Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
The Perpendicular Transversal Theorem
One of the interesting things we do in math is look at lines and angles, and we try to figure out a way to prove whether they are the same or have any other similarities. Even if two things look the same, mathematicians will never say they are unless they can prove it mathematically. That is where our theorems come in. A theorem is a proven statement. By using a theorem, mathematicians can prove a certain situation is definitely true or not. When mathematicians can prove that a certain situation is true, then they can move on and build on that information. This is how the field of mathematics has grown over the past hundreds of years. All the mathematics that we learn is based on layers and layers of mathematical proofs and theorems.
In this lesson, we'll look at the perpendicular transversal theorem and how you can use this theorem to help you find parallel lines. This particular theorem tells you that in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. What does this mean? Take a look at these two lines.
This theorem is telling you that line t is perpendicular to line k, and lines k and l are parallel, then line t is also perpendicular to line l.
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Two Congruent Angles
You can actually use this theorem to help you prove two angles are congruent, or two angles that are the same. How? Well, think about what it means to be perpendicular. When two lines are perpendicular, it means that they intersect at a right angle. So, the angles formed by the intersection of the two lines are right angles. If we have two intersections that are both perpendicular, then the corresponding angles are congruent to each other. Looking at our image, the two angles that are marked are congruent to each other.
The Converse of the Theorem
This particular theorem also has a converse statement. You get the converse statement when your conclusion and phrases are switched. So, the converse perpendicular transversal theorem is in a plane - if two lines are perpendicular to the same line, then they are parallel. We basically took the original theorem and wrote it backwards. By taking the converse of our theorem, we get even more use out of the theorem.
Two Parallel Lines
This converse statement now allows us to prove that two lines are parallel if the line that crosses both of them is perpendicular to both. In math, we need to be able to prove everything! So, sometimes you're given a problem where they tell you that these two angles are perpendicular. Then it might ask you if this information tells you anything about a pair of lines. If the two angles are corresponding angles of a line that crosses two other lines, then you can use our converse theorem to prove that the two lines are parallel.
Let's review what we've learned. The perpendicular transversal theorem tells you that in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. The converse perpendicular transversal theorem tells you in a plane, if two lines are perpendicular to the same line, then they are parallel. These theorems are used to help you prove that two angles are congruent or that two lines are parallel.
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The Perpendicular Transversal Theorem & Its Converse
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