Back To CourseGeometry: High School
15 chapters | 160 lessons
As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.Try it risk-free
Amy has a master's degree in secondary education and has taught math at a public charter high school.
To begin, we know that a pair of parallel lines is a pair that never intersect and are always the same distance apart. Think of the tracks on a roller coaster ride. Do you see how they never intersect each other and are always the same distance apart? This is what parallel lines are about.
Along with parallel lines, we are also dealing with converse statements. Don't worry, it's nothing complicated. If we had a statement such as 'If a square is a rectangle, then a circle is an oval,' then its converse would just be the same statement but in reverse order, like this: 'If a circle is an oval, then a square is a rectangle.' We started with 'If this, then that,' and we ended up with 'If that, then this.' So just think of the converse as flipping the order of the statement.
Now, with parallel lines, we have our original statements that tell us when lines are parallel.
We have four original statements we can make. But in order for the statements to work, for us to be able to prove the lines are parallel, we need a transversal, or a line that cuts across two lines. This line creates eight different angles that we can compare with each other. So if you're still picturing the tracks on a roller coaster ride, now add in a straight line that cuts across the tracks. You will see that it forms eight different angles. This is your transversal. You need this to prove parallel lines because you need the angles it forms because it's the properties of the angles that either make or break a pair of parallel lines.
What are the properties that the angles must have if the lines are parallel?
1. If the lines are parallel, then the corresponding angles are congruent. These are the angles that are on the same corner at each intersection. You will see that the transversal produces two intersections, one for each line. So, a corresponding pair of angles will both be at the same corner at their respective intersections. So if one angle was at the top left corner at one intersection, the corresponding angle at the other intersection will also be at the top left. For parallel lines, these angles must be equal to each other.
2. If the lines are parallel, then the alternate interior angles are congruent. The word 'alternate' means that you will have one angle on one side of the transversal and the other angle on the other side of the transversal. 'Interior' means that both angles are between the two lines that are parallel. These angles must be equal to each other for parallel lines.
3. If the lines are parallel, then the alternate exterior angles are congruent. This is similar to the one we just went over except now the angles are outside the pair of parallel lines. So these angles must likewise be equal to each for parallel lines.
4. Last but not least, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary. Here, the angles are the ones between the two lines that are parallel, but both angles are not on the same side of the transversal. These must add up to 180 degrees.
Now let's look at how our converse statements will look like and how we can use it with the angles that are formed by our transversal. All I need is for one of these to be satisfied in order to have a successful proof.
1. If the corresponding angles are congruent, then the lines are parallel. To use this statement to prove parallel lines, all we need is to find one pair of corresponding angles that are congruent. That is all we need. So we look at both intersections and we look for matching angles at each corner. For example, if we found that the top-right corner at each intersection is equal, then we can say that the lines are parallel using this statement.
2. If the alternate interior angles are congruent, then the lines are parallel. All we need here is also just one pair of alternate interior angles to show that our lines are parallel. So, for example, if we found that the angle located at the bottom-left corner at the top intersection is equal to the angle at the top-right corner at the bottom intersection, then we can prove that the lines are parallel using this statement.
3. If the alternate exterior angles are congruent, then the lines are parallel. Yes, here too we only need to find one pair of angles that is congruent. So, if my angle at the top right corner of the top intersection is equal to the angle at the bottom left corner of the bottom intersection, then by means of this statement I can say that the lines are parallel.
4. If the interior angles on the same side of the transversal are supplementary, then the lines are parallel. So, if the interior angles on either side of the transversal add up to 180 degrees, then I can use this statement to prove the lines are parallel. For example, if I added the angle at the bottom left of the top intersection to the angle at the top left of the bottom intersection and I got 180 degrees, then I can use this statement to prove my lines are parallel.
To prove any pair of lines is parallel, all you need is to satisfy one of the above.
What have we learned? We know that in order to prove a pair of parallel lines, lines that never intersect and are always the same distance apart, are indeed parallel, we need a transversal, which is a line that intersects two other lines. This transversal creates eight angles that we can compare with each other to prove our lines parallel. When the lines are indeed parallel, the angles have four different properties.
We can use the converse of these statements to prove that lines are parallel by saying that if the angles show a particular property, then the lines are parallel. These properties are:
If any of these properties are met, then we can say that the lines are parallel.
The process of studying this video lesson could allow you to:
To unlock this lesson you must be a Study.com Member.
Create your account
Already a member? Log InBack
Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.
To learn more, visit our Earning Credit Page
Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.
Back To CourseGeometry: High School
15 chapters | 160 lessons