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Consider when three lines are drawn. Eight angles are formed when two or fewer of those lines are parallel. The angles formed at one intersection point are related to the angles formed at the other. In the given diagram, there are no parallel lines.
### Catch-Up and Review

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.**

Challenge

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.

a Which angles are congruent to $∠7?$

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b Which angles are congruent to $∠1?$

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Discussion

When a line $ℓ$ intersects two lines at two different points, the line $ℓ$ receives a special name.

Concept

A transversal is a line that intersects two or more lines at different points.

When a pair of lines are cut by a transversal, eight different angles are created. There are four angles around each point of intersection.

Discussion

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.

Concept

When two lines $ℓ_{1}$ and $ℓ_{2}$ 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 $ℓ_{1}$ and $ℓ_{2}.$

- Interior angles: the angles that are
*between*the lines $ℓ_{1}$ and $ℓ_{2}.$ - Exterior angles: the angles that are
*outside*the lines $ℓ_{1}$ and $ℓ_{2}.$

In the diagram, all the even-numbered angles are interior angles, and all the odd-numbered angles are exterior angles.

Interior Angles | Exterior Angles |
---|---|

$∠2,$ $∠4,$ $∠6,$ $∠8$ | $∠1,$ $∠3,$ $∠5,$ $∠7$ |

Discussion

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.

Concept

Consider a pair of lines cut by a transversal. The pairs of interior angles with different vertices that lie on opposite sides of the transversal are called alternate interior angles.

In the diagram, two pairs of angles can be identified as alternate interior angles.

$Pair1:Pair2: ∠3and∠5∠4and∠6 $

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel. If | Then |
---|---|

$ℓ_{1}∥ℓ_{2}$ | $∠3≅∠5$ and $∠4≅∠6$ |

$∠3≅∠5$ or $∠4≅∠6$ | $ℓ_{1}∥ℓ_{2}$ |

Example

a Dylan really wants to makes art every minute of his free time. He finishes fixing the fence and runs to grab his canvas. It is just him and his canvas 🎨🖌️💗. He will continue painting a mosaic he began days ago. There are only eight regions left to be painted.

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b Dylan's mosaic reminded his parents of some of the paintings they saw in the Louvre museum in Paris. They took out their photo album and Dylan excitedly flipped through it. Dylan saw the photo of the Louvre's entrance. Immediately, he could identify some parallel lines.

External credits: Diego Lopez Sebastian

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a The transversal is the line that goes from the top edge to the bottom edge of the canvas. The alternate interior angles are the interior angles with different vertices that lie on opposite sides of the transversal. Remember that the angles must also be acute.

b When the lines are parallel, the alternate interior angles are congruent.

a Let's start by identifying which of the lines in the mosaic is the transversal. Due to the position of the regions, the transversal is the line that goes from the top edge to the bottom edge of the canvas. Let $ℓ_{1}$ and $ℓ_{2}$ be the lines that the transversal intersects.

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.$Regions1and8Regions3and5 $

Dylan only wants the alternate interior angles that are acute. The regions $1$ and $8$ correspond to acute angles. The regions $3$ and $5$ are obtuse angles. These can be verified with the help of a protractor. Dylan is now ready to paint the regions $1$ and $8$ blue.
Nicely done! Now Dylan has six regions remaining to paint.

b The line $t$ intersects the lines $ℓ_{1}$ and $ℓ_{2}.$ Therefore, it is a transversal. Now, ignore the photo and focus on the three lines to have a clear picture of the lines. Also, label some points on the lines.

$Alternate Interior Angles∠ABDand∠CAB $

Since the lines $ℓ_{1}$ and $ℓ_{2}$ are parallel, the mentioned angles are congruent. As a result, they have the same measure. That is enough information for the value of $x$ to be identified.
$∠ABD≅∠CAB⇓x=116 $

- It is the most visited art museum in the world.
- It is the largest museum in the world. It covers over $15$ acres.
- Originally, it was constructed as a fortress by King Phillip of France in $1190.$ In the $14th$ century, the Louvre was turned into a royal residence. Since $1793,$ it was opened to the public as
*Muséum central des arts de la République*. - From $1803$ to $1814$ it was named Napoleon Museum.
- The Mona Lisa was stolen from the Louvre in $1911.$ This incident rose the popularity of the painting across the globe. It was recovered two years later.
- During World War II, the Louvre was emptied to protect all of the art pieces.
- There are five pyramids in the Louvre.
- In $2016,$ the second Louvre of the world was opened. It is located in Abu Dhabi.

Discussion

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.

Concept

Consider a pair of lines cut by a transversal. The pairs of exterior angles with different vertices that lie on opposite sides of the transversal are called alternate exterior angles.

In the diagram, two pairs of angles can be identified as alternate exterior angles.

$Pair1:Pair2: ∠1and∠7∠2and∠8 $

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel. If | Then |
---|---|

$ℓ_{1}∥ℓ_{2}$ | $∠1≅∠7$ and $∠2≅∠8$ |

$∠1≅∠7$ or $∠2≅∠8$ | $ℓ_{1}∥ℓ_{2}$ |

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.

Example

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.

a Assume $∠BCH$ has a measure of $129_{∘}$ and $∠FEG$ has a measure of $3x_{∘}.$ What is the value of $x?$

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b Assume $m∠HDE=141_{∘}$ and $m∠GBA=3y_{∘}.$ What is the value of $y?$

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a Use the fact that the given angles are alternate exterior angles. Also, the streets are parallel to each other. Note that the names of the streets are shown in its original form of Spanish. The diagrams of the Solution also show translations to English. The word

callein Spanish means

streetin English.

b Notice that streets named Calle $42$ and Calle $43$ are parallel to each other. The involved angles are alternate exterior angles. Note that the names of the streets are shown in its original form of Spanish. The diagrams of the Solution also show translations to English. The word

callein Spanish means

streetin English.

a It is important to have a clear picture of the mentioned angles. Draw the angles on the map to get started.

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.$m∠BCH=m∠FEG$

SubstituteII

$m∠BCH=129$, $m∠FEG=3x$

$129=3x$

DivEqn

$LHS/3=RHS/3$

$43=x$

RearrangeEqn

Rearrange equation

$x=43$

b Again, it is important to have a clear vision of the angles drawn on the map.

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.Discussion

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.

Concept

Consider a pair of lines cut by a transversal. The pairs of interior angles with different vertices that lie on the same side of the transversal are called same-side interior angles or consecutive interior angles.
**co-interior angles**. In the diagram, two pairs of same-side interior angles can be identified.

Alternatively, same-side interior angles are called

$Pair1:Pair2: ∠3and∠6∠4and∠5 $

If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary. The same logic in reverse can be applied. If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel. If | Then |
---|---|

$ℓ_{1}∥ℓ_{2}$ | $m∠3+m∠6=180_{∘}$ and $m∠4+m∠5=180_{∘}$ |

$m∠3+m∠6=180_{∘}$ or $m∠4+m∠5=180_{∘}$ | $ℓ_{1}∥ℓ_{2}$ |

Example

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.

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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.$(4x+8)_{∘}+76_{∘}=180_{∘} $

Finally, solve the obtained equation to find the value of $x.$
- The Pyramids of Giza
- Hanging Gardens of Babylon
- Temple of Artemis
- Statue of Zeus
- Mausoleum at Halicarnassus
- Colossus of Rhodes
- Lighthouse of Alexandria

Discussion

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.

Concept

Consider a pair of lines cut by a transversal. The pairs of exterior angles with different vertices that lie on the same side of the transversal are called same-side exterior angles or **co-exterior angles**.

In the diagram, two pairs of same-side exterior angles can be identified.

$Pair1:Pair2: ∠1and∠8∠2and∠7 $

If two parallel lines are cut by a transversal, then the same-side exterior angles are supplementary. The same logic in reverse can be applied. If two lines and a transversal form same-side exterior angles that are supplementary, then the lines are parallel. If | Then |
---|---|

$ℓ_{1}∥ℓ_{2}$ | $m∠1+m∠8=180_{∘}$ and $m∠2+m∠7=180_{∘}$ |

$m∠1+m∠8=180_{∘}$ or $m∠2+m∠7=180_{∘}$ | $ℓ_{1}∥ℓ_{2}$ |

Example

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? |

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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$(8x−2)_{∘}+(3x−16)_{∘}=180_{∘} $

Solve the previous equation to determine the value of $x.$
$8x−2+3x−16=180$

AddSubTerms

Add and subtract terms

$11x−18=180$

AddEqn

$LHS+18=RHS+18$

$11x=198$

DivEqn

$LHS/11=RHS/11$

$x=18$

$The number of countries inAfrica is three timesx. $

Since $x=18,$ the number of countries in Africa is $three$ times $18.$ Performing this multiplication gives $3⋅18=54.$ There are $54$ countries in Africa.
Discussion

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.

Concept

Consider a pair of lines cut by a transversal. The pairs of angles with different vertices that lie on the same position relative to the vertex are called corresponding angles. In the diagram, $∠1$ and $∠5$ are corresponding angles because they are both northeast of their vertices.

Alternatively, two angles are corresponding angles if they meet the following conditions.

- One angle is exterior and the other is interior.
- The angles have different vertices.
- The angles lie on the same side of the transversal.

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 |
---|---|

$ℓ_{1}∥ℓ_{2}$ | $∠1≅∠5,$ $∠2≅∠6,$ $∠3≅∠7,$ and $∠4≅∠8$ |

$∠1≅∠5,$ $∠2≅∠6,$ $∠3≅∠7,$ or $∠4≅∠8$ | $ℓ_{1}∥ℓ_{2}$ |

Example

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.

a Dylan began to recognize things about the bridge. He notices how some of the diagonal beams of the bridge footings are parallel.

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b Next, Dylan checks out a photo of their sailboat. He notices something in the middle of the sails. There are two bars parallel to each other.

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a Draw the sides of the angles labeled. The vertical sides of each angle lie over the vertical beam. As a result of that, they lie on the same line. The vertical beam acts like a transversal. Notice that the angles are corresponding angles because they lie on the same position relative to their vertices.

b Both angles are northwest to their vertices. Note these angles are corresponding angles.

a Try to find a relationship about the given angles to find the measure of the lower angle. It is a good idea to begin by drawing the sides of each angle.

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.

The diagram shows that the angles lie on the same side of the transversal. One of them is an exterior angle while the other is an interior angle. This means that these angles are corresponding angles. The angles have the same measure because the lines are parallel.$(2x+6)_{∘}=(4x−28)_{∘} $

Next, solve the equation for $x.$
$2x+6=4x−28$

SubEqn

$LHS−2x=RHS−2x$

$6=2x−28$

AddEqn

$LHS+28=RHS+28$

$34=2x$

DivEqn

$LHS/2=RHS/2$

$17=x$

RearrangeEqn

Rearrange equation

$x=17$

$(4x−28)_{∘}$

Substitute

$x=17$

$(4⋅17−28)_{∘}$

Multiply

Multiply

$(68−28)_{∘}$

SubTerm

Subtract term

$40_{∘}$

b Start by drawing the lines containing the sides of the angles labeled.

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.

Those characteristics suggest that these two angles are corresponding angles. As a result, they have the same measure.$(6y+2)_{∘}=(7(y−1)−3)$