Triangle Introduction
Triangles are polygons convex with as few sides as possible. Among the most diverse applications, the main results involving triangles are trigonometry and the Theorem of Pythagoras.
Representation
In order to facilitate solving exercises, it is common to indicate the vertices of a triangle in capital letters.
In the figure below, we have the triangle
A
B
C
.
Classification
We can classify a triangle in two ways: either by the measurement of the sides or by the measurement of its sides. angles internal.
A triangle, in relation to the measurements of its sides, can be classified as:
1. Equilateral triangle: if the three sides have equal measurements.
2. Isosceles triangle: if two sides have the same measurement.
3. Scalene triangle: if all sides have different measurements.
In the case of the equilateral triangle, we can show that the three interior angles also have the same measure.
The side with different measurements in an isosceles triangle is called the base of the triangle and the base angles have the same measurement, as shown in the following figure.
Classification according to angle measurements
Among the values of the internal angles of a triangle, we have the following classifications:
1. Acute triangle: is one with all acute internal angles, that is, less than 90°.
2. Right triangle: occurs when an internal angle is right (worth 90°).
3. Obtuse angle triangle: is one with an obtuse angle, that is, whose measure is greater than 90°.
In any polygon, the side opposite the largest internal angle is also the largest side. So, in a triangle rectangle, the side opposite the right angle is the side of greatest measurement and is called the hypotenuse and the other two remaining sides are defined as the legs of the triangle.
Angles of a triangle
Internal angles of a triangle
It is possible to show that the sum of the interior angles of a triangle is 180°. This result applies to any triangle, regardless of its shape or the measurements of its sides.
It is because of this result that it is impossible for there to be more than one interior angle that is either right or obtuse in a triangle; in other words, we will always have only three possibilities: either there are only acute angles or a single right angle and two acute angles or a single obtuse angle and two acute angles, which are nothing more than the classifications listed previously regarding the measurements of the internal angles.
From the result of the sum of the internal angles of a triangle, let us consider an equilateral triangle. We have that the measures of its internal angles are equal to each other:
And how the sum is equal to 180°, we have:
x+x+x=180º⇒3x=180º
In other words, in every equilateral triangle, the measure of each internal angle is 60°.
If we consider an isosceles right triangle, then the two acute angles will be equal to each other:
Thus, the measure of each acute angle in this case will be 45°.
Let us now take any right-angled triangle :
As the sum of the interior angles is equal to 180°, then:
x+y+90º=180º⇒x+y=90º
In other words, we conclude that the sum of the acute angles of a right-angled triangle is always 90°, that is, they are complementary.
Exterior angle of a triangle
An external angle of any triangle is the one that is formed when we extend one of its sides, as shown in the figure below.
It is possible to observe that the sum between an external angle and the internal angle adjacent to it is worth 180°, that is, they are supplementary.
Using this result with the fact that the sum of the internal angles of a triangle is equal to 180°, we have that the measure of an external angle of a triangle is equal to the sum of the two internal angles not adjacent to it :
Area of a triangle
Area depending on base and height
If the base of a triangle measures b and its height measures h, then the area is given in this image below:
