Angles of Parallel Lines Cut by Transversals
Basics on the topic Angles of Parallel Lines Cut by Transversals
Understanding Parallel Lines
In this geometry lesson, we're delving into the world of Parallel Lines. These lines are everywhere around us, from the straight paths of train tracks to the crisp edges of your tablet, and even in the architectural lines of buildings and bridges. Parallel lines are unique because they follow a simple rule: no matter how far they extend, they never cross.
This lesson will uncover the angles and patterns formed when these parallel lines are crossed by another line, known as a transversal. It's about seeing the math in our everyday world and understanding the geometry that shapes it. Ready to see how? Let’s get started!
Parallel Lines, Perpendicular Lines, and Transversals
- Parallel Lines: Parallel lines are like train tracks, never touching or crossing each other, and they stay the same distance apart forever.
Perpendicular Lines: Perpendicular lines cross each other and always form a 90-degree angle, making corners like the letter 'L' or the corners of a square.
Transversal Lines: A transversal line is a line that crosses at least two other lines. When it crosses parallel lines, it creates equal angles at the points of intersection. With non-parallel lines, it forms various angles.
Understanding Parallel Lines – Definition
Parallel Lines are lines on a plane that are always the same distance apart and never intersect.
Congruent means having the exact size and shape. In geometry, congruent angles have equal measures.
There are also some angle relationships that are important to know when learning about parallel lines.
Parallel Lines – Angle Relationships
Angle Relationships:
| Type of Angles | Explanation |
|---|---|
| Vertical Angles | Angles opposite each other when two lines intersect. They are always congruent. |
| Supplementary Angles | Two angles that add up to 180 degrees. They often appear when lines intersect. |
| Corresponding Angles | When a transversal crosses two parallel lines, these angles are in matching positions. They are congruent in parallel lines. |
| Alternate Interior Angles | Angles inside the parallel lines on opposite sides of the transversal. They are congruent in parallel lines. |
| Alternate Exterior Angles | Angles outside the parallel lines on opposite sides of the transversal. They are congruent in parallel lines. |
Parallel Lines – Guided Practice
Parallel Lines – Exercises
Vertical Angles: $\angle{X}$ and $\angle{Z}$, $\angle{Y}$ and $\angle{W}$, $\angle{A}$ and $\angle{C}$, $\angle{B}$ and $\angle{D}$
Corresponding Angles: $\angle{X}$ and $\angle{A}$, $\angle{W}$ and $\angle{D}$, $\angle{Y}$ and $\angle{B}$, $\angle{Z}$ and $\angle{C}$
Alternate Exterior Angles: $\angle{X}$ and $\angle{C}$, $\angle{Y}$ and $\angle{D}$
Alternate Interior Angles: $\angle{W}$ and $\angle{B}$, $\angle{Z}$ and $\angle{A}$
Using the illustration above, answer the following questions to check your understanding.
Parallel Lines – Summary
Key Learnings:
Parallel lines remain the same distance apart and never intersect.
A transversal creates various angle types, including corresponding, alternate interior, and alternate exterior angles, which are congruent in parallel lines.
Understanding these angle relationships is essential for mastering geometry concepts.
If you are ready and confident with this topic, why not apply it to the following topic Calculations with Supplementary, Complementary, and Vertical Angles
Explore more interactive and engaging geometry lessons on our website, complete with practice problems, videos, and worksheets!
Parallel Lines – Frequently Asked Questions
Transcript Angles of Parallel Lines Cut by Transversals
On their nightly food run, the three raccoons crashed their shopping cart... AGAIN. It's time to go back to the drawing stump. They decide to practice going around the sharp corners and tight angles during the day, before they get their loot. To put this surefire plan into action they'll have to use their knowledge of parallel lines and transversals. Let's look at this map of their city. All the HORIZONTAL roads are parallel lines. They DON'T intersect. But there are several roads which CROSS the parallel ones. These lines are called TRANSVERSALS. The raccoons crashed HERE at angle 1. The measure of angle 1 is 60 degrees. Can you see any other angles that are also 60 degrees? There are a few such angles, and one of them is angle 3. That's because angle 1 and angle 3 are vertical angles, and vertical angles are always equal in measure. Do we have enough information to determine the measure of angle 2? Since angles 1 and 2 are angles on a line, they sum to 180 degrees. That means angle 2 is 120 degrees. And since angles 2 and 4 are vertical, angle 4 must also be 120 degrees. Now we know all of the angles around this intersection, but what about the angles at the other intersection? Let's take a look at angle 5. If we translate angle 1 along the transversal until it overlaps angle 5, it looks like they are congruent. And they are! That means angle 5 is also 60 degrees. Angle 1 and angle 5 are examples of CORRESPONDING angles. Corresponding angles are pairs of angles that are in the SAME location around their respective vertices. And whenever two PARALLEL lines are cut by a transversal, pairs of corresponding angles are CONGRUENT. That means you only have to know the measure of one angle from the pair, and you automatically know the measure of the other! Can you see other pairs of corresponding angles here? Angles 2 and 6 are also corresponding angles. So are angles 3 and 7 and angles 4 and 8. That means the measure of angle 2 equals the measure of angle 6, the measure of angle 3 equals the measure of angle 7, and the measure of angle 4 equals the measure of angle 8. We already know that angles 4 and 6 are both 120 degrees, but is it ALWAYS the case that such angles are congruent? It is! Let's show this visually. Now, let's use our knowledge of vertical and corresponding angles to prove it. We are going to use angle 2 to help us compare the two angles. Angle 4 must be equal to angle 2 because they are vertical angles. And angle 6 must be equal to angle 2 because they are corresponding angles. Since angle 6 and angle 4 are both equal to the same angle, they also must be equal to each other! We call angle pairs like angle 6 and angle 4 alternate interior angles because they are found on ALTERNATE sides of the transversal and they are both INTERIOR to the two parallel lines. Can you see another pair of alternate interior angles? 3 and 5 are ALSO alternate interior. If two parallel lines are cut by a transversal, alternate interior angles are always congruent. We just looked at alternate interior angles, but we also have pairs of angles that are called alternate EXTERIOR angles. Based on the name, which angle pairs do you think would be called alternate exterior angles? Well, they need to be EXTERIOR to the parallel lines and on ALTERNATE sides of the transversal. 1 and 7 are a pair of alternate exterior angles and so are 2 and 8. Notice that the measure of angle 1 equals the measure of angle 7 and the same is true for angles 2 and 8. If two parallel lines are cut by a transversal, alternate exterior angles are always congruent. In fact, when parallel lines are cut by a transversal, there are a lot of congruent angles. Look at what happens when this same transversal intersects additional parallel lines. We can use congruent angle pairs to fill in the measures for THESE angles as well. The raccoons only need to practice driving their shopping cart around ONE corner to be ready for ALL the intersections along this transversal. For each transversal, the raccoons only have to measure ONE angle. They can then use their knowledge of corresponding angles, alternate interior angles, and alternate exterior angles to find the measures for ALL the angles along that transversal. Now it's time for some practice before they do a little...um... shopping. While they are riding around, let's review what we've learned. When parallel lines are cut by a transversal, congruent angle pairs are created. Corresponding angles are in the SAME position around their respective vertices and there are FOUR such pairs. Alternate interior angles are on ALTERNATE sides of the transversal and INTERIOR to the parallel lines and there are two such pairs. Alternate EXTERIOR angles are on alternate sides of the transversal and EXTERIOR to the parallel lines and there are also two such pairs. The raccoons are trying to corner the market on food scraps, angling for a night-time feast! Well, THAT was definitely a TURN for the worse!
Angles of Parallel Lines Cut by Transversals exercise
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Identify the measurement of a missing angle.
HintsVertical Angles are opposite each other when two lines intersect. They are always congruent.
Congruent Angles have the same angle measurement.
So if, $\angle{x}=100^\circ$ and $\angle{y}$ was known to be congruent, then $\angle{y}$ would also equal $100^\circ$.
Solution$\angle{b}$ and $\angle{d}$ are vertical angles and therefore they are congruent.
$\angle d = 150^\circ$
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Identify angle relationships.
HintsCorresponding angles are in the same position around their respective vertices.
Interior angles are the ones found between the parallel lines.
Exterior angles are the ones found on the outside of the parallel lines.
The word alternate, found in the angle relationships Alternate Interior Angles, and Alternate Exterior Angles, refers to the angles on opposite sides of the transversal.
SolutionCorresponding Angles
$\angle$4 and $\angle$8
$\angle$2 and $\angle$6
${}$ Alternate Interior Angles
$\angle$3 and $\angle$5
$\angle$4 and $\angle$6
${}$ Alternate Exterior Angles
$\angle$2 and $\angle$8
$\angle$1 and $\angle$7
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Understanding Supplementary Angles
HintsSupplementary Angles are two angles next to each other on a line that have a sum of $180^\circ$.
The sum of $130^\circ$ and $50^\circ$ is $180^\circ$.
SolutionAngle Z is equal to $155^\circ$ because $25+155=180^\circ$.
Angle Y is equal to $25^\circ$ for a few reasons:
- Angle Y is verticle to the existing $25^\circ$
- Angle Y + Angle Z must equal $180^\circ$
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Understand the angle relationships when there are two parallel lines cut by a transversal.
HintsHere you see an example of parallel lines.
Alternate Interior Angles -- Angles inside the parallel lines on opposite sides of the transversal. They are congruent.
Alternate Exterior Angles -- Angles outside the parallel lines on opposite sides of the transversal. They are congruent.
SolutionLines $m$ and $n$ are parallel lines and they will never cross. Line $n$ is called a transversal and when it crosses the parallel lines it forms eight angles. All of the angles formed have different angle relationships. For example, $\angle{2}$ and $\angle{4}$ are across from each other and are vertical angles. $\angle{3}$ and $\angle{5}$ are the same measurement because they are alternate interior angles, while $\angle{1}$ and $\angle{7}$ are the same measurement because they are alternate exterior angles. The word we use to describe angles that are the same measurement is: congruent.
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Understand the relationship between parallel lines and their transversals.
HintsThis image will show you what a set of parallel lines and transversals look like.
This image will show you what perpendicular lines look like.
SolutionParallel Lines: Lines on a plane that are always the same distance apart and never intersect.
Transversal: Line that crosses at least two other lines and when it crosses parallel lines, it creates equal angles at the points of intersection.
Perpendicular Lines: Lines cross each other and always form a 90-degree angle, making corners like the letter 'L' or the corners of a square.
Congruent: Having the same angle measurement.
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Identify missing angle values using your knowledge of angle relationships.
HintsTo find missing angles is sort of like solving a puzzle. It is helpful to start with the ones you are most confident with which can then make more difficult problems more approachable.
Supplementary Angles -- two angles that have a sum of $180^\circ$.
SolutionTo find the missing measurements, you have to determine the relationship between the known angle and the missing angle(s).
Also, remember that a straight line has angles that have a sum of $180^\circ$, which can be helpful when finding missing angles.
