$$ a $$ 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, 4, 6 = 65 °; 2, 5, 7 = 115 °

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

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

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

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

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

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!

What is the size of the unlabelled angle?

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Understanding Sides, Vertices, and Angles in Geometry

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 segment that connects two adjacent vertices of a polygon. 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 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 formed where two sides of a polygon meet at a vertex. The measure of an angle indicates the degree of rotation between the two sides. Angles can be acute (less than 90°), right (90°), obtuse (greater than 90°), or straight (180°).

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

$41^o$

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Test your knowledge

Question 1

What is the size of the missing angle?

Question 2

What is the size of the missing angle?

Question 3

$$ a $$ is parallel to

$$ b $$

Determine which of the statements is correct.

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

$4x^2+y^2$

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:

$4$

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

Lines a and b are parallel.

Which of the following angles are co-interior?

Question 3

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

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

What is the size of the missing angle?

Video Solution

Step-by-Step Solution

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is $180^\circ$ . Given that one angle is $80^\circ$ , we can calculate the missing angle using the following steps:

Step 1: Recognize that the given angle $\alpha = 80^\circ$ and the missing angle $\beta$ form a straight line.
Step 2: Use the angle sum property for a straight line: $\alpha + \beta = 180^\circ$
Step 3: Substitute the known value: $80^\circ + \beta = 180^\circ$
Step 4: Solve for the missing angle $\beta$ : $\beta = 180^\circ - 80^\circ = 100^\circ$

Therefore, the size of the missing angle is $100^\circ$ .

Answer

100°

Exercise #2

$a$ 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 #3

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 #4

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

$\gamma1,\gamma2$

Exercise #5

a is parallel to b.

Calculate the angles shown in the diagram.

Video Solution

Step-by-Step Solution

Given that according to the definition, the vertex angles are equal to each other, it can be argued that:

$115=2$ Now we can calculate the second pair of vertex angles in the same circle:

$1=3$

Since the sum of a plane angle is 180 degrees, angle 1 and angle 3 are complementary to 180 degrees and equal to 65 degrees.

We now notice that between the parallel lines there are corresponding and equal angles, and they are:

$115=4$

Since angle 4 is opposite to angle 6, it is equal to it and also equal to 65 degrees.

Another pair of alternate angles are angle 1 and angle 5.

We have proven that: $1=3=65$

Therefore, angle 5 is also equal to 65 degrees.

Since angle 7 is opposite to angle 5, it is equal to it and also equal to 115 degrees.

That is:

$115=2=4=6$

$65=1=3=5=7$

Answer

1, 3 , 5, 7 = 65°; 2, 4 , 6 = 115°

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.

Check your understanding

Question 1

a is parallel to b.

Calculate the angles shown in the diagram.

Question 2

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

Question 3

Look at the angles shown in the figure below.

What is their relationship?

$$

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.

Do you think you will be able to solve it?

Question 1

Line a is parallel to line b.

What is the relationship between the angles?

Question 2

Look at the angles in the figure.

What is their relationship?

Question 3

a is parallel to b.

Which of the angles in the drawing are adjacent to each other?

Start practice

Sides, Vertices, and Angles

Understanding Sides, Vertices, and Angles in Geometry

Definitions:

Side

Vertex

Angle

Test yourself on angles!

Understanding Sides, Vertices, and Angles in Geometry

Definitions:

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

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

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