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

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

Calculate the missing angle: 120 120 120

It is not possible to calculate.

$$ a $$ 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 Practice Problems: Sides, Vertices & Geometry

Understanding Angles

Complete explanation with examples

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

Practice Angles

Test your knowledge with 51 quizzes

What is the size of the missing angle?

Examples with solutions for Angles

Step-by-step solutions included

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:

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

Video Solution

Frequently Asked Questions

Everything you need to know about Angles

How do you find missing angles in triangles?

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Use the triangle angle sum theorem: all angles in a triangle add up to 180°. If you know two angles, subtract their sum from 180° to find the third angle. For example, if two angles are 53° and 86°, the third angle is 180° - 53° - 86° = 41°.

What is the relationship between sides, vertices, and angles in polygons?

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In any polygon, the number of sides equals the number of vertices and angles. A triangle has 3 sides, 3 vertices, and 3 angles. A square has 4 of each. This relationship helps identify and classify different geometric shapes.

How do you calculate angles with parallel lines?

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When parallel lines are cut by a transversal, several angle relationships form: 1) Corresponding angles are equal, 2) Alternate interior angles are equal, 3) Same-side interior angles are supplementary (add to 180°). Use these properties to find unknown angles.

What are the different types of angles and their measures?

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Angles are classified by their measures: Acute angles are less than 90°, right angles equal exactly 90°, obtuse angles are greater than 90° but less than 180°, and straight angles equal 180°. These classifications help solve geometry problems.

How do you identify vertices in geometric shapes?

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A vertex is where two or more lines meet, forming a corner. To identify vertices: 1) Look for intersection points of sides, 2) Count the corners of the shape, 3) Remember that each vertex creates an angle. Triangles have 3 vertices, squares have 4.

What is the difference between adjacent and opposite angles?

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Adjacent angles share a common vertex and side but don't overlap. They often form linear pairs that add to 180°. Opposite angles (vertical angles) are across from each other when two lines intersect and are always equal in measure.

How do you solve angle problems step by step?

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Follow these steps: 1) Identify what type of angle relationship exists, 2) Write an equation using angle properties (sum = 180° for triangles), 3) Substitute known values, 4) Solve for the unknown angle, 5) Check your answer makes geometric sense.

Why do triangle angles always sum to 180 degrees?

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This is a fundamental geometric theorem. When you draw any triangle and extend one side, the exterior angle equals the sum of the two non-adjacent interior angles. This relationship proves that all three interior angles must total 180°, making it a reliable problem-solving tool.

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Angles Practice Problems: Sides, Vertices & Geometry

Understanding Angles

Understanding Sides, Vertices, and Angles in Geometry

Definitions:

Side

Vertex

Angle

Practice Angles

Examples with solutions for Angles

Frequently Asked Questions

How do you find missing angles in triangles?

What is the relationship between sides, vertices, and angles in polygons?

How do you calculate angles with parallel lines?

What are the different types of angles and their measures?

How do you identify vertices in geometric shapes?

What is the difference between adjacent and opposite angles?

How do you solve angle problems step by step?

Why do triangle angles always sum to 180 degrees?

More Angles Questions

Sum and Difference of Angles

Angles

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