We have now discussed that when we have parallel lines cut by a transversal, several angle pairs end up being congruent and other angle pairs are supplementary. We will now discuss the converse of each of the statements to see if the converse also holds true.
Let’s begin with the following situation:
Let’s begin with the following situation:
1) What kind of angle pair is shown in the diagram?
2) What is the relationship between those angles?
3) What do you notice is true about \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)?
4) Move around any of the points to see if the above conclusions still hold true.This activity brings us to the converse of the corresponding angles postulate we learned in Parallel & Perpendicular Lines Target B:
2) What is the relationship between those angles?
3) What do you notice is true about \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\)?
4) Move around any of the points to see if the above conclusions still hold true.This activity brings us to the converse of the corresponding angles postulate we learned in Parallel & Perpendicular Lines Target B:
Converse of Corresponding Angles Postulate: If two lines are cut by a transversal where the corresponding angles are congruent, then the lines are parallel.
If corresponding angle are congruent, then parallel lines.
If corresponding angle are congruent, then parallel lines.
Example 1: Determine if any of the following lines are parallel.
a) b)
a) b)
Solution:
a) Notice that the are corresponding angles are \(92^{\circ}\) and \(91^{\circ}\). Since they do not have the same measure, the corresponding angles are not congruent. Therefore we cannot say that the lines are parallel.
b) Notice that the angles marked are corresponding angles. Since they have the same measure (they both measure \(x^{\circ}\)), we know that the corresponding angles ARE congruent. Therefore, we can say that because if two lines are cut by a transversal so that the corresponding angles are congruent, then they are parallel.
Now we can actually prove the rest of the theorems using the converse of the corresponding angles postulate.
a) Notice that the are corresponding angles are \(92^{\circ}\) and \(91^{\circ}\). Since they do not have the same measure, the corresponding angles are not congruent. Therefore we cannot say that the lines are parallel.
b) Notice that the angles marked are corresponding angles. Since they have the same measure (they both measure \(x^{\circ}\)), we know that the corresponding angles ARE congruent. Therefore, we can say that because if two lines are cut by a transversal so that the corresponding angles are congruent, then they are parallel.
Now we can actually prove the rest of the theorems using the converse of the corresponding angles postulate.
Solution:
Let’s write the proof out as a paragraph proof:
We know that \(\angle HCA\cong\angle CAB\) since it is given. Now because \(\angle HCA\)and \(\angle ECD\) are vertical angles, then they are congruent. Since \(\angle CAB\) and \(\angle ECD\) are congruent to the same angle, \(\angle HCA\), they are congruent to each other by the transitive property. These are congruent corresponding angles; therefore \(\overleftrightarrow{HD}\parallel\overleftrightarrow{GB}\)
This proves our next theorem. Notice \(\angle CAB\) and \(\angle ECD\) are alternate interior angles. We just proved that:
Let’s write the proof out as a paragraph proof:
We know that \(\angle HCA\cong\angle CAB\) since it is given. Now because \(\angle HCA\)and \(\angle ECD\) are vertical angles, then they are congruent. Since \(\angle CAB\) and \(\angle ECD\) are congruent to the same angle, \(\angle HCA\), they are congruent to each other by the transitive property. These are congruent corresponding angles; therefore \(\overleftrightarrow{HD}\parallel\overleftrightarrow{GB}\)
This proves our next theorem. Notice \(\angle CAB\) and \(\angle ECD\) are alternate interior angles. We just proved that:
Theorem: If two lines are cut by a transversal where the alternate interior angles are congruent, then the lines are parallel.
If alternate interior angles are congruent, then lines are parallel.
If alternate interior angles are congruent, then lines are parallel.
Similar to the proof above, we can use the corresponding angles postulate to prove the theorems below.
Theorem: If two lines are cut by a transversal where the alternate exterior angles are congruent, then the lines are parallel.
If alternate exterior angles are congruent, then the lines are parallel.
If alternate exterior angles are congruent, then the lines are parallel.
Theorem: If two lines are cut by a transversal where the same side interior or consecutive interior angles (which have also been called same side interior angles) are supplementary, then the lines are parallel.
If same side interior angles are supplememtary, then the lines are parallel.
or
If consecutive angles are supplementary, then the lines are parallel.
If same side interior angles are supplememtary, then the lines are parallel.
or
If consecutive angles are supplementary, then the lines are parallel.
Now that we have several ways to prove lines to be parallel, it will be our task to figure out which of the above reasons (if any) will allow us to prove lines are parallel. Remember, just because the lines look parallel, it doesn’t mean that they actually are!
Example 3:
Determine if any lines are parallel. Make sure to justify your reasoning.
Example 3:
Determine if any lines are parallel. Make sure to justify your reasoning.
Solution:
a) The angles marked are NOT a special angle pair. We can, however, use a linear pair to help us find some other angles. Since a linear pair of angles are supplementary, we can find the measure of the adjacent angle: \(180^{\circ} - 70^{\circ} = 110^{\circ}\) Now you can see that the alternate exterior angles are NOT congruent, as one has measure of \(70^{\circ}\) and the other \(110^{\circ}\). Therefore we can say that \(b\nparallel c\). |
b) The angles marked are same side interior or consecutive angles. Our theorem states that if the same side interior angles are supplementary, then the lines are parallel. Let’s see if those angles are supplementary:
\(34^{\circ} + 156^{\circ} = 190^{\circ}\).
These angles are NOT supplementary, so \(g\nparallel h\).
c) The angles marked in this diagram are alternate exterior angles. Since both of the alternate exterior angles have the same measure, we know that they are congruent. Therefore \(e\parallel f\), because if alternate exterior angles are congruent, then parallel lines.
d) The angles marked are alternate interior angles. Since the alternate interior angles have the same measure, they are congruent. Therefore \(n\parallel p\), because if alternate interior angles are congruent, then parallel lines.
\(34^{\circ} + 156^{\circ} = 190^{\circ}\).
These angles are NOT supplementary, so \(g\nparallel h\).
c) The angles marked in this diagram are alternate exterior angles. Since both of the alternate exterior angles have the same measure, we know that they are congruent. Therefore \(e\parallel f\), because if alternate exterior angles are congruent, then parallel lines.
d) The angles marked are alternate interior angles. Since the alternate interior angles have the same measure, they are congruent. Therefore \(n\parallel p\), because if alternate interior angles are congruent, then parallel lines.
e) Watch this video.
f) Watch this video
Example 4: Determine the value of \(x\) that would make \(e\parallel f\).
Solution:
Quick Check
1) Determine if any lines are parallel. Justify your reasoning.
1) Determine if any lines are parallel. Justify your reasoning.
2) Describe and correct the error in solving the following problem:
Determine the value of \(x\) that will make \(b\parallel c\).
Determine the value of \(x\) that will make \(b\parallel c\).
Student’s solution:
These angles are alternate interior angles, so they have to be congruent:
\(\begin{align*}
3x - 10 &= 2x + 20\\
x - 10 &= 20\\
x &= 30
\end{align*}\)
Quick Check Solutions
These angles are alternate interior angles, so they have to be congruent:
\(\begin{align*}
3x - 10 &= 2x + 20\\
x - 10 &= 20\\
x &= 30
\end{align*}\)
Quick Check Solutions