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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

When you have a pair of parallel lines and a transversal, something very interesting happens to the angles that are formed. You can see this happen in real life at street intersections and such. Watch this video lesson to learn about all of this.

Before we have a transversal, we need a pair of **parallel lines**. These types of lines will never intersect and are always the same distance apart. In some cities, where the blocks are rectangles or squares, the streets generally run parallel to each other. If you are one street, there is no way you can get to the street parallel to yours without using a street that runs in a different direction.

This street that runs in a different direction can be likened to the **transversal**, a straight line that crosses two other lines. In this particular video lesson, we are going to talk about our transversal crossing a pair of parallel lines. I want to show you how, in this situation, something interesting happens to the angles that are formed by it.

When a transversal crosses a pair of parallel lines, you end up with a total of 8 angles. Before I tell you what that interesting thing is, let me label these angles so that we can talk about them better. I'm going to label each angle with a number starting with the number 1 on the upper left and then working my way down to angle 8 on the lower right.

You might already be able to guess what that interesting thing is when you look carefully at the different angles.

What happens is that when you have a transversal crossing a pair of parallel lines, you end up with only two different-sized angles. I know there are 8 angles in total, but you can match them up into two groups of four equal angles. There are also four different ways to match the equal angles to each other.

Let me show the groups of angles now.

The first group of equal angles consists of angles 1, 4, 5, and 8. All of these angles measure one size. The other group consists of angles 2, 3, 6, and 7. These angles all measure the same and the measurement is usually different from the first group. The only circumstance where both groups will have the same measurement is when the transversal is perpendicular to both parallel lines.

Do you see how the angles belonging to the two groups are equal to each other? You can kind of visually estimate it, as they do look like they're the same.

As I mentioned earlier, there are four ways to match our angles together. Each way has its own name.

The first way is called **corresponding angles**. These are the angles that are on the same corner at each intersection. So angles 1 and 5 are corresponding angles. Do you see how they are located at the same corner at their respective intersections? Another pair of corresponding angles consists of angles 2 and 6. The other pairs are angles 3 and 7 and angles 4 and 8. All of these pairs are made of equal angles. So angles 1 and 5 are equal; angles 2 and 6 are equal; angles 3 and 7 are equal; and angles 4 and 8 are equal. You can think of corresponding angles as the angles that are at the same corner at each intersection as a way to help you remember.

The second way is called **alternate interior angles**, which are the angles that are between the two parallel lines but on opposite sides of the transversal. The pairs that belong here are angles 3 and 6 and angles 4 and 5. Each pair is equal to each other. So angles 3 and 6 are equal to each other and angles 4 and 5 are equal to each other. To help you remember alternate interior angles, think of the first word (alternate) as telling you that you need one angle on one side of the transversal and another angle on the other side. Then, think of the second word (interior) as telling you that you have to stay between the two parallel lines.

The third way is called **alternate exterior angles**. These are the angles that are outside the pair of parallel lines but on opposite sides of the transversal. Here again, we only have two pairs that belong here. They are angles 1 and 8 and angles 2 and 7. Also, each pair is made of equal angles. So angles 1 and 8 are equal to each other and angles 2 and 7 are equal to each other. To help you remember alternate exterior angles, think of the first word (alternate) as telling you that you need one angle on one side of the transversal and another angle on the other side. Then, think of the second word (exterior) as telling you that you have to stay outside the two parallel lines.

The fourth way is called **vertical angles**, meaning the angles that are kitty corner to each other at each intersection. Again, each pair consists of equal angles. So for vertical angles, we have angles 1 and 4 equal to each other, angles 2 and 3 equal to each other, angles 5 and 8 equal to each other, and angles 6 and 7 equal to each other. You can think of vertical angles as the angles whose tips are touching each other. If you took a stick and pinned it to each intersection and spun the stick around, the vertical angles would be the angles that the stick touches at any given position.

We have reached the end of our lesson, so let's review what we've learned. We know that **parallel lines** are lines that never meet each other and are always the same distance apart and the **transversal** is the line that crosses the two lines. When a transversal crosses a pair of parallel lines it produces 8 angles in total, which can be matched together into groups of equal angles. We know that angles 1, 4, 5, and 8 are equal, as well as angles 2, 3, 6, and 7. There are four ways we can match our equal angles.

They are 1) **corresponding angles**, the angles that are on the same corner at each intersection; 2) **alternate interior angles**, the angles that are between the two parallel lines but on opposite sides of the transversal; 3) **alternate exterior angles**, the angles that are outside the parallel lines but on opposite sides of the transversal; and 4) **vertical angles**, the angles that are kitty corner to each other at each intersection.

Upon finishing this lesson, you should be able to:

- Identify the angles caused by a transversal of two parallel lines
- Point out the congruent angles
- Demonstrate how to match the four angles of the intersecting transversal

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Geometry: High School15 chapters | 160 lessons

- Angles Formed by a Transversal 7:40
- Using Converse Statements to Prove Lines Are Parallel 6:46
- Constructing a Parallel Line Using a Point Not on the Given Line 5:15
- The Parallel Postulate: Definition & Examples 4:25
- What Are Polygons? - Definition and Examples 4:25
- Regular Polygons: Definition & Parts 6:01
- How to Find the Number of Diagonals in a Polygon 4:49
- Finding the Perimeter of Polygons 5:19
- Measuring the Area of Regular Polygons: Formula & Examples 4:15
- Measuring the Angles of Triangles: 180 Degrees 5:14
- How to Measure the Angles of a Polygon & Find the Sum 6:00
- Go to High School Geometry: Parallel Lines and Polygons

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