Determine the value of the α- and -β- angles shown in the below diagram: α α α 104 104 104 81 81 81 β β β a a a b b b

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

Calculate the missing angle: 120 120 120

It is not possible to calculate.

Calculate the missing angle: 120 120 120

It is not possible to calculate.

lines a and b are parallel. 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

$$ \alpha,\beta $$ are verticals, while $$ \gamma,\delta $$ are adjacent.

$$ \alpha,\beta $$ are co-interior, while $$ \gamma,\delta $$ are corresponding.

$$ \alpha,\beta $$ are verticals, while $$ \gamma,\delta $$ are adjacent.

$$ \alpha,\beta $$ are corresponding, while $$ \gamma,\delta $$ are co-interior.

$$ \beta,\delta $$ are corresponding, while $$ \alpha,\gamma $$ are co-interior.

$$ a $$ 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$$ \gamma,\delta $$ Adjacent

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

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

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

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

Sides, Vertices, and Angles

Triangle

Understanding Sides, Vertices, and Angles in Geometry

In geometry, shapes are defined by three key components: sides, vertices, and angles. These elements work together to form polygons and other figures, helping us understand their properties and relationships.

The number of sides in a polygon equals the number of vertices and angles. For example, a hexagon has six sides, six vertices, and six angles.

Definitions:

Side

A side is the straight line that lies between two points called vertices. An angle is formed between two lines. Sides form the edges of a polygon. For example, a triangle has three sides, while a square has four. The length and arrangement of sides determine the size and shape of a figure.

Vertex

A vertex is the point of origin where two or more straight lines meet, thus creating an angle. These vertices are often referred to as the "corners" of a shape. A triangle has three vertices, a square has four, and a pentagon has five.

Angle

An angle is created when two lines originate from the same vertex. The measure of an angle indicates the degree of rotation between the two sides. Angles can be acute (less than $90^\circ$ ), right ( $90^\circ$ ), obtuse (greater than $90^\circ$ ), or straight ( $180^\circ$ ).

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

Detailed explanation

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Test yourself on angles!

Identify the angle shown in the figure below?

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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

Identify the angles shown in the diagram below?

Question 2

Which type of angles are shown in the figure below?

Question 3

Which type of angles are shown in the diagram?

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

Identify the angle shown in the figure below?

Step-by-Step Solution

Remember that adjacent angles are angles that are formed when two lines intersect one another.

These angles are created at the point of intersection, one adjacent to the other, and that's where their name comes from.

Adjacent angles always complement one another to one hundred and eighty degrees, meaning their sum is 180 degrees.

Answer

Adjacent

Exercise #2

Identify the angles shown in the diagram below?

Step-by-Step Solution

Let's remember that vertical angles are angles that are formed when two lines intersect. They are are created at the point of intersection and are opposite each other.

Answer

Vertical

Exercise #3

Which type of angles are shown in the figure below?

Step-by-Step Solution

Alternate angles are a pair of angles that can be found on the opposite side of a line that cuts two parallel lines.

Furthermore, these angles are located on the opposite level of the corresponding line that they belong to.

Answer

Alternate

Exercise #4

Which type of angles are shown in the diagram?

Step-by-Step Solution

First let's remember that corresponding angles can be defined as a pair of angles that can be found on the same side of a transversal line that intersects two parallel lines.

Additionally, these angles are positioned at the same level relative to the parallel line to which they belong.

Answer

Corresponding

Exercise #5

Determine the value of the α-and-β- angles shown in the below diagram:

Video Solution

Step-by-Step Solution

In the question, we can observe that there are two pairs of parallel lines, lines a and b.

When a line crosses two parallel lines, different angles are formed

Angles alpha and the given angle of 104 are on different sides of the transversal line, but both are in the interior region between the two parallel lines,

This means they are alternate angles, and alternate angles are equal.

Therefore,

Angle beta and the second given angle of 81 degrees are both on the same side of the transversal line, but each is in a different position relative to the parallel lines, one in the exterior region and one in the interior. Therefore, we can conclude that these are corresponding angles, and corresponding angles are equal.

Therefore,

Answer

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

Do you know what the answer is?

Question 1

Determine the value of the α-and-β- angles shown in the below diagram:

Question 2

Calculate the missing angle:

Question 3

Calculate the missing angle:

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

Calculate the size of the missing angle given that lines a and b are parallel.

Question 2

Which angles are marked in the figure?

Question 3

The lines a and b are parallel.

What are the corresponding angles?

Start practice

Sides, Vertices, and Angles

Understanding Sides, Vertices, and Angles in Geometry

Definitions:

Side

Vertex

Angle

Test yourself on angles!

Exercises on Sides, Vertices, and Angles

Exercise 1

Exercise 2

Exercise 3

Exercise 4

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

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

How to Get Ready Quickly for a Surprise Exam?

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

More Questions

Sum and Difference of Angles

Angles