# Sides, Vertices, and Angles

🏆Practice 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:

## Test yourself on angles!

The sum of the adjacent angles is 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: $180^o$

$X+86+53=180$

$X+139=180$

$X=180-139$

$X=41$

$41^o$

### 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: $4x^2+y^2$

$4x^2+y^2$

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### 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

$4$

### 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?

$3$

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

### Exercise #1

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.

Alternate

### Exercise #2

a is parallel to b.

Calculate the angles shown in the diagram.

### 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:

$115=2=4=6$

$65=1=3=5=7$

1, 3 , 5, 7 = 65°; 2, 4 , 6 = 115°

### Exercise #3

The lines a and b are parallel.

What are the corresponding angles?

### Step-by-Step Solution

Given that line a is parallel to line b, let's remember the definition of corresponding angles between parallel lines:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

Corresponding angles are equal in size.

According to this definition $\alpha=\beta$and therefore the corresponding angles

$\alpha,\beta$

### Exercise #4

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

### 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

$\gamma1,\gamma2$

### Exercise #5

Lines a and b are parallel.

Which of the following angles are co-interior?

### 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.

$\beta,\gamma$

Do you know what the answer is?

## How to Get Ready Quickly for a Surprise Exam?

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.

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.
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