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Signs, constituent elements and properties of an isosceles triangle. Math I like

Lesson topic

The purpose of the lesson

Introduce students to an isosceles triangle;
Continue to develop skills in constructing right triangles;
Expand schoolchildren’s knowledge about the properties of isosceles triangles;
Strengthen theoretical knowledge when solving problems.

Lesson Objectives

Be able to formulate, prove and use the theorem about the properties of an isosceles triangle in the process of solving problems;
Continue to develop conscious perception educational material, logical thinking, self-control and self-esteem skills;
Arouse cognitive interest in mathematics lessons;
Foster activity, curiosity and organization.

Lesson Plan

1. General concepts and definitions of an isosceles triangle.
2. Properties of an isosceles triangle.
3. Signs of an isosceles triangle.
4. Questions and tasks.

Isosceles triangle

An isosceles triangle is a triangle that has two equal sides, called the sides of an isosceles triangle, and its third side is called the base.

The top of a given figure is the one located opposite its base.

The angle that lies opposite the base is called the vertex angle of this triangle, and the other two angles are called the base angles of an isosceles triangle.

Types of isosceles triangles

An isosceles triangle, like other figures, can have different types. Among isosceles triangles there are acute, rectangular, obtuse and equilateral triangles.

Acute triangle has all the sharp corners.
A right triangle has a straight angle at its apex and sharp angles at its base.
An obtuse angle has an obtuse angle at the apex, and the angles at its base are acute.
An equilateral object has all its angles and sides equal.

Properties of an isosceles triangle

Opposite angles in relation to equal sides of an isosceles triangle are equal to each other;

Bisectors, medians and altitudes drawn from angles opposite equal sides of a triangle are equal to each other.

The bisector, median and height, directed and drawn to the base of the triangle, coincide with each other.

The centers of the inscribed and circumscribed circles lie at the altitude, bisector and median (they coincide) drawn to the base.

Angles opposite equal sides of an isosceles triangle are always acute.

These properties of an isosceles triangle are used in solving problems.

Homework

1. Define an isosceles triangle.
2. What is special about this triangle?
3. How does an isosceles triangle differ from a right triangle?
4. Name the properties of an isosceles triangle that you know.
5. Do you think it is possible in practice to check the equality of angles at the base and how to do this?

Exercise

Now let's conduct a short survey and find out how you learned the new material.

Listen carefully to the questions and answer whether the following statement is true:

1. Can a triangle be considered isosceles if its two sides are equal?
2. A bisector is a segment that connects the vertex of a triangle with the middle opposite side?
3. A bisector is a segment that bisects an angle that connects a vertex with a point on the opposite side?

Tips for solving isosceles triangle problems:

1. To determine the perimeter of an isosceles triangle, it is enough to multiply the length of the side by 2 and add this product with the length of the base of the triangle.
2. If the perimeter and length of the base of an isosceles triangle are known in the problem, then to find the length of the side it is enough to subtract the length of the base from the perimeter and divide the found difference by 2.
3. And to find the length of the base of an isosceles triangle, knowing both the perimeter and the length of the side, you just need to multiply the side by two and subtract this product from the perimeter of our triangle.

Tasks:

1. Among the triangles in the figure, identify one extra one and explain your choice:

2. Determine which of the triangles shown in the figure are isosceles, name their bases and sides, and also calculate their perimeter.

3. The perimeter of an isosceles triangle is 21 cm. Find the sides of this triangle if one of them is 3 cm larger. How many solutions can it have? this task?

4. It is known that if the lateral side and the angle opposite to the base of one isosceles triangle are equal to the lateral side and the angle of another, then these triangles will be equal. Prove this statement.

5. Think and say whether any isosceles triangle is equilateral? And will any equilateral triangle be isosceles?

6. If the sides of an isosceles triangle are 4 m and 5 m, then what will be its perimeter? How many solutions can this problem have?

7. If one of the angles of an isosceles triangle is equal to 91 degrees, then what are the other angles equal to?

8. Think and answer, what angles should a triangle have in order for it to be both rectangular and isosceles?

How many of you know what Pascal's triangle is? The problem of constructing Pascal's triangle is often asked to test basic programming skills. In general, Pascal's triangle relates to combinatorics and probability theory. So what kind of triangle is this?

Pascal's triangle is an infinite arithmetic triangle or triangle-shaped table that is formed using binomial coefficients. In simple words, the vertex and sides of this triangle are units, and it itself is filled with the sums of the two numbers that are located above. Such a triangle can be folded ad infinitum, but if we outline it, we will get an isosceles triangle with symmetrical lines relative to its vertical axis.

Think about where in Everyday life Have you ever come across isosceles triangles? Isn't it true, the roofs of houses and ancient architectural structures do they look a lot like them? And remember what the basis is Egyptian pyramids? Where else have you come across isosceles triangles?

Since ancient times, isosceles triangles have helped the Greeks and Egyptians in determining distances and heights. For example, the ancient Greeks used it to determine from afar the distance to a ship at sea. And the ancient Egyptians determined the height of their pyramids based on the length of the cast shadow, because... it was an isosceles triangle.

Since ancient times, people already appreciated the beauty and practicality of this figure, since the shapes of triangles surround us everywhere. Moving through different villages, we see the roofs of houses and other buildings that remind us of an isosceles triangle; going into a store, we see packages with triangular-shaped products and juices and even some human faces have the shape of a triangle. This figure is so popular that you can see it at every step.

Subjects > Mathematics > Mathematics 7th grade

This lesson will cover the topic “Isosceles triangle and its properties.” You will learn what isosceles and equilateral triangles look like and how they are characterized. Prove the theorem on the equality of angles at the base of an isosceles triangle. Consider also the theorem about the bisector (median and altitude) drawn to the base of an isosceles triangle. At the end of the lesson, you will solve two problems using the definition and properties of an isosceles triangle.

Definition:Isosceles is called a triangle whose two sides are equal.

Rice. 1. Isosceles triangle

AB = AC - sides. BC - foundation.

The area of an isosceles triangle is equal to half the product of its base and its height.

Definition:Equilateral is called a triangle in which all three sides are equal.

Rice. 2. Equilateral triangle

AB = BC = SA.

Theorem 1: In an isosceles triangle, the base angles are equal.

Given: AB = AC.

Prove:∠B =∠C.

Rice. 3. Drawing for the theorem

Proof: triangle ABC = triangle ACB according to the first sign (two equal sides and the angle between them). From the equality of triangles it follows that all corresponding elements are equal. This means ∠B = ∠C, which is what needed to be proven.

Theorem 2: In an isosceles triangle bisector drawn to the base is median And height.

Given: AB = AC, ∠1 = ∠2.

Prove:ВD = DC, AD perpendicular to BC.

Rice. 4. Drawing for Theorem 2

Proof: triangle ADB = triangle ADC according to the first sign (AD - general, AB = AC by condition, ∠BAD = ∠DAC). From the equality of triangles it follows that all corresponding elements are equal. BD = DC since they are opposite equal angles. So AD is the median. Also ∠3 = ∠4, since they lie opposite equal sides. But, besides, they are equal in total. Therefore, ∠3 = ∠4 = . This means that AD is the height of the triangle, which is what we needed to prove.

In the only case a = b = . In this case, the lines AC and BD are called perpendicular.

Since the bisector, height and median are the same segment, the following statements are also true:

The altitude of an isosceles triangle drawn to the base is the median and bisector.

The median of an isosceles triangle drawn to the base is the altitude and bisector.

Example 1: In an isosceles triangle, the base is half the size of the side, and the perimeter is 50 cm. Find the sides of the triangle.

Given: AB = AC, BC = AC. P = 50 cm.

Find: BC, AC, AB.

Solution:

Rice. 5. Drawing for example 1

Let us denote the base BC as a, then AB = AC = 2a.

2a + 2a + a = 50.

5a = 50, a = 10.

Answer: BC = 10 cm, AC = AB = 20 cm.

Example 2: Prove that in an equilateral triangle all angles are equal.

Given: AB = BC = SA.

Prove:∠A = ∠B = ∠C.

Proof:

Rice. 6. Drawing for example

∠B = ∠C, since AB = AC, and ∠A = ∠B, since AC = BC.

Therefore, ∠A = ∠B = ∠C, which is what needed to be proven.

Answer: Proven.

In today's lesson we looked at an isosceles triangle and studied its basic properties. In the next lesson we will solve problems on the topic of isosceles triangles, on calculating the area of an isosceles and equilateral triangle.

Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Education.
Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. Geometry 7. 5th ed. - M.: Enlightenment.
Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.

Dictionaries and encyclopedias on Academician ().
Festival of Pedagogical Ideas " Public lesson» ().
Kaknauchit.ru ().

1. No. 29. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.

2. The perimeter of an isosceles triangle is 35 cm, and the base is three times smaller than the side. Find the sides of the triangle.

3. Given: AB = BC. Prove that ∠1 = ∠2.

4. The perimeter of an isosceles triangle is 20 cm, one of its sides is twice as large as the other. Find the sides of the triangle. How many solutions does the problem have?

Isosceles triangle is a triangle in which two sides are equal in length. Equal sides are called lateral, and the last one is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

Properties

Angles opposite equal sides of an isosceles triangle are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.
The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.
Angles opposite equal sides are always acute (follows from their equality).

Let a- the length of two equal sides of an isosceles triangle, b- length of the third side, α And β - corresponding angles, R- radius of the circumscribed circle, r- radius of inscribed .

The sides can be found as follows:

Angles can be expressed in the following ways:

The perimeter of an isosceles triangle can be calculated in any of the following ways:

The area of a triangle can be calculated in one of the following ways:

(Heron's formula).

Signs

Two angles of a triangle are equal.
The height coincides with the median.
The height coincides with the bisector.
The bisector coincides with the median.
The two heights are equal.
The two medians are equal.
Two bisectors are equal (Steiner-Lemus theorem).

Properties

Angles opposite equal sides of an isosceles triangle are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.
The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.
Angles opposite equal sides are always acute (follows from their equality).

Let a- the length of two equal sides of an isosceles triangle, b- length of the third side, α And β - corresponding angles, R- radius of the circumscribed circle, r- radius of inscribed .

The sides can be found as follows:

Angles can be expressed in the following ways:

The perimeter of an isosceles triangle can be calculated in any of the following ways:

The area of a triangle can be calculated in one of the following ways:

(Heron's formula).

Signs

Two angles of a triangle are equal.
The height coincides with the median.
The height coincides with the bisector.
The bisector coincides with the median.
The two heights are equal.
The two medians are equal.
Two bisectors are equal (Steiner-Lemus theorem).

Definition 7. Any triangle whose two sides are equal is called isosceles.
Two equal sides are called lateral, the third is called the base.
Definition 8. If all three sides of a triangle are equal, then the triangle is called equilateral.
It is a special type of isosceles triangle.
Theorem 18. The height of an isosceles triangle, lowered to the base, is at the same time the bisector of the angle between equal sides, the median and the axis of symmetry of the base.
Proof. Let us lower the height to the base of the isosceles triangle. It will divide it into two equal (along the leg and hypotenuse) right triangles. Angles A and C are equal, and the height also divides the base in half and will be the axis of symmetry of the entire figure under consideration.
This theorem can also be formulated as follows:
Theorem 18.1. The median of an isosceles triangle, lowered to the base, is also the bisector of the angle between equal sides, the height and the axis of symmetry of the base.
Theorem 18.2. The bisector of an isosceles triangle, lowered to the base, is simultaneously the height, median and axis of symmetry of the base.
Theorem 18.3. The axis of symmetry of an isosceles triangle is simultaneously the bisector of the angle between equal sides, the median and the altitude.
The proof of these consequences also follows from the equality of the triangles into which an isosceles triangle is divided.

Theorem 19. The angles at the base of an isosceles triangle are equal.
Proof. Let us lower the height to the base of the isosceles triangle. It will divide it into two equal (along the leg and hypotenuse) right triangles, which means the corresponding angles are equal, i.e. ∠ A=∠ C
The criteria for an isosceles triangle come from Theorem 1 and its corollaries and Theorem 2.
Theorem 20. If two of the indicated four lines (height, median, bisector, axis of symmetry) coincide, then the triangle will be isosceles (which means all four lines will coincide).
Theorem 21. If any two angles of a triangle are equal, then it is isosceles.

Proof: Similar to the proof of the direct theorem, but using the second criterion for the equality of triangles. The center of gravity, the centers of the circumcircle and incircle, and the point of intersection of the altitudes of an isosceles triangle all lie on its axis of symmetry, i.e. on high.
An equilateral triangle is isosceles for each pair of its sides. Due to the equality of all its sides, all three angles of such a triangle are equal. Considering that the sum of the angles of any triangle is equal to two right angles, we see that each of the angles of an equilateral triangle is equal to 60°. Conversely, to ensure that all sides of a triangle are equal, it is enough to check that two of its three angles are equal to 60°.
Theorem 22 . In an equilateral triangle, all the remarkable points coincide: the center of gravity, the centers of the inscribed and circumscribed circles, the point of intersection of the altitudes (called the orthocenter of the triangle).
Theorem 23 . If two of the indicated four points coincide, then the triangle will be equilateral and, as a consequence, all four named points will coincide.
Indeed, such a triangle will turn out, according to the previous one, isosceles with respect to any pair of sides, i.e. equilateral. An equilateral triangle is also called a regular triangle. The area of an isosceles triangle is equal to half the product of the square of the side side and the sine of the angle between the sides
Consider this formula for an equilateral triangle, then the alpha angle will be equal to 60 degrees. Then the formula will change to this:

Theorem d1 . In an isosceles triangle, the medians drawn to the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its medians. Then triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have common side AB, sides AL and BK are equal as halves of the lateral sides of an isosceles triangle, and angles LAB and KBA are equal as the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB are equal. But AK and LB are the medians of an isosceles triangle drawn to its lateral sides.
Theorem d2 . In an isosceles triangle, the bisectors drawn to the lateral sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its bisectors. Triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have a common side AB, angles LAB and KBA are equal as the base angles of an isosceles triangle, and angles LBA and KAB are equal as half the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB - the bisectors of triangle ABC - are congruent. The theorem has been proven.
Theorem d3 . In an isosceles triangle, the heights lowered to the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its altitudes. Then angles ABL and KAB are equal, since angles ALB and AKB are right angles, and angles LAB and ABK are equal as the base angles of an isosceles triangle. Consequently, triangles ALB and AKB are equal according to the second criterion for the equality of triangles: they have common side AB, angles KAB and LBA are equal according to the above, and angles LAB and KBA are equal as the base angles of an isosceles triangle. If the triangles are congruent, their sides AK and BL are also congruent. Q.E.D.

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Signs, constituent elements and properties of an isosceles triangle. Math I like

Lesson topic

The purpose of the lesson

Lesson Objectives

Lesson Plan

Isosceles triangle

Types of isosceles triangles

Properties of an isosceles triangle

Homework

Properties

Signs

see also

See what an “Isosceles triangle” is in other dictionaries:

Properties

Signs

see also

See what an “Isosceles triangle” is in other dictionaries:

What else to read

THE LAST NOTES

Signs, constituent elements and properties of an isosceles triangle. Math I like

Lesson topic

The purpose of the lesson

Lesson Objectives

Lesson Plan

Isosceles triangle

Types of isosceles triangles

Properties of an isosceles triangle

Homework

Properties

Signs

see also

See what an “Isosceles triangle” is in other dictionaries:

Properties

Signs

see also

See what an “Isosceles triangle” is in other dictionaries:

What else to read

THE LAST NOTES

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