Back To CourseGeometry: High School
15 chapters | 160 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Picture a railroad track and a road crossing the tracks. You know that the railroad tracks are parallel; otherwise, the train wouldn't be able to run on them without tipping over. But, how can you prove that they are parallel? First, you recall the definition of parallel lines, meaning they are a pair of lines that never intersect and are always the same distance apart. Then you think about the importance of the transversal, the line that cuts across two other lines. At this point, you link the railroad tracks to the parallel lines and the road with the transversal. Now what?
Now you get to look at the angles that are formed by the transversal with the parallel lines. There are four different things you can look for that we will see in action here in just a bit. Just remember that when it comes to proving two lines are parallel, all you have to look at are the angles. Specifically, we want to look for pairs of:
If we find just one pair that works, then we know that the lines are parallel. Also, you will see that each pair has one angle at one intersection and another angle at another intersection. When I say intersection, I mean the point where the transversal cuts across one of the parallel lines. So, since there are two lines in a pair of parallel lines, there are two intersections.
Corresponding angles are the angles that are at the same corner at each intersection. This means that if my first angle is at the top left corner of one intersection, the matching angle at the other intersection is also at the top left. Since there are four corners, we have four possibilities here:
We can match the corners at top left, top right, lower left, or lower right.
What we are looking for here is whether or not these two angles are congruent or equal to each other. If they are, then the lines are parallel. So, if both of these angles measured 60 degrees, then you know that the lines are parallel. But, if the angles measure differently, then automatically, these two lines are not parallel.
Alternate interior angles is the next option we have. These angles are the angles that are on opposite sides of the transversal and inside the pair of parallel lines. So, you will have one angle on one side of the transversal and another angle on the other side of the transversal. And, both of these angles will be inside the pair of parallel lines. The inside part of the parallel lines is the part between the two lines. So, for the railroad tracks, the inside part of the tracks is the part that the train covers when it goes over the tracks. We have two possibilities here:
We can match top inside left with bottom inside right or top inside right with bottom inside left.
Also here, if either of these pairs is equal, then the lines are parallel. So, say the top inside left angle measures 45, and the bottom inside right also measures 45, then you can say that the lines are parallel.
Next is alternate exterior angles. These are the angles that are on opposite sides of the transversal and outside the pair of parallel lines. Going back to the railroad tracks, these pairs of angles will have one angle on one side of the road and the other angle on the other side of the road. But, both of these angles will be outside the tracks, meaning they will be on the part that the train doesn't cover when it goes over the tracks. We also have two possibilities here:
We can have top outside left with the bottom outside right or the top outside right with the bottom outside left.
If either of these is equal, then the lines are parallel. So, if my top outside right and bottom outside left angles both measured 33 degrees, then I can say for sure that my lines are parallel.
The last option we have is to look for supplementary angles or angles that add up to 180 degrees. For parallel lines, there are four pairs of supplementary angles. All of these pairs match angles that are on the same side of the transversal. So, if you were looking at your railroad track with the road going through it, the angles that are supplementary would both be on the same side of the road. One pair would be outside the tracks, and the other pair would be inside the tracks. You would have the same on the other side of the road. So, you have a total of four possibilities here:
If you find that any of these pairs is supplementary, then your lines are definitely parallel. So, say that my top outside left angle is 110 degrees, and my bottom outside left angle is 70 degrees. They add up to 180 degrees, which means that they are supplementary. And, since they are supplementary, I can safely say that my lines are parallel.
Proving that lines are parallel is quite interesting. We've learned that parallel lines are lines that never intersect and are always at the same distance apart. We also know that the transversal is the line that cuts across two lines. We learned that there are four ways to prove lines are parallel.
The first is if the corresponding angles, the angles that are on the same corner at each intersection, are equal, then the lines are parallel. The second is if the alternate interior angles, the angles that are on opposite sides of the transversal and inside the parallel lines, are equal, then the lines are parallel. The third is if the alternate exterior angles, the angles that are on opposite sides of the transversal and outside the parallel lines, are equal, then the lines are parallel. And, fourth is to see if either the same side interior or same side exterior angles are supplementary or add up to 180 degrees. All you have to do is to find one pair that fits one of these criteria to prove a pair of lines is parallel.
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Back To CourseGeometry: High School
15 chapters | 160 lessons