
Here you can find geometry lessons that will help you understand by explaining in detail and clarifying with some examples.


Geometry

# Basic Geometric Terms: Point, Line, Ray, Line Segment

Geometry is a branch of mathematics, just as numbers are the basic building block of algebra, so geometry has its basic building block, which is the point.

The point is the basic component of the line segment, ray, line and all geometric shapes.

In this lesson, we will try to define the concept of line segment, line, and ray, with mention of their mathematical symbols.

We will also see the relative positions of two lines in the plane: parallelism, perpendicularity and intersection.

## Point

The point is the basic structural unit in geometry, and it has **no dimensions**, and we often represent it as a single point with the tip of a pen or a circular point using a computer.

<Image src="/imgs/lessons/geometry/point_a.png" alt="plotting a point in a plane" width='318px' height='182px'/>

## Line

A Line is an infinite set of compact collinear points, and it has **one dimension**.

Geometrically, we take two different points on this line so that both sides are not bound by these two points.

<Image src="/imgs/lessons/geometry/line_ab.png" alt="plotting a line in a plane" width='318px' height='182px'/>

In the above figure, the line is denoted by the symbol: <Tex>\overleftrightarrow{AB}</Tex> or <Tex>\overleftrightarrow{BA}</Tex>

<Alert type="infos">
In some countries, the Line is denoted by the symbole <Tex>(AB)</Tex> or <Tex>(BA)</Tex>.

</Alert>

## Ray

The Ray is part of the line, with one endpoint, and the other side is unfinished, so it starts at one point and goes on forever in some direction. And it has one dimension.

Geometrically, we take two different points so that the first point is the endpoint and the second point is on the infinite side.

<Image src="/imgs/lessons/geometry/ray_ab.png" alt="plotting a ray in a plane" width='318px' height='182px'/>

In the above figure, the ray is denoted by the symbol: <Tex>\overrightarrow{AB}</Tex>

<Alert type="infos">

In some countries, the ray is denoted by the symbole <Tex>[AB)</Tex>.

</Alert>

## Line Segment

The line segment is a part of the Line so that it have two endpoints, that is, its length is finite. And it has one dimension.

Geometrically, we take two different points so that the first point is on the first endpoint and the second point is on the second endpoint.

<Image src="/imgs/lessons/geometry/segment_ab.png" alt="plotting a line segment in a plane" width='318px' height='182px'/>

In the above figure, the line segment is denoted by the symbol: <Tex>\overline{AB}</Tex> or <Tex>\overline{BA}</Tex>

<Alert type="infos">
In some countries, the Line is denoted by the symbole <Tex>[AB]</Tex> or <Tex>[BA]</Tex>.

</Alert>

## Plane

A plane is a flat surface such that if we take two points from this plane, we always get a straight line belonging to this plane.

<Image src="/imgs/lessons/geometry/plane_p.png" alt="the plane" width='318px' height='182px'/>

In the above figure, the plane is denoted by the symbol: <Tex>\textit{\textbf{(P)}}</Tex>

## Parallel Lines

We say that two lines are parallel if they have no common point, that never intersect.

The following Lines are parallel:

<Image src="/imgs/lessons/geometry/parallel_lines.png" alt="plotting a parallel lines in a plane" width='318px' height='182px'/>

In the above figure, the parallel lines denoted by the symbol: <Tex>(AB)\parallel{(CD)}</Tex> or <Tex>(CD)\parallel{(AB)}</Tex>

## Intersecting Lines

We say that two lines intersect if they share only one point.

The following Lines are intersect at point I:

<Image src="/imgs/lessons/geometry/intersect_lines.png" alt="plotting a intersect lines in a plane" width='318px' height='182px'/>

## Perpendicular Lines

We say that two lines are perpendicular if they intersect and form a right angle at their point of intersection.

The following Lines are perpendicular:

<Image src="/imgs/lessons/geometry/perp_lines.png" alt="plotting a perpendicular lines in a plane" width='318px' height='182px'/>

In the above figure, the intersecting lines denoted by the symbol: <Tex>(AB)\perp{(CD)}</Tex> or <Tex>(CD)\perp{(AB)}</Tex>

## Collinear Points

We say that three or more points are collinear if they belong to the same line.

In the example below, the points A, B and C are collinear:

<Image src="/imgs/lessons/geometry/collinear_points.png" alt="plotting a collinrear points in a plane" width='318px' height='182px'/>

But, in the example below, the points A, B and C are non-collinear:

<Image src="/imgs/lessons/geometry/noncoll_points.png" alt="plotting a non-collinrear points in a plane" width='318px' height='182px'/>

basic Geometric Terms are Point, Straight Line, Ray and Line Segment with the parallelism and perpendicularity of two Lines.

Basic Geometric Terms


# Types of Angles

From the title it comes to your mind that there are different kinds of angles, right?

Yes, there are many types of angles that we will learn about in this lesson with the definition and measurement of each type of angle.

But what is the angle? What is it made of? An angle is two rays (legs) that have the same endpoint. This common endpoint is called the **vertex**, and the space between the two legs is the measure of the angle. The unit of measure for angles is the **degree** (there are other units).

In this lesson you will find each angle with its name, description and the geometric representation.

## Acute Angle

Each angle that measure is less than 90°.

<Image src="/imgs/lessons/geometry/acute_angle.png" alt="Acute Angle" width='318px' height='182px'/>

## Right Angle
An angle that measure is exactly 90°.

<Image src="/imgs/lessons/geometry/right_angle.png" alt="Right Angle" width='318px' height='182px'/>

## Obtuse Angle

Each angle that measure is greater than 90° and less than 180°.

<Image src="/imgs/lessons/geometry/obtuse_angle.png" alt="Obtuse Angle" width='318px' height='182px'/>

## Straight Angle

An angle that measure is exactly 180°.

<Image src="/imgs/lessons/geometry/straight_angle.png" alt="Straight Angle" width='318px' height='182px'/>

## Reflex Angle

Each angle that measure is greater than 180° and less than 360°.

<Image src="/imgs/lessons/geometry/reflex_angle.png" alt="Reflex Angle" width='318px' height='182px'/>

## Full Angle

An angle that measure is exactly 360°.

<Image src="/imgs/lessons/geometry/full_angle.png" alt="Full Angle" width='318px' height='182px'/>

------------

## Complementary Angles

Two angles that add up 90°.

<Image src="/imgs/lessons/geometry/complem_angles.png" alt="Complementary Angles" width='312px' height='210px'/>

## Supplementary Angles

Two angles that add up 180°.

<Image src="/imgs/lessons/geometry/supplem_angles.png" alt="Supplementary Angles" width='312px' height='210px'/>

## Adjacent Angles

Angles with a common sdie and a common vertex.

The angles <Tex>\angle{1}</Tex> and <Tex>\angle{2}</Tex> are adjacent angles.

<Image src="/imgs/lessons/geometry/adj_angles.png" alt="Adjacent Angles" width='312px' height='210px'/>

## Opposite Angles

Angles that have a common vertex and whose arms are formed by the same lines. Opposite angles are equal

The angles <Tex>\angle{1}</Tex> and <Tex>\angle{2}</Tex> are opposite angles, therefore: <Tex>\angle{1}=\angle{2}</Tex>

The angles <Tex>\angle{3}</Tex> and <Tex>\angle{4}</Tex> are opposite angles, therefore: <Tex>\angle{3}=\angle{4}</Tex>

<Image src="/imgs/lessons/geometry/opp_angles.png" alt="Opposite Angles" width='312px' height='210px'/>

<Related>

<Link to="../geometry/angle-bisector" as="/lessons/geometry/angle-bisector">Angle Bisector</Link>

</Related>

This lesson show you the types of angles (6 types) and the  Complementary, Supplementary, Adjacent and Opposite angles.

Types of Angles


# Angle Bisector

<Related>

<Link to="../geometry/types-of-angles" as="/lessons/geometry/types-of-angles">Types of Angles</Link>

</Related>

Angle Bisector is a ray passing through the vertex of the angle and it divides the angle into two congruent angles.

**Properties of Angle Bisector:**

- Each point belongs to the bisector and is located at the same distance between the two sides of the angle;
- The two angles obtained are congruent and adjacent;

In the following figure, we have the angle <Tex>\angle{ABC}</Tex> and its bisector <Tex>\overrightarrow{BE}</Tex>.

<Image src="/imgs/lessons/geometry/angle_bisc.png" alt="Angle Bisector" width='382px' height='264px'/>

## How to draw an Angle Bisector

To draw the angle bisector of the angle <Tex>\angle{ABC}</Tex> using compass, we follow the following steps:

**Step1:**

We draw two arcs of a circle with center <Tex>B</Tex> (the vertex of <Tex>\angle{ABC}</Tex>) and radius <Tex>R_1</Tex> so that the first arc intersects with the leg <Tex>\overrightarrow{BA}</Tex> and the second arc intersects with the leg <Tex>\overrightarrow{BC}</Tex>.

<Image src="/imgs/lessons/geometry/angle_step1.png" alt="Angle Bisector" width='382px' height='364px'/>

**Step2:**

Draw arcs of two circles with radius <Tex>R_2</Tex>:

- The center of the first circle is the intersection of the first arc with the leg <Tex>\overrightarrow{BA}</Tex> we obtained in step 1;
- The center of the second circle is the intersection of the second arc with the leg <Tex>\overrightarrow{BC}</Tex> we obtained in step 1;
<Image src="/imgs/lessons/geometry/angle_step2.png" alt="Angle Bisector" width='382px' height='264px'/>

**Step3:**

We connect by a ray the vertex <Tex>B</Tex> of the angle <Tex>\angle{ABC}</Tex> with the intersection of the arcs obtained in step 2.

<Image src="/imgs/lessons/geometry/angle_bisc.png" alt="Angle Bisector" width='382px' height='264px'/>

## Angles Bisectors of a Triangle

We saw that each angle had an angle bisector. So a triangle that has three angles can have three angle bisectors.

Following the same steps above, we draw the angle bisectors of triangle <Tex>ABC</Tex> below:

<Image src="/imgs/lessons/geometry/tr_abc.png" alt="Angle Bisectors of triangle" width='419px' height='307px'/>

- The angle bisector of <Tex>\angle{BAC}</Tex>
<Image src="/imgs/lessons/geometry/angle_bisc01.png" alt="Angle Bisectors of triangle" width='487px' height='442px'/>
- The angle bisector of <Tex>\angle{ABC}</Tex>
<Image src="/imgs/lessons/geometry/angle_bisc02.png" alt="Angle Bisectors of triangle" width='536px' height='440px'/>
- The angle bisector of <Tex>\angle{ACB}</Tex>
<Image src="/imgs/lessons/geometry/angle_bisc03.png" alt="Angle Bisectors of triangle" width='506px' height='417px'/>
The three angle bisectors intersect at a one point, this point being the center of a **circle inscribed** with triangle <Tex>ABC</Tex>.

<Image src="/imgs/lessons/geometry/angle_bisc04.png" alt="Angle Bisectors of triangle and inscribed circle" width='520px' height='385px'/>

Angle Bisector is a ray passing through the vertex of the angle and it divides the angle into two congruent angles. The triangle have three angle bisectors.

Angle Bisector


# Perpendicular Bisector: Bisector of a Line Segment

Perpendicular Bisector is a straight line passing through the midpoint of the line segment and perpendicular to it.

**Properties of Perpendicular Bisector:**

- Divides the line segment into two equal lines segments;
- Each point belongs to the bisector of the segment, so it is located at the same distance with the two endpoints of the segment;
- The Perpendicular Bisector of the segment forms two right angles at the midpoint;

In the following figure, we have the line <Tex>\overleftrightarrow{AB}</Tex> and the median of the line segment <Tex>\overline{MN}</Tex>, and we notice that two right angles formed at the midpoint <Tex>O</Tex>.

<Image src="/imgs/lessons/geometry/perpend_bisc.png" alt="Perpendicular Bisector" width='500px' height='309px'/>

## How to draw a Perpendicular Bisector

To draw the perpendicular bisector of the line segment <Tex>\overline{MN}</Tex> using compass, we follow the following steps:

**Step1:**

Draw an arc on the top side and an arc on the bottom side of a circle with center <Tex>M</Tex> (endpoint of <Tex>\overline{MN}</Tex>) and radius <Tex>R</Tex> so that <Tex>R>\frac{\overline{MN}}{2}</Tex> (the radius should be a little bigger than the middle of the line segment).

<Image src="/imgs/lessons/geometry/perpend_step1.png" alt="Perpendicular Bisector" width='500px' height='309px'/>

**Step2:**

Without changing the radius <Tex>R</Tex> which we took in step 1, draw an arc on the top side and an arc on the bottom side of a circle with center <Tex>N</Tex> (endpoint of <Tex>\overline{MN}</Tex>).

<Image src="/imgs/lessons/geometry/perpend_step2.png" alt="Perpendicular Bisector" width='500px' height='309px'/>

**Step3:**

At the last step, we connect the intersection of the arches on the upper and lower sides by a straight line.

<Image src="/imgs/lessons/geometry/perpend_step3.png" alt="Perpendicular Bisector" width='500px' height='309px'/>

## Perpendicular Bisectors of a Triangle

We saw that each segment had a perpendicular bisector. So a triangle that has three sides can have three perpendicular bisectors.

Following the same steps above, we draw the perpendicular bisectors of triangle <Tex>ABC</Tex> below:

<Image src="/imgs/lessons/geometry/triangle_abc.png" alt="Perpendicular Bisectors of triangle" width='419px' height='307px'/>

- The perpendicular bisector of <Tex>\overline{AB}</Tex>
<Image src="/imgs/lessons/geometry/triangle_step1.png" alt="Perpendicular Bisectors of triangle" width='649px' height='287px'/>
- The perpendicular bisector of <Tex>\overline{AC}</Tex>
<Image src="/imgs/lessons/geometry/triangle_step2.png" alt="Perpendicular Bisectors of triangle" width='539px' height='318px'/>
- The perpendicular bisector of <Tex>\overline{BC}</Tex>
<Image src="/imgs/lessons/geometry/triangle_step3.png" alt="Perpendicular Bisectors of triangle" width='574px' height='305px'/>

The three perpendicular bisectors intersect at a one point, this point being the center of a **circle circumscribed** to triangle <Tex>ABC</Tex>.

<Image src="/imgs/lessons/geometry/triangle_circle.png" alt="Perpendicular Bisectors of triangle and circumscribed circle" width='510px' height='376px'/>

<Related>

<Link to="../geometry/basic-geometric-terms" as="/lessons/geometry/basic-geometric-terms">Basic Geometry Terms</Link>

</Related>

Perpendicular Bisector is a straight line passing through the midpoint of the line segment and perpendicular to it. The triangle have three Perpendicular Bisectors.

Perpendicular Bisector


# Types of Triangles

As the lesson title indicates, there are many types of triangles that are classified by their sides, angles, or both. Each type is distinguished by some properties from the other types.

It is worth noting that triangles are among the important shapes in geometry and algebra together and have many uses for solving mathematical and physical problems.

In this lesson, we will try to explain each type of triangle with the shapes, definitions and properties, and at the end you will be able to understand the difference between them.

## Classification of the Different Types of Triangles

Triangles are classified according to the measure of sides and measures of angles. The two properties common to all triangles are that the sum of the angles of a triangle is 180 degrees, and the sum of the lengths of the two sides of a triangle is greater than the length of the third side. The following are the distinguishing characteristics of each type of triangle according to classification.

**By Side:**

1. Equilateral Triangle

2. Isosceles Triangle

3. Scalene Triangle

**By Angle:**

1. Acute-Angled Triangle

2. Right-Angled Triangle

3. Obtuse-Angled Triangle

## Types of Triangles Based on their Sides

For this type of triangle, we'll only be concerned with the sides of the triangle. As you well know, any triangle is determined from three sides, and these sides can be all isometric, or only two isometric sides, or the three sides of different lengths.

**Equilateral Triangle:** the three sides are equal (isometric).

<Image src="/imgs/lessons/geometry/equilateral_triangle.jpeg" alt="Equilateral Triangle Shape" width={250} height={250}/>

**Isoscele Triangle:** only two of the sides, called the **legs**, must have the same measure. The other side is called the **base**.

<Image src="/imgs/lessons/geometry/isocele_triangle.jpeg" alt="Isocele Triangle Shape" width={250} height={250}/>

**Scalene Triangle:** is uneven, no side is the same length as any other. To be more precise,“scalene” means “not
equilateral or isosceles.

<Image src="/imgs/lessons/geometry/scalene_triangle.jpeg" alt="Scalene Triangle Shape" width={250} height={250}/>

## Types of Triangles Based on their Angles

In this classification, we will only study the <Link to="../geometry/types-of-angles" as="/lessons/geometry/types-of-angles">angles</Link> of a triangle. As you know, any triangle has three angles. All of these angles can be acute, one obtuse and the two other acute, or one right angle and the two other acute.

**Acute Triangle:** all angles are acute (less than 90°).

<Image src="/imgs/lessons/geometry/acute_triangle.jpeg" alt="Acute Triangle Shape" width={250} height={250}/>

**Obtuse Triangle:** has an angle obtuse (more than 90°), and two acute angles.

<Image src="/imgs/lessons/geometry/obtuse_triangle.jpeg" alt="Obtuse Triangle Shape" width={250} height={250}/>

**Right Triangle:** has a right angle (90°), and two acute angles.

<Image src="/imgs/lessons/geometry/right_triangle.jpeg" alt="Right Triangle Shape" width={250} height={250}/>

## Types of Triangle Based on Sides and Angles

Triangles can also be categorized by both angles and sides. In this case, we will have seven different states all derived from the above classifications.

**Equilateral or Regular Triangle:** In this triangle, all the sides are congruent and all the angles are congruent. Each angle measures 60°.

**Isosceles Right Triangle:** This triangle has two congruent sides and a right angle. The two acute angles at the *base* are congruent, and their measure is 45°.

**Obtuse Isosceles Triangle:** This triangle has two congruent sides and an obtuse angle. The two acute angles at the *base* are congruent, measuring less than 45°.

**Acute Isosceles Triangle:** This triangle has two congruent sides and an acute angle. The two acute angles at the *base* are congruent, measuring greater than 45°.

**Right Scalene Triangle:** This triangle has a right angle and all sides are not congruent.

**Obtuse Scalene Triangle:** This triangle has an obtuse angle and all sides are not congruent.

**Acute Scalene Triangle:** This triangle has an acute angle and all sides are not congruent.

---------------------

<Related>

<Link to="../geometry/types-of-angles" as="/lessons/geometry/types-of-angles">Types of Angles</Link><br/>

<Link to="../geometry/angle-bisector" as="/lessons/geometry/angle-bisector">Angle Bisector</Link><br/>

<Link to="../geometry/angle-bisector" as="/lessons/geometry/basic-geometric-terms">Basic Geometric Terms</Link><br/>

<Link to="../geometry/angle-bisector" as="/lessons/geometry/perpendicular-bisector">Perpendicular Bisector</Link>

</Related>

different types of triangles based on thier sides and angles. Learn all about classification of triangles with shape, definition and example for each type.

Explorer all lessons of Geometry