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| 16 Theory slides |
| 13 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
This lesson will focus on finding relations between these eight angles under the condition that two lines are parallel.
Here is a recommended reading before getting started with this lesson.
It is a Friday afternoon. Dylan just arrive at home 🏠 from school 🏫. He begins the weekend by mending the fence in his backyard. It broke last night because of extremely strong wind 💨.
Things were going great. While nailing up some boards 🔨, he noticed something. All of the vertical boards are parallel. Also, the horizontal stringers make four angles with each vertical board. Which angles are congruent? Dylan wonders.
When a line ℓ intersects two lines at two different points, the line ℓ receives a special name.
The eight angles formed by a transversal are classified into different groups. These grouping are based on the angle positions relative to the lines involved. As an example, focus on the angles between the two lines and those that are not.
When two lines ℓ1 and ℓ2 are cut by a transversal t, eight different angles are created — four around each point of intersection. These angles are classified into two groups according to their positions with respect to lines ℓ1 and ℓ2.
In the diagram, all the even-numbered angles are interior angles, and all the odd-numbered angles are exterior angles.
Interior Angles | Exterior Angles |
---|---|
∠2, ∠4, ∠6, ∠8 | ∠1, ∠3, ∠5, ∠7 |
Of the eight angles formed by a transversal that intersects two lines, the four interior angles are also classified according to their position with respect to the transversal.
If | Then |
---|---|
ℓ1∥ℓ2 | ∠3≅∠5 and ∠4≅∠6 |
∠3≅∠5 or ∠4≅∠6 | ℓ1∥ℓ2 |
The interior angles are those angles that lie between the two lines that are cut. In the mosaic, the regions 1, 3, 5, and 8 correspond to interior angles. Focus on these regions.
Nicely done! Now Dylan has six regions remaining to paint.
It was just taught that the four interior angles can be divided into two pairs of alternate interior angles. The same can be said for exterior angles. They can also be grouped into two pairs.
If | Then |
---|---|
ℓ1∥ℓ2 | ∠1≅∠7 and ∠2≅∠8 |
∠1≅∠7 or ∠2≅∠8 | ℓ1∥ℓ2 |
So far, the eight angles formed by a transversal and two parallel lines have been divided into two groups. In addition, each of these two groups has been divided into a subgroup.
Dylan continued to flip through the photo book. A picture his parents took in La Plata, Argentina fascinated him. He did an internet search to learn more. He discovered that it is one of the best-planned cities of the world. It is shaped like a square grid with some diagonal avenues, squares, and traffic circles.
Astonished by the structure of the city, he zoomed in on the map and saw that many interior streets are parallel to each other. For example, he noticed that the streets named Calle 8 and Calle 9 are parallel.
callein Spanish means
streetin English.
callein Spanish means
streetin English.
Notice that FE and CB lie over the street named Diagonal 77. This implies that these two segments lie over the same line. Also, GE lies over the street named Calle 8, and CH lies over Calle 9. That means the lines containing the segments are parallel since the two streets are parallel.
The diagram shows two parallel lines cut by a transversal. This means that ∠BCH and ∠FEG are alternate exterior angles.
m∠BCH=129, m∠FEG=3x
LHS/3=RHS/3
Rearrange equation
Note that AB and DE lie over the street named Diagonal 77. That means these two segments lie over the same line. Additionally, BG lies over Calle 42 and DH lies over Calle 43. These two streets are parallel. This implies that the lines containing the segments are also parallel.
The diagram shows two parallel lines cut by a transversal. That characteristic means that ∠GBA and ∠HDE are alternate exterior angles.
The alternate interior angles lie on opposite sides of the transversal. Now it is time to get to know the interior angles with different vertices that lie on the same side of the transversal.
If | Then |
---|---|
ℓ1∥ℓ2 | m∠3+m∠6=180∘ and m∠4+m∠5=180∘ |
m∠3+m∠6=180∘ or m∠4+m∠5=180∘ | ℓ1∥ℓ2 |
Dylan is still mesmerized by the travel album. He is staring at a photo of the Giza Pyramids in Egypt. His parents teach him about them. They were built as tombs of Egyptian pharaohs. The tallest pyramid is called the Great Pyramid. There are seven wonders of the ancient world — only the pyramids are still standing.
Draw the sides of the angles marked in the photo. Notice that these angles are consecutive interior angles.
Start by drawing the sides of the angles marked in the photo to relate these two angles.
Notice that the non-common sides of the angles are parallel to each other. Additionally, the common side can be seen as a transversal. The marked angles lie between the parallel lines and they are on the same side of the transversal.
It is now time to learn about the grouping of exterior angles that have different vertices but are on the same side of the transversal.
If | Then |
---|---|
ℓ1∥ℓ2 | m∠1+m∠8=180∘ and m∠2+m∠7=180∘ |
m∠1+m∠8=180∘ or m∠2+m∠7=180∘ | ℓ1∥ℓ2 |
Dylan was loving the stories his parents were telling. He asked for them to tell another. One of his parents looked through the photos and saw an elephant 🐘. What a beautiful safari they had while in Zimbabwe.
Dylan's curiosity peaked. He asked for more details about Zimbabwe. His parents told him that it is a country in South East Africa whose flag is made up of many parallel lines. The flag reminds them of a riddle they were challenged with during the trip. Dylan's parent drew some angles on the flag to prepare the riddle.
His parent begins the riddle.
Consider what I drew on Zimbabwe's flag. The number of countries in all of Africa equals three times x. How many countries are there in Africa? |
Draw the sides of the angles marked on the flag. The horizontal sides are parallel to each other. The other sides can be connected to form a transversal. Notice that the marked angles are same-side exterior angles.
Try to relate the two marked angles in a way that the value of x can be found. It seems like a good idea to start by drawing the sides of these angles.
Notice that the horizontal sides are parallel to each other. Additionally, the sides lying on the border of the triangular region lie on the same line. For this reason, those two sides can be connected to form a transversal.
Dylan's parent drew the angles on top of the flag. Draw the angles now without the flag for a clearer view.
Add and subtract terms
LHS+18=RHS+18
LHS/11=RHS/11
When a transversal cuts two lines, the interior angles can be paired in two different ways: as alternate angles or same-side angles. Something similar happens with exterior angles. Notice that each of these pairs is made up of either two interior angles or two exterior angles.
It is worth asking if an interior angle can be related to an exterior angle. The following concept establishes that type of relationship.
In the diagram, four pairs of angles can be identified as corresponding angles.
Pair | Position Relative to the Vertex | Position Relative to the Transversal |
---|---|---|
∠1 and ∠5 | Northeast | Right |
∠2 and ∠6 | Northwest | Left |
∠3 and ∠7 | Southwest | Left |
∠4 and ∠8 | Southeast | Right |
If two parallel lines are cut by a transversal, then the corresponding angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.
If | Then |
---|---|
ℓ1∥ℓ2 | ∠1≅∠5, ∠2≅∠6, ∠3≅∠7, and ∠4≅∠8 |
∠1≅∠5, ∠2≅∠6, ∠3≅∠7, or ∠4≅∠8 | ℓ1∥ℓ2 |
Dylan is having an incredible weekend full of stories and learning. He is impressed with all the countries his parents have visited. So far, they have talked about France, Argentina, Egypt, and Zimbabwe. With a huge smile, Dylan's parents tell him that they have been to Japan too.
In Japan they sailed ⛵ along the longest suspension bridge in the world at the time. It is called Akashi Kaikyo Bridge. In disbelief, Dylan looked for those photos in the album.
Notice that the vertical sides lie on the left vertical beam. That implies they lie on the same line. This line acts like a transversal cutting the two parallel diagonal beams.
Draw the given information without the photo in the background for a clearer visual of the angles, lines, and transversal.
LHS−2x=RHS−2x
LHS+28=RHS+28
LHS/2=RHS/2
Rearrange equation
The two diagonal bars are parallel to each other according to what Dylan sees. The upper angle is exterior while the lower angle is an interior angle. Additionally, both angles lie on the same side of the transversal.
Distribute 7
Subtract term
LHS−6y=RHS−6y
LHS+10=RHS+10
Rearrange equation
The Akashi Kaikyo Bridge was the longest suspension bridge in the world from April 1998 to March 2022. It connects the city of Kobe with Iwaya. In March 2022, the 1915 Çanakkale Bridge was opened in Türkiye. This bridge is 2023 meters long, only 32 meters longer than the Akashi Kaikyo Bridge.
Consider two lines ℓ1 and ℓ2 that are cut by the transversal t. Identify the pair of angles labeled.
Dylan was fixing the fence of his home. He noticed that all the vertical boards were parallel, and the horizontal stringers formed angles with each vertical board.
Making this discovery about the fence made him wonder about two things.
These two questions can be answered thanks to all of the information learned in this lesson. To start, focus on ∠7.
The diagram shows the lines of two vertical boards and the horizontal stringer. Notice that ∠7 can be related to some of the other angles according to its position relative to the transversal and to the parallel lines.
Angles | Relation |
---|---|
∠7 and ∠4 | Alternate Interior Angles |
∠7 and ∠3 | Consecutive Interior Angles |
∠7 and ∠2 | Corresponding Angles |
∠7 and ∠5 | Vertical Angles |
Since ∠7 and ∠5 are vertical, they are congruent. However, there is more! Because the lines ℓ1 and ℓ2 are parallel, some of the first three pairs of angles are congruent and some are supplementary.
Angles | Relation | Relationship Between Measures |
---|---|---|
∠7 and ∠4 | Alternate Interior Angles | Congruent |
∠7 and ∠3 | Consecutive Interior Angles | Supplementary |
∠7 and ∠2 | Corresponding Angles | Congruent |
∠7 and ∠5 | Vertical Angles | Congruent |
The angles congruent to ∠1 can be determined following a similar method. Again, study the given diagram. This time, write the relations for ∠1 in relation to ∠8, ∠5, ∠6, and ∠7.
Angles | Relation | Relationship Between Measures |
---|---|---|
∠1 and ∠8 | Alternate Exterior Angles | Congruent |
∠1 and ∠5 | Same-Side Exterior Angles | Supplementary |
∠1 and ∠6 | Corresponding Angles | Congruent |
∠1 and ∠3 | Vertical Angles | Congruent |
It is time to update the diagram. Recall that congruent angles are denoted with the same number of angle markers.
Classify the marked angles according to their positions for each of the following diagrams.
The angles the lines form with the transversal can be classified although the lines are not parallel. Here, the two marked angles are between l_1 and l_2. This means that the angles are interior angles.
We can also see that they are on opposite sides of the transversal.
Let's sum up what we know. The two angles are interior angles with different vertices that lie on opposite sides of the transversal. These characteristics allow for us to say that the marked angles are alternate interior angles.
Notice that both angles are outside the lines l_1 and l_2. This means that the marked angles are exterior angles.
These two angles are on the same side of the transversal. That is different than what we found in Part A.
The two marked angles are exterior angles with different vertices that are on the same side of the transversal. We conclude that the angles are same-side exterior angles.
One of the first things we can see is that the lines are parallel. Be aware that this does not change the way of classifying the angles. This time, we can see that one angle is between the lines and the other is outside them.
We can also see that the angles are in the same position relative to their vertices — both are southeast of their vertices. This means that the marked angles are corresponding angles.
Commercial airplanes usually take off at an angle between 5∘ to 15∘. The exact angle depends on the airplane and the sky conditions. This morning, Cameron's plane took off at an angle of 15∘. A few seconds passed. It then crossed a trail made by a plane that flew by earlier.
Let's begin by making a diagram that illustrates the described situation. Remember, the trail left by the other plane and the airport runway are parallel.
The plane's path is a transversal that cuts the old trail and the airport runway. Also, the two labeled angles are between the horizontal lines and on the same side of the transversal. That means the two angles are same-side interior angles. The two angles are supplementary since the old trail and the runway are parallel. x + 15^(∘) &= 180^(∘) &⇓ x &= 165^(∘) We conclude that the angle formed by the plane's path and the trail left by the other plane has a measure of 165^(∘).
Consider the following diagram. Lines ℓ1 and ℓ2 are parallel.
In the given diagram, notice that the required angle along with ∠ FDG and ∠ FDE form a straight angle.
The sum of the measures of these three angles is equal to 180^(∘). m∠ HDG + m∠ FDG + m∠ FDE = 180^(∘) We might wonder why we set the equation above if we do not know the measure of ∠ FDE. Well, notice that ∠ FDE is northwest of D, and ∠ ABD is also northwest of B. This implies that these two angles are corresponding angles. Corresponding Angles ∠ FDE & ∠ ABD The corresponding angles are congruent because lines l_1 and l_2 are parallel and t is a transversal. For this reason, we conclude that ∠ FDE and ∠ ABD have the same measure. m∠ FDE &= m∠ ABD &⇓ m∠ FDE &= 40^(∘) Next, let's substitute the measure of ∠ FDE into the equation written at the beginning. Then, we will solve it for m∠ HDG. Recall that the diagrams shows m∠ FDG=60^(∘).
We calculated the measure of the required angle to be 80^(∘). There is one last thing to do before finishing. Let's update the diagram by adding the measures we found.
Consider the following diagram where AB is parallel to DE.
Notice that the labeled angles are not formed by the same transversal. Then, we cannot relate them directly. However, we can try to find a connection between them by drawing an auxiliary line. Draw a line that passes through C and is parallel to AB and DE. Also, consider a point F on this line.
The diagram shows that CF divides the angle ACD into two angles, ∠ ACF and ∠ FCD. Therefore, the sum of the measures of these two new angles is equal to x. x = m∠ ACF + m∠ FCD That method caused us to have two more unknown angles! Why did we do this? Well, we can relate these two new angles to the two given angle measures. Begin by focusing on DE, CF, and the angles formed by the transversal CD.
The angles FCD and CDE are between the lines DE and CF. They are also on opposite sides of the transversal. These are the characteristics of alternate interior angles. Even more, they are congruent since the lines are parallel. As a result, they have the same measure. ∠ FCD &≅ ∠ CDE &⇓ m∠ FCD &= m∠ CDE In the given diagram, we can see that m∠ CDE=30^(∘). Then, m∠ FCD=30^(∘). Let's repeat the same reasoning but focus on the two lines below and the transversal AC.
The angles BAC and ACF are between the lines and on the same side of the transversal. For this reason, they are consecutive interior angles. The consecutive interior angles are supplementary because AB and CF are parallel, . Supplementary Angles ∠ ACF & ∠ BAC The sum of their measures is equal to 180^(∘) according to the definition of supplementary angles. m∠ ACF + m∠ BAC = 180^(∘) We see that m∠ BAC = 135^(∘). Let's substitute this into the equation to find the measure of ∠ ACF.
We now know the measures of ∠ ACF and ∠ FCD. They can be substituted into the first equation we wrote. This will determine the value of x.
The measure of ∠ ACD is 75^(∘).
Determine whether the lines ℓ1 and ℓ2 are parallel to each other.
Notice that the two labeled angles are between the lines l_1 and l_2. They are interior angles.
Additionally, the angles are on the same side of the transversal. This means that the angles are consecutive interior angles. Let's recall a fact about a situation just like this which will help us determine if the lines are parallel.
If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.
Now, we will verify whether the labeled angles are supplementary. Let's add their measures. 100^(∘) + 81^(∘) = 181^(∘) The sum is not equal to 180^(∘). This means that the angles are not supplementary. As a result of that, the lines are not parallel.
Notice that the two labeled angles are outside the lines l_1 and l_2. They are exterior angles.
In addition, the angles are on opposite sides of the transversal. This means that the angles are alternate exterior angles. Next, let's recall a fact about this type of situation which will help us determine whether the lines are parallel.
If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.
The two labeled angles have the same measure according to the diagram. That means the angles are congruent. Then, we can conclude that the lines are parallel.
Consider the following diagram.
Notice that ∠ 3 and ∠ 7 have different vertices and lie in the same position relative to their vertices. Both are west of their vertices. This means that these two angles are corresponding angles.
Additionally, the lines r and s have triangular hatch marks. This means that they are parallel. Now, remember that when two parallel lines are cut by a transversal, the corresponding angles are congruent. In other words, corresponding angles have the same measure. ∠ 3 ≅ ∠ 7 ⇓ m∠ 3 = m∠ 7 Since we were told that the measure of ∠ 3 is 50^(∘), we can say that the measure of ∠ 7 is also 50^(∘).
This time, let's relate ∠ 4 and ∠ 8. Take a look at the diagram. They have different vertices and lie on the exterior of lines r and s. This means that both are exterior angles. Also, they are on the same side of the transversal. As a result, they are same-side exterior angles.
As in Part A, we note that the lines r and s are parallel. Remember that when two parallel lines are cut by a transversal, the same-side exterior angles are supplementary. This means that the measures of the angles add up to 180^(∘). m∠ 4 + m∠ 8 = 180^(∘) We are told that the measure of ∠ 4 is 120^(∘). Let's substitute it into the equation and solve it for m∠ 8.
Finally, let's relate ∠ 6 and ∠ 5. Notice that both have different vertices and lie between the lines. They are interior angles. Additionally, both are on opposite sides of the transversal. For this reason, we conclude that the two angles are alternate interior angles.
As before, remember that the lines r and s are parallel. Even more, when two parallel lines are cut by a transversal, the alternate interior angles are congruent. In other words, alternate interior angles have the same measure. ∠ 6 ≅ ∠ 5 ⇓ m∠ 6 = m∠ 5 Since the measure of ∠ 6 is 125^(∘), we conclude that the measure of ∠ 5 is also 125^(∘).