Types of triangles. Triangle angles
The simplest polygon that is studied in school is a triangle. It is more comprehensible for students and encounters fewer difficulties. Despite the fact that there are different types of triangles that have special properties.
What shape is called a triangle?
Formed by three points and segments. The first are called vertices, the second - sides. Moreover, all three segments should be connected to form angles between them. Hence the name of the figure "triangle".
Differences in the names of the corners
Since they can be sharp, blunt and straight, then the types of triangles are determined by these names. Accordingly, there are three groups of such figures.
- The first. If all corners of the triangle are acute, then it will have the name of an acute angle. Everything is logical.
- The second. One of the corners is obtuse, which means a triangle is obtuse. There is simply no place.
- Third. There is an angle of 90 degrees, which is called straight. The triangle becomes rectangular.
Differences in names on the sides
Depending on the features of the sides, these types of triangles are distinguished:
the general case is versatile, in which all parties have an arbitrary length;
isosceles, the two sides of which have the same numerical values;
equilateral, the lengths of all its sides are the same.
If the task does not specify a specific type of triangle, then an arbitrary one should be drawn. In which all angles are sharp, and the sides have different lengths.
Properties common to all triangles
- If you add up all the corners of the triangle, you get a number equal to 180º. It does not matter what he looks like. This rule is always valid.
- The numerical value of either side of the triangle is less than the other two added together. At the same time, it is more than their difference.
- Each external corner has a value that is obtained by adding two internal, not adjacent to it. Moreover, it is always more than the inner one adjacent to it.
- Opposite the smaller side of the triangle is always the smallest angle. Conversely, if the side is large, then the angle will be the largest.
These properties are always valid, no matter what kind of triangles are considered in problems. All the others follow from specific features.
Properties of an isosceles triangle
- Angles that are adjacent to the base are equal.
- The height that is held to the base is also the median and the bisector.
- The heights, medians, and bisectors that are built to the sides of the triangle are respectively equal to each other.
Properties of an equilateral triangle
If there is such a figure, then all the properties described above will be true. Because the equilateral will always be isosceles. But not the other way around, an isosceles triangle will not necessarily be equilateral.
- All its angles are equal to each other and have a value of 60º.
- Any median of an equilateral triangle is its height and a bisector. And they are all equal to each other. To determine their values, there is a formula that consists of the product of a side and the square root of 3 divided by 2.
Properties of a right triangle
- Two acute angles give a total value of 90º.
- The length of the hypotenuse is always greater than that of any of the legs.
- The numerical value of the median conducted to the hypotenuse is half of it.
- The same value is equal to the leg, if it lies opposite the angle of 30º.
- The height, which is drawn from the top with a value of 90º, has a certain mathematical dependence on the legs: 1 / n2= 1 / a2+ 1 / in2. Here: a, c - legs, n - height.
Tasks with different types of triangles
№1. An isosceles triangle is given. Its perimeter is known and is 90 cm. It is required to know its sides. As an additional condition: the side is less than the base by 1.2 times.
The value of the perimeter is directly dependent on the values that need to be found. The sum of all three sides will give 90 cm. Now we need to recall the sign of the triangle, in which it is isosceles. That is, the two sides are equal. You can make an equation with two unknowns: 2a + b = 90. Here a is the side and in the base.
It is the turn of an additional condition. Following it, the second equation is obtained: в = 1,2а. You can substitute this expression in the first. It turns out: 2a + 1.2a = 90. After the transformations: 3.2a = 90. Hence, a = 28.125 (cm). Now it's easy to find out the foundation. This is best done from the second condition: c = 1.2 * 28.125 = 33.75 (cm).
To check, you can add three values: 28.125 * 2 + 33.75 = 90 (cm). That's right.
Answer: the sides of the triangle are 28.125 cm, 28.125 cm, 33.75 cm.
№2. The side of an equilateral triangle is 12 cm. It is necessary to calculate its height.
Decision. To search for an answer, it is enough to return to the point where the properties of the triangle were described.This is the formula for finding the height, median and bisector of an equilateral triangle.
n = a * √3 / 2, where n is the height and a is the side.
Substitution and calculation give the following result: n = 6 √3 (cm).
This formula does not need to be remembered. Suffice it to recall that the height divides the triangle into two rectangular ones. And it turns out to be a leg, and the hypotenuse in it is the side of the original, the second leg is half of the known side. Now you need to write the Pythagorean theorem and derive a formula for height.
Answer: height is 6 √3 cm.
No. 3. Given MCR - a triangle, 90 degrees in which makes an angle K. The sides of MP and CU are known, they are 30 and 15 cm, respectively. It is necessary to know the value of the angle R.
Decision. If you make a drawing, it becomes clear that MR is a hypotenuse. And it is twice the leg of the KR. Again you need to refer to the properties. One of them is associated with angles. From it it is clear that the angle of the CMR is equal to 30º. So the desired angle P will be equal to 60º. This follows from another property, which states that the sum of two acute angles should be 90º.
Answer: P angle is 60º.
№4. It is necessary to find all the angles of an isosceles triangle. About him it is known that the external angle from the angle at the base is 110º.
Decision. Since only the outer corner is given, this should be used. It forms with an internal angle unfolded. So in total, they will give 180º. That is, the angle at the base of the triangle will be equal to 70º. Since it is isosceles, the second angle has the same meaning. It remains to calculate the third angle. By the property common to all triangles, the sum of the angles is 180º. So, the third is defined as 180º - 70º - 70º = 40º.
Answer: the angles are 70º, 70º, 40º.
№5. It is known that in an isosceles triangle the angle lying opposite the base is 90º. Based on the marked point. The segment connecting it with a right angle divides it in the ratio of 1 to 4. You need to know all the angles of the smaller triangle.
Decision. One of the corners can be identified immediately. Since the triangle is rectangular and isosceles, those that lie at its base will be at 45º, that is, at 90º / 2.
The second of them will help to find the known in the condition relation. Since it is equal to 1 to 4, then the parts into which it is divided is obtained only 5. Therefore, to find out the smaller angle of the triangle, you need 90º / 5 = 18º. It remains to learn the third. To do this, from 180º (the sum of all the angles of the triangle) you need to subtract 45º and 18º. The calculations are simple, and it will turn out: 117º.