زوايا المثلثات المشهورة

زوايا المثلثات المشهورة , مما لا شك فيه أن هذا الموضوع من أهم وأفضل الموضوعات التي يمكن أن أتحدث عنها اليوم، حيث أنه موضوع شيق ويتناول نقاط حيوية، تخص كل فرد في المجتمع، وأتمنى من الله عز وجل أن يوفقني في عرض جميع النقاط والعناصر التي تتعلق بهذا الموضوع.

The angles of the famous triangles, any triangle contains three angles, where the sum of the angles of the famous triangles is equal to 180 degrees, regardless of the type of triangle. Our next lines on the reference site on the concept of the triangle and the answer to the question posed, will also be addressed to the Pythagorean theorem, its importance and how to prove it.

Triangle Definition

A triangle is known as a closed two-dimensional and three-sided shape, it consists of three straight segments whose sides intersect and form the angles and vertices. interior angle, and also the longest side of the triangle corresponds to the largest interior angle, and terms related to the triangle are:[1][2]

Vertex: It is the angle of the triangle, since each triangle has three vertices.
Base: Which side of the triangle forms a base.
The median of the triangle: It is a line extending from the vertex of the triangle to the middle of the opposite side, since the triangle has three of them and they intersect at one point called the central point of the triangle.
Altitude: It is the column extending from the base to the vertex of the triangle opposite it. There are three different heights and they intersect at a point called the area of the altitudes or the right center.

Famous Triangles Angles

A triangle is a shape with three sides and consists of three straight segments. The type of triangle depends on the size of the angle and the length of its side. It is divided based on the angles it has, which will be shown as follows:[3]

Right-angled triangle: This triangle has one 90-degree angle and two acute angles.
Obtuse angled triangle: This triangle consists of two acute angles and an obtuse angle, which measure more than 90 degrees.
Acute Angle Triangle: This triangle consists of three acute angles, and an acute angle is one whose measure is less than 90 degrees.

Types of triangles and their properties

Dividing triangles according to the length of the sides into three types, where the sides of equal length can be distinguished by placing a slash mark on them, and the following will show these types:[3]

Scaledan triangle: A triangle that has no sides of equal length or equal angles of measure.
Equilateral Triangle: It is a triangle with three equal sides and three equal angles, each of which measures 60 degrees.
Isosceles Triangle: It is a triangle that has two sides of equal length and two angles of equal measure, which are the base angles.

properties of triangles

The triangle is considered a polygon with three sides, three angles and three vertices. The most important general properties of the triangle can be summarized as follows:
[4]

The sum of the three angles of a triangle is 180 degrees.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
The difference between the lengths of any two sides of a triangle is less than the length of the third side.
The side opposite the major angle in a triangle is considered to be the longest side.
The exterior angle of a triangle is equal to the sum of the two remote interior angles, and is known as the exterior angle property.
Triangles are similar if the opposite angles of the two triangles are congruent and the lengths of their sides are proportional.
The formula for the area of a triangle and the perimeter of a triangle can be stated as follows:
- The formula for the area of a triangle = ½ x base x height.
- Perimeter of a triangle = sum of all three sides.
A triangle in which all angles are less than 90 degrees is known as an acute triangle.
A triangle that has one angle of greater than 90 degrees is called an obtuse triangle.

A function is a relationship that defines only one output for a single input

How to calculate the height of a triangle

The height of a triangle is calculated if its area and base length are known by the triangle area law, for all types of triangles, by rearranging the triangle area law = (1/2 x base x height) and resulting in triangle height = (2 x area) / base , where it can be applied through the example. If there is a triangle with an area of 20 cm² and a base length of 4 cm, it can be calculated as follows:[5][6]

Putting the formula for height of a triangle = (2 x area) / base
Substitute the values given in the law: height = (2 x 20) / 4 = 40 / 4
Height = 10 cm.

How many triangles are in the picture

Phitagors theory

The Pythagorean Theorem is known as one of the oldest known theorems, as it was named after the Greek mathematician and philosopher Pythagoras. In the following, the Pythagorean theorem and its importance will be explained:

Pythagorean Theory Law

The Pythagorean Theorem theorem states that: The sum of the squares of the lengths of the two sides of the right, which are the two shortest sides in a right-angled triangle, is equal to the square of the length of the hypotenuse, and it is considered to be the longest side in the triangle. (A, B) are the two sides of the right triangle AB C, and (C) is the hypotenuse of the right triangle AB C, as it is considered to be the longest side in it, and it is worth mentioning that the inverse of the theorem is true, as the triangle to which the Pythagorean theorem applies is: A² + B²=C², it is necessarily a right-angled triangle, and there are many applications of the Pythagorean Theorem, which are as follows:[7]

Indicate whether the triangle is right-angled or not.
Find the length of the diagonal in the square or rectangle.
Find the length of any side of a triangle if the length of the other two sides of a right-angled triangle is known.

Types of triangles according to sides and angles

The importance of the Pythagorean theorem

The Pythagorean theorem is considered one of the important theorems in mathematics, as it has many uses. The importance of the theorem will be explained as follows:[8]

Clarify the shape and type of a triangle. When the square of the hypotenuse is equal to the sum of the squares of the other two sides, it is a right-angled triangle, and when the square of the hypotenuse is longer than the square of the other two sides, the triangle is obtuse, and when the square of the hypotenuse is less than the square of the other two sides, it is called an acute-angled triangle.
The theorem helps to calculate the lengths of the hidden sides, not only in triangles, but also in squares and rectangles.
This theory helps construction workers maintain correct measurements of angles in the construction of homes and buildings.

Photo of Pythagorean trigonometry theory

Known as the Pythagorean theorem, it was named after the scientist Pythagoras, a scientist from ancient Greece, where he explained that the opposite of the Pythagorean theorem is that if there is a square of one side in a triangle equal to the sum of the squares of the other two sides in a triangle, then the angle opposite the large side is a right-angled that is equal to 90 degrees .[7]

A triangle whose measures of angles are 100°, 45°, and 35° are classified as,

Examples of the famous Pythagorean triangles

There are some examples that show how to calculate the length of the side or the hypotenuse in the Pythagorean theorem, in addition to checking whether the triangle is right-angled or not. Here are some examples:[7][9]

First example: A right-angled triangle has the length of the first side equal to 12 cm and the length of the second side equal to 5 cm. What is the length of its hypotenuse?
- Substitute the value of the side lengths into the Pythagorean equation as follows: (a² + b² = c²).
- It results in: (12) ² + (5) ² = c², where c² = 169.
- Taking the square root of both sides results in c = 13, and the length of the hypotenuse = 13 cm.
The second example: showing the diameter of a square with an area of 1 cm, and the length of the hypotenuse?
- The diagonal of the square is divided into two congruent triangles.
- Thus, the lengths of the sides of the square = the lengths of the sides of a right-angled triangle, it is equal to 1 cm.
- Substitute the value of the side lengths into the Pythagorean equation, to get that: a² + b² = g², and gives c² = 2.
- Taking the square root of both sides results in c = 1.414.
- The length of the hypotenuse = the length of the diagonal of the square = 1.414 cm.

A triangle containing a right angle is

Calculating the Angles of Famous Triangles

If the value of two angles in a triangle is known and the measure of the third angle is unknown, its measure can be calculated by subtracting the sum of the two angles from 180 degrees, as the sum of the measure of the interior angles of a triangle = 180, and the following are the methods that contribute to finding the value of the angles of a triangle of various kinds:[10]

Calculating the angles of a right-angled triangle: A right-angled triangle can be known when one of its angles is 90 degrees. The equation can be explained as follows:
- x + y + 90 = 180.
- x + y = 90, since (x, y) are the angles of a right triangle.
Calculating the Angles of an Isosceles Triangle: The isosceles triangle is called by this name because the base angles are equal in measure, so the sum of the angles of this triangle can be stated as follows:
- 2 x x + y = 180, where x is the measure of the base angles, and y is the measure of the vertex angle.
Calculating the angles of an equilateral triangle: An equilateral triangle can be explained as an equilateral triangle with angles, since the measure of each of its angles is always equal to 60 degrees, and therefore:
- C + C + S = 180. Accept 3% C = 180.
- Divide both sides by the number 3.
- It turns out that the value of x = 60 degrees.

How are the angles of famous triangles measured?

The angles of popular triangles can be measured with engineering tools such as using a protractor, or through a digital angle finder, but this finder is not suitable as a technical drawing tool, because the axis will not be flat on paper unlike a protractor, in addition to that it is a stainless steel machine that may be a machine Sharp and not suitable for children’s use.[11]

Exterior angles of a triangle

An exterior angle is defined as the result of a flat geometric figure as an angle between one of its sides and it is an extension of another side, since the sum of the three exterior angles of any triangle is 360 degrees, and angles outside the triangle can also be obtained by drawing a ray or a straight line extending from One of the sides, and in this way the exterior angle is the angle between this extension and the side of the triangle next to it.[12]

Calculating the sides of a right-angled triangle

The right-angled triangle is one of the most important types of triangles in trigonometry, as the right-angled triangle consists of three sides and one right angle, and the right angle of measure 90 is denoted by a small square on the corner, while one of the last two angles is denoted by the symbol x, as the other two sides are Perpendicular to each other, and each of them is called the leg of a right-angled triangle or the right side. The length of the sides of a right-angled triangle can be calculated using the Pythagorean theorem. The three sides of the triangle are classified as follows:[5]

The adjacent side: It is the side that is adjacent to or close to the angle x.
Opposite side: It is considered to be the side opposite the angle x.
Hypotenuse: It is the longest side of the triangle, and it is the side opposite the right angle.

What is the formula for the area of a triangle?

Basic Trigonometric Identities

There are the most important trigonometric ratios or functions of right-angled triangles in trigonometry:[13]

Sine: where it is denoted (S), and its law in a right-angled triangle is Jas = the side opposite the angle x ÷ the hypotenuse of the triangle.
Cosine: It is symbolized by (cos), and its law in a right-angled triangle is cos x = the side adjacent to the angle x ÷ the hypotenuse of the triangle.
The tangent: It is denoted by (za), and its law in a right-angled triangle is tan x = the side opposite the angle x ÷ the side adjacent to the angle, which can be abbreviated as follows x = sin (x) / cos (x).
The secant: It is denoted by (Q), but its law in a right-angled triangle is s x = hypotenuse of the triangle ÷ the side adjacent to the angle x = 1 ÷ cos x.

Any triangle of given side lengths and a right-angled triangle

And here we have come to the conclusion of our article that raised the question about the angles of the famous triangles, where we answered this question, where the concept of the triangle and its most prominent general characteristics were explained, and the Pythagorean theorem, its importance, text and some examples of it, as well as clarifying the sides of a right-angled triangle.

خاتمة لموضوعنا زوايا المثلثات المشهورة ,وفي نهاية الموضوع، أتمنى من الله تعالى أن أكون قد استطعت توضيح كافة الجوانب التي تتعلق بهذا الموضوع، وأن أكون قدمت معلومات مفيدة وقيمة.