Angles - Examples, Exercises and Solutions

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

Can these angles form a triangle?

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

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

Can these angles make a triangle?

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.

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

Can these angles form a triangle?

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.

In a right triangle, the sum of the two non-right angles is...?

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

90 degrees

What is the value of the void angle?

The empty angle is an angle adjacent to 160 degrees.

Remember that the sum of adjacent angles is 180 degrees.

Therefore, the value of the empty angle will be:

$180-160=20$

20

Calculate the size of angle X given that the triangle is equilateral.

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

60

Which type of angles are shown in the figure below?

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

What angles are described in the drawing?

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

Vertices

$a$ is parallel to

$b$

Determine which of the statements is correct.

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

Indicates which angle is greater

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Indicates which angle is greater

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Indicates which angle is greater

Note that in drawing A, the angle is more acute, meaning it's smaller:

While in drawing B, the angle is more obtuse, meaning it's larger:

Indicates which angle is greater

Note that in drawing A, the angle is a straight angle equal to 180 degrees:

While in drawing B, we are given a right angle, equal to 90 degrees:

Therefore, the angle in drawing A is larger.

Indicates which angle is greater

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Can a triangle have more than one obtuse angle?

If we try to draw two obtuse angles and connect them to form a triangle (i.e., only 3 sides), we will see that it is not possible.

Therefore, the answer is no.

No