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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.
Alternate interior angles are interior angles with different vertices that lie on opposite sides of the transversal. Of the regions highlighted before, the following pairs correspond to alternate interior angles.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.
The alternate exterior angles are congruent because the lines GE and CH are parallel. That means they have the same measure. An equation in terms of x will be set and solved using this information.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 exterior angles are congruent because the lines BG and DH are parallel. This means that ∠GBA≅∠HDE. Additionally, because ∠GBA≅∠HDE, they have the same measure. Set and solve an equation in terms of y using what information is now known.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.
Dylan's father says that the angle at the top of the Great Pyramid is about 76∘. The non-common sides of the marked angles are parallel to each other. What is the value of x?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.
The angle's characteristics suggest that they are consecutive interior angles. Additionally, the two angles are supplementary because the two lines are parallel. This means that the sum of their measures is equal to 180∘. The following equation can be set using the given information.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.
Notice that the angles lie on the exterior of two parallel lines and on the same side of the transversal. This means the angles are same-side exterior angles. These two angles are supplementary and their measures add up to 180∘ because of that.Add and subtract terms
LHS+18=RHS+18
LHS/11=RHS