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

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

$$ a $$ 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

$$ \alpha,\beta $$ 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 , 5, 7 = 65 °; 2, 4 , 6 = 115 °

1, 3, 4, 6 = 65 °; 2, 5, 7 = 115 °

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

Angles - Examples, Exercises and Solutions

Understanding Angles

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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:

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Indicates which angle is greater

Examples with solutions for Angles

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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:

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:

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

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Angles - Examples, Exercises and Solutions

Understanding Angles

Understanding Sides, Vertices, and Angles in Geometry

Definitions:

Side

Vertex

Angle

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