examples
Question 5
Consider the following diagram:
Show that $\angle CST$∠CST is congruent to $\angle XQP$∠XQP.
Think: It is always a good idea to first translate the markings on the diagram into geometrical statements, as this is the given information in this problem. Here we can see that $\overleftrightarrow{AX}$›‹AX and $\overleftrightarrow{BY}$›‹BY are parallel, and $\overleftrightarrow{BY}$›‹BY and $\overleftrightarrow{CZ}$›‹CZ are also parallel. This means $\overleftrightarrow{PT}$›‹PT is a transversal for two sets of parallel lines, so we will use the theorems relating to the angles made by such transversals.
Do: Write what we want to show at the top, enter the given information as lines, and then use the correct steps and correct reasons to arrive at the conclusion.
| To prove: $\angle CST\cong\angle XQP$∠CST≅∠XQP | |
| Statement | Reason |
| $\overleftrightarrow{AX}\parallel\overleftrightarrow{BY}$›‹AX∥›‹BY | Given |
| $\overleftrightarrow{BY}\parallel\overleftrightarrow{CZ}$›‹BY∥›‹CZ | Given |
| $\angle XQP\cong\angle YRQ$∠XQP≅∠YRQ | Corresponding angles theorem |
| $\angle YRQ\cong\angle ZSR$∠YRQ≅∠ZSR | Corresponding angles theorem |
| $\angle XQP\cong\angle ZSR$∠XQP≅∠ZSR | Transitive property of congruence |
| $\angle ZSR\cong\angle CST$∠ZSR≅∠CST | Vertical angles congruence theorem |
| $\angle XQP\cong\angle CST$∠XQP≅∠CST | Transitive property of congruence |
Reflect: Could we have proved this another way? If we didn't use the corresponding angles theorem, could we have still proved it? Is the property of being parallel transitive?
question 6
Consider the following diagram:
Show that $\overline{AD}$AD and $\overline{BC}$BC are parallel.
Think: Here we have given that $\overline{AB}\parallel\overline{DC}$AB∥DC and $\angle ABC$∠ABC is congruent to $\angle CDA$∠CDA. We will need to use the properties of the lines we know are parallel to tell us about the angles on them, and the properties of the angles we know are congruent to tell us about the lines. In other words, we will need to use both a parallel line theorem and its converse.
Do: Write what we want to show at the top, enter the given information as lines, and then use the correct steps and correct reasons to arrive at the conclusion.
| To prove: $\overleftrightarrow{AD}\parallel\overleftrightarrow{BC}$›‹AD∥›‹BC | |
| Statement | Reason |
| $\overleftrightarrow{AB}\parallel\overleftrightarrow{DC}$›‹AB∥›‹DC | Given |
| $\angle ABC\cong\angle CDA$∠ABC≅∠CDA | Given |
| $\angle ABC$∠ABCand$\angle BCD$∠BCD are supplementary | Consecutive interior angles theorem |
| $\angle CDA$∠CDAand$\angle BCD$∠BCD are supplementary | Substitution |
| $\overleftrightarrow{AD}\parallel\overleftrightarrow{BC}$›‹AD∥›‹BC | Consecutive interior angles converse |
Reflect: Notice that we have used a theorem and its converse here - we use the pair of parallel lines we were given to talk about the interior angles of its transversal and then used one of the original lines as a transversal to the lines we wanted to show were parallel.
Practice questions
Question 7
This two-column proof shows that $\overleftrightarrow{BS}\parallel\overleftrightarrow{CR}$›‹BS∥›‹CR in the attached diagram, but it is incomplete.
| To prove: $\overleftrightarrow{BS}\parallel\overleftrightarrow{CR}$›‹BS∥›‹CR | |
|---|---|
| Statements | Reasons |
| $\angle BMN$∠BMN and $\angle QNR$∠QNR are supplementary | Given |
| $\left[\text{__________}\right]$[__________] | $\left[\text{__________}\right]$[__________] |
|
$\left[\text{__________}\right]$[__________] |
$\left[\text{__________}\right]$[__________] |
|
$\overleftrightarrow{BS}\parallel\overleftrightarrow{CR}$›‹BS∥›‹CR |
Alternate exterior angles converse |
Select the correct pair of reasons to complete the proof.
$\angle CNQ$∠CNQ and $\angle QNR$∠QNR are supplementary Linear pair theorem and $\angle CNQ\cong\angle BMN$∠CNQ≅∠BMN Congruent supplements theorem A$\angle BMN\cong\angle AMS$∠BMN≅∠AMS Vertical angles congruence theorem and $\angle AMS$∠AMS and $\angle QNR$∠QNR are supplementary Substitution B$\angle BMN$∠BMN and $\angle NMS$∠NMS are supplementary Linear pair theorem and $\angle QNR\cong\angle NMS$∠QNR≅∠NMS Congruent supplements theorem C$\angle BMN$∠BMN and $\angle BMA$∠BMA are supplementary Linear pair theorem and $\angle BMA\cong\angle QNR$∠BMA≅∠QNR Congruent supplements theorem D
Question 8
This two-column proof shows that $\overleftrightarrow{AD}\parallel\overleftrightarrow{BC}$›‹AD∥›‹BC in the attached diagram, but it is incomplete.
| Statements | Reasons |
|---|---|
| $\overleftrightarrow{AB}\parallel\overleftrightarrow{DC}$›‹AB∥›‹DC | Given |
| $\angle NBA$∠NBA and $\angle ADC$∠ADC are supplementary | Given |
| $\left[\text{_____}\right]$[_____] | $\left[\text{_____}\right]$[_____] |
|
$\left[\text{_____}\right]$[_____] |
$\left[\text{_____}\right]$[_____] |
|
$\overleftrightarrow{AD}\parallel\overleftrightarrow{BC}$›‹AD∥›‹BC |
Alternate interior angles converse |
Select the correct pair of reasons to complete the proof.
$\angle NBA$∠NBA and $\angle DCR$∠DCR are supplementary Corresponding angles theorem and $\angle ADC\cong\angle DCR$∠ADC≅∠DCR Congruent supplements theorem A$\angle ADC\cong\angle ABC$∠ADC≅∠ABC Consecutive interior angles theorem and $\angle ABC$∠ABC and $\angle NBA$∠NBA are supplementary Linear pair theorem B$\angle NBA$∠NBA and $\angle DCR$∠DCR are supplementary Consecutive exterior angles theorem and $\angle ADC\cong\angle DCR$∠ADC≅∠DCR Congruent supplements theorem C$\angle NBA$∠NBA and $\angle DCR$∠DCR are supplementary Consecutive exterior angles theorem and $\angle ADC\cong\angle DCR$∠ADC≅∠DCR Alternate interior angles theorem D