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:

## Practice Angles

### Exercise #1

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

Can these angles make a triangle?

### 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.

No.

### Exercise #2

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

Can these angles form a triangle?

### 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.

Yes

### Exercise #3

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

Can these angles form a triangle?

### Step-by-Step Solution

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

$90+115+35=240$
The sum of the given angles is not equal to 180, so they cannot form a triangle.

No.

### Exercise #4

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 #5

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 #1

$a$ is parallel to

$b$

Determine which of the statements is correct.

### 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.

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

### Exercise #2

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.

$\beta,\gamma$

### Exercise #3

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 #4

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 #5

What angles are described in the drawing?

### Step-by-Step Solution

Since the angles are not on parallel lines, none of the answers are correct.

Ninguna de las respuestas

### Exercise #1

What is the value of X given that the angles are between parallel lines?

### Step-by-Step Solution

The angle X given to us in the drawing corresponds to an angle that is adjacent to an angle equal to 154 degrees. Therefore, we will mark it with an X

Now we can calculate:

$x+154=180$

$x=180-154=26$

26°

### Exercise #2

Look at the parallelogram in the diagram. Calculate the angles indicated.

### Step-by-Step Solution

$a$is an alternate angle to the angle that equals 30 degrees. That means$\alpha=30$Now we can calculate: $\beta$

As they are adjacent and theredore complementary angles to 180:

$180-30=150$

Angle$\gamma$Is on one side with an angle of 20, which means:

$\gamma=20$

$\alpha=30$ $\beta=150$ $\gamma=20$

### Exercise #3

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

### Step-by-Step Solution

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

$180-50=130$

$130:2=65$

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65

$65$°

### Exercise #4

What kind of triangle is shown in the diagram below?

### Step-by-Step Solution

We calculate the sum of the angles of the triangle:

$117+53+21=191$

It seems that the sum of the angles of the triangle is not equal to 180°,

Therefore, the figure can not be a triangle and the drawing is incorrect.

The triangle is incorrect.

### Exercise #5

Given the parallelogram.

What are alternate angles?

### Step-by-Step Solution

To solve the question, first we must remember that the property of a parallelogram is that it has two pairs of opposite sides that are parallel and equal.

That is, the top line is parallel to the bottom one.

From this, it is easy to identify that angle X is actually an alternate angle of angle δ, since both are on different sides of parallel straight lines.

$\delta,\chi$