$$ a $$ is parallel to $$ b $$ Determine which of the statements is correct. α α α β β β γ γ γ δ δ δ a a a b b b

$$ \beta,\gamma $$ Colaterales$$ \alpha,\delta $$ Alternates

$$ \alpha,\beta $$ Colaterales$$ \gamma,\delta $$ Adjacent

$$ \alpha,\beta $$ Adjacent $$ \gamma,\delta $$ Alternates

a is parallel to b. Calculate the angles shown in the diagram. 115 115 115 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 a a a b b b

1, 3 , 5, 7 = 65 °; 2, 4 , 6 = 115 °

1, 3, 4, 6 = 65 °; 2, 5, 7 = 115 °

1, 3 = 65 °; 2, 4, 6 = 115 °; 5, 7 = 75 °

a.b parallel. Find the angles marked α α α 104 104 104 81 81 81 β β β a a a b b b

$$ \alpha=104 $$ $$ \beta=81 $$

$$ \alpha=104 $$ $$ \beta=104 $$

Line a is parallel to line b. What is the relationship between the angles? a a a b b b α β γ δ

$$ \alpha,\beta $$ Vertices; $$ \gamma,\delta $$ Adjacent

$$ \alpha,\beta $$ Vertices; $$ \gamma,\delta $$ co-interior

$$ \alpha,\delta $$ co-interior; $$ \beta,\gamma $$ Adjacent

$$ \alpha,\beta $$ Vertices; $$ \gamma,\delta $$ Adjacent

$$ \alpha,\beta $$ Adjacent; $$ \gamma,\delta $$ Vertices

Look at the angles in the figure. What is their relationship? α β γ δ

$$ \alpha,\delta $$ are adjacents that add up to 180; $$ \beta,\delta $$ are opposite at the vertex and equal.

$$ \alpha,\beta $$ are opposite at the vertex and add up to 180; $$ \gamma,\delta $$ are adjacent and equal.

$$ \alpha,\delta $$ are adjacents that add up to 180; $$ \beta,\delta $$ are opposite at the vertex and equal.

$$ \beta,\gamma $$ are adjacent and equal; $$ \beta,\delta $$ are opposite at the vertex and equal.

$$ \gamma,\delta $$ are adjacents that add up to 180; $$ \alpha,\delta $$ are adjacent and equal.

a is parallel to b. Which of the angles in the drawing are adjacent to each other? a a a b b b α1 α2 β1 β2 γ1 γ2 δ1 δ2

$$ \alpha1,\delta1 $$ | $$ \beta2,\gamma2 $$

$$ \alpha1,\beta1 $$ | $$ \alpha1,\delta1 $$

$$ \alpha1,\delta1 $$ | $$ \beta2,\gamma2 $$

$$ \beta1,\delta2 $$ | $$ \alpha1,\gamma2 $$

$$ \delta1,\delta2 $$ | $$ \beta1,\gamma1 $$

Look at the straight lines and angles in the diagram. Which angles are adjacent? β3 α3 α1 β1 δ1 γ1 γ3 δ3 α2 γ2

$$ \alpha1,\delta1 $$ $$ \delta1,\beta1 $$ $$ \beta1,\gamma1 $$ $$ \alpha1,\gamma1 $$ $$ \alpha2,\gamma2 $$ $$ \gamma3,\delta3 $$

$$ \gamma1,\delta1 $$ $$ \alpha1,\beta1 $$ $$ \alpha3,\beta3 $$

$$ \alpha2,\beta1 $$ $$ \beta1,\gamma1 $$ $$ \alpha1,\beta3 $$ $$ \alpha3,\gamma3 $$

$$ \alpha2,\delta3 $$ $$ \alpha2,\delta1 $$ $$ \delta1,\beta3 $$ $$ \gamma2,\gamma3 $$ $$ \gamma1,\beta3 $$

Sides, Vertices, and Angles

A side is the straight line that lies between two points called vertices. An angle is formed between two lines.

A vertex is the point of origin where two or more straight lines meet, thus creating an angle.

An angle is created when two lines originate from the same vertex.

To clearly illustrate these concepts, we will represent them in the following drawing:

Detailed explanation

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Test yourself on angles!

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

Practice more now

Exercises on Sides, Vertices, and Angles

Exercise 1

Assignment

Given the angles between parallel lines:

What is the value of: $X$ ?

Solution

We will mark the angle adjacent to the angle equal to $94^o$ with the letter $Z$ and find its value through the following calculation:

$Z=180-94=86$

Now we will focus on the triangle to find $X$ and remember that the sum of the angles in a triangle is equal to: $180^o$

$X+86+53=180$

$X+139=180$

$X=180-139$

$X=41$

Answer

$41^o$

Exercise 2

Assignment

At the vertices of a square with a side length of $Y$ cm, $4$ squares each with a side length of $X$ cm are drawn

What is the area of the entire shape?

Solution

The area of the entire shape is composed of the area of $4$ small squares and the area of one large square.

Let's calculate the area of a small square

$x\times x=x^2$

Therefore, the area of $4$ squares will be equal to: $4x^2$

The area of the large square is equal to: $y\times y=y^2$

Thus, the total area of the shape will be equal to: $4x^2+y^2$

Answer

$4x^2+y^2$

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Test your knowledge

Question 1

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

Question 2

$$ a $$ is parallel to

$$ b $$

Determine which of the statements is correct.

Question 3

Lines a and b are parallel.

Which of the following angles are co-interior?

Exercise 3

Prompt

Given that $A,B$ are two vertices in a rectangle.

How many rectangles can be drawn so that $A,B$ are adjacent vertices?

Solution

Answer:

$4$

Exercise 4

Assignment

Given that $B,D$ are two bisectors in a rectangle.

How many rectangles can be drawn so that $BD$ is a diagonal in them?

Answer

$3$

Examples and Exercises with Solutions on Sides, Vertices, and Angles

Exercise #1

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they equal 180 degrees:

$30+60+90=180$
The sum of the angles equals 180, so they can form a triangle.

Answer

Yes

Exercise #2

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

$56+89+17=162$

The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Exercise #3

$a$ is parallel to

$b$

Determine which of the statements is correct.

Video Solution

Step-by-Step Solution

Let's review the definition of adjacent angles:

Adjacent angles are angles formed where there are two straight lines that intersect. These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Now let's review the definition of collateral angles:

Two angles formed when two or more parallel lines are intersected by a third line. The collateral angles are on the same side of the intersecting line and even are at different heights in relation to the parallel line to which they are adjacent.

Therefore, answer C is correct for this definition.

Answer

$\beta,\gamma$ Colaterales $\gamma,\delta$ Adjacent

Exercise #4

Lines a and b are parallel.

Which of the following angles are co-interior?

Video Solution

Step-by-Step Solution

Let's remember the definition of consecutive angles:

Consecutive angles are, in fact, a pair of angles that can be found on the same side of a straight line when this line crosses a pair of parallel straight lines.

These angles are on opposite levels with respect to the parallel line to which they belong.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the angles $\beta+\gamma=180$

are consecutive.

Answer

$\beta,\gamma$

Exercise #5

Which angles in the drawing are co-interior given that a is parallel to b?

Video Solution

Step-by-Step Solution

Given that line a is parallel to line b, the angles $\alpha_2,\beta_1$ are equal according to the definition of corresponding angles.

Also, the angles $\alpha_1,\gamma_1$ are equal according to the definition of corresponding angles.

Now let's remember the definition of collateral angles:

Collateral angles are actually a pair of angles that can be found on the same side of a line when it crosses a pair of parallel lines.

These angles are on opposite levels with respect to the parallel line they belong to.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the angles

γ1+γ2=180

are the collateral angles

Answer

$\gamma1,\gamma2$

Do you know what the answer is?

Question 1

Lines a and b are parallel.

Which of the following angles are co-interior?

Question 2

Which angles in the drawing are co-interior given that a is parallel to b?

Question 3

a is parallel to b.

Calculate the angles shown in the diagram.

How to Get Ready Quickly for a Surprise Exam?

The answer is quite simple.
Many students fear pop quizzes, but in reality, they are an opportunity to exercise and demonstrate your knowledge.
As long as you study throughout the year and not just before exams.

Knowing there will be a quiz usually motivates you to do your homework.
Avoid falling behind with the study material, and stay up-to-date with the latest classes.
Quizzes often test your knowledge on just one topic. For example: calculating the area of a trapezoid.
Quizzes are calculated into an annual average, so it's in your best interest to obtain the best possible grade on each one.

As long as you pay attention in class and do your homework, you have no reason to fear exams.

How to Realize We're Falling Behind with the Study Material?

Is there an area of geometry that you don't understand? That's normal, as there are topics you'll learn easily, and others that will be more challenging for you.

Important: don't fall behind with the study material, because in mathematics, the pace of learning is very fast.
The problem is that many topics are based on what was taught before. Therefore, the moment your understanding of a certain topic is partial, you will struggle to grasp the next topic.
How do you know if you've fallen behind with the study material?

You find it difficult to concentrate in class because you struggle to understand the teacher.
You have difficulty solving homework assignments.

You received a very low grade on a test, which reflects your level.

What can you do in this case?

You can ask a classmate to explain what you don't understand.
Ask your math teacher for help with the topic you haven't understood.
You can take lessons with a private tutor to explain the topic you haven't understood, from the beginning.

Study mathematics with a private tutor

There are students who struggle to keep up with the learning pace in class.
It's important to understand that the ability to quickly learn what is taught is not necessarily related to the student's ability to understand different topics taught, and even to pass exams with good grades.
Sometimes math teachers teach very quickly to cover all the topics of the annual program. This way, there are students who fail to properly understand the different explanations and formulas, and gradually fall behind.

With a private math tutor, you can not only learn all the topics you haven't understood, but also assimilate the material effectively.
A private tutor can help you pass high school exams, and of course, prepare you for college.
It is also possible to take classes with a private tutor through your computer, with our online study program.
This way, you can enjoy private lessons with high-level teachers, without leaving your home.

This platform offers a wide variety of private tutors. You can read different opinions and comments about each teacher.
This means that you can quickly get an idea about the profile of each teacher, and thus you can easily choose the tutor who will accompany you in the learning process.

Check your understanding

Question 1

Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.

Can these angles form a triangle?

Question 2

a.b parallel. Find the angles marked

Question 3

Look at the angles shown in the figure below.

What is their relationship?

$$

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