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

Properties

• The angles opposite the equal sides of an isosceles triangle are equal to each other. Bisectors, medians and heights 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 be a is the length of two equal sides of an isosceles triangle, b- the length of the third side, α and β - corresponding angles, R- radius of the circumscribed circle, r- the radius of the inscribed .

The sides can be found like this:

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

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

see also

Wikimedia Foundation. 2010 .

See what the "Isosceles Triangle" is in other dictionaries:

ISOSHELES TRIANGLE, A TRIANGLE having two sides equal in length; the angles at these sides are also equal ... Scientific and technical encyclopedic dictionary

And (simple) triangle, triangle, husband. 1. A geometric figure bounded by three mutually intersecting straight lines forming three internal angles (mat.). Obtuse triangle. Acute triangle. Right triangle.… … Explanatory Dictionary of Ushakov

ISOSHELES, oy, oy: an isosceles triangle with two equal sides. | noun isosceles, and, wives. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov

triangle- ▲ a polygon having, three, angle triangle is the simplest polygon; is given by 3 points that do not lie on the same straight line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language

triangle- TRIANGLE1, a, m of which or with def. An object that has the shape of a geometric figure bounded by three intersecting straight lines forming three internal angles. She sorted through her husband's letters, yellowed front-line triangles. TRIANGLE2, a, m ... ... Explanatory dictionary of Russian nouns

This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is a geometric figure formed by three line segments that connect three non-linear points. Three dots, ... ... Wikipedia

Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, polygon with 3 sides. Sometimes under... Illustrated Encyclopedic Dictionary

encyclopedic Dictionary

triangle- a; m. 1) a) A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles triangle/flax. Calculate the area of ​​the triangle. b) resp. what or with def. A figure or object of such a form. ... ... Dictionary of many expressions

BUT; m. 1. A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles m. Calculate the area of ​​the triangle. // what or with def. A figure or object of such a shape. T. roof. T.… … encyclopedic Dictionary

Lesson topic

Isosceles triangle

The purpose of the lesson

Introduce students to the isosceles triangle;
Continue to form the skills of building right triangles;
To expand the knowledge of schoolchildren about the properties of isosceles triangles;
To consolidate theoretical knowledge in solving problems.

Lesson objectives

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

Lesson Plan

1. General concepts and definitions about 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, which are called the sides of an isosceles triangle, and its third side is called the base.

The top of this figure is the one that is located opposite its base.

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

Types of isosceles triangles

An isosceles triangle, like other shapes, can have different types. Isosceles triangles include acute, right, obtuse, and equilateral triangles.

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

Properties of an isosceles triangle

Opposite angles with respect to the equal sides of an isosceles triangle are equal to each other;

Bisectors, medians and heights 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 height, bisector and median, (they coincide) drawn to the base.

The angles opposite the 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 the peculiarity of this triangle?
3. What is the difference between an isosceles triangle and a right triangle?
4. Name the properties of an isosceles triangle known to you.
5. Do you think it is possible in practice to check the equality of angles at the base and how to do it?

Exercise

And now let's take a short quiz 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 midpoint of the opposite side?
3. Is a bisector a segment that divides the angle that bisects 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 to the length of the base of the triangle.
2. If in the problem the perimeter and length of the base of an isosceles triangle are known, then to find the length of the lateral 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, determine one extra 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 this problem have?

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 the other, then these triangles will be equal. Prove this statement.

5. Think and say, is any isosceles triangle 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 so that it is both rectangular and isosceles at the same time?

Do you know what Pascal's triangle is? Pascal's triangle is often asked to test basic programming skills. In general, Pascal's triangle refers to combinatorics and probability theory. So what is this triangle?

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 is filled with the sums of the two numbers that are located above. You can add such a triangle to infinity, but if you outline it, then we get an isosceles triangle with symmetrical lines about its vertical axis.

Think about where in everyday life you had to meet isosceles triangles? Isn't it true that the roofs of houses and ancient architectural structures are very reminiscent of them? And remember, what is the basis of the Egyptian pyramids? Where else have you seen isosceles triangles?

Isosceles triangles from ancient times helped the Greeks and Egyptians in determining distances and heights. So, for example, the ancient Greeks used it to determine from afar the distance to the ship in the sea. And the ancient Egyptians determined the height of their pyramids due to the length of the cast shadow, because. it was an isosceles triangle.

Since ancient times, people have 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 structures that remind us of an isosceles triangle. When we go to a store, we see triangular-shaped packages of food and juices, and even some human faces have the shape of a triangle. This figure is so popular that it can be found at every turn.

Subjects > Mathematics > Mathematics Grade 7

A triangle with two equal sides is called an isosceles triangle. These sides are called the sides, and the third side is called the base. In this article, we will tell you about the properties of an isosceles triangle.

Theorem 1

The angles near the base of an isosceles triangle are equal to each other

Proof of the theorem.

Suppose we have an isosceles triangle ABC whose base is AB. Let's look at triangle BAC. These triangles, by the first sign, are equal to each other. So it is, because BC = AC, AC = BC, angle ACB = angle ACB. It follows from this that angle BAC = angle ABC, because these are the corresponding angles of our triangles equal to each other. Here is the property of the angles of an isosceles triangle.

Theorem 2

The median in an isosceles triangle drawn to its base is also the height and bisector

Proof of the theorem.

Let's say we have an isosceles triangle ABC whose base is AB and CD is the median we drew to its base. In triangles ACD and BCD, angle CAD = angle CBD, as the corresponding angles at the base of an isosceles triangle (Theorem 1). And side AC = side BC (by definition of an isosceles triangle). Side AD \u003d side BD, After all, point D divides segment AB into equal parts. Hence it follows that triangle ACD = triangle BCD.

From the equality of these triangles, we have the equality of the corresponding angles. That is, angle ACD = angle BCD and angle ADC = angle BDC. Equation 1 implies that CD is a bisector. And angle ADC and angle BDC are adjacent angles, and from equality 2 it follows that they are both right angles. It turns out that CD is the height of the triangle. This is the property of the median of an isosceles triangle.

And now a little about the signs of an isosceles triangle.

Theorem 3

If two angles in a triangle are congruent, then the triangle is isosceles.

Proof of the theorem.

Let's say we have a triangle ABC in which angle CAB = angle CBA. Triangle ABC = triangle BAC by the second criterion of equality between triangles. So it is, because AB = BA; angle CBA = angle CAB, angle CAB = angle CBA. From such an equality of triangles, we have the equality of the corresponding sides of the triangle - AC = BC. Then it turns out that triangle ABC is isosceles.

Theorem 4

If in any triangle its median is also its height, then such a triangle is isosceles

Proof of the theorem.

In the triangle ABC we draw the median CD. It will also be height. Right triangle ACD = right triangle BCD, since leg CD is common to them, and leg AD = leg BD. From this it follows that their hypotenuses are equal to each other, as the corresponding parts of equal triangles. This means that AB = BC.

Theorem 5

If three sides of a triangle are equal to three sides of another triangle, then these triangles are congruent

Proof of the theorem.

Suppose we have a triangle ABC and a triangle A1B1C1 such that the sides are AB = A1B1, AC = A1C1, BC = B1C1. Consider the proof of this theorem by contradiction.

Assume that these triangles are not equal to each other. Hence we have that the angle BAC is not equal to the angle B1A1C1, the angle ABC is not equal to the angle A1B1C1, the angle ACB is not equal to the angle A1C1B1 at the same time. Otherwise, these triangles would be equal according to the above criterion.

Assume that triangle A1B1C2 = triangle ABC. The vertex C2 of a triangle lies with the vertex C1 relative to the line A1B1 in the same half-plane. We assumed that the vertices C2 and C1 do not coincide. Assume that point D is the midpoint of segment C1C2. So we have isosceles triangles B1C1C2 and A1C1C2, which have a common base C1C2. It turns out that their medians B1D and A1D are also their heights. This means that line B1D and line A1D are perpendicular to line C1C2.

B1D and A1D have different points B1 and A1 and therefore cannot coincide. But after all, through the point D of the straight line C1C2 we can draw only one straight line perpendicular to it. We've got a contradiction.

Now you know what are the properties of an isosceles triangle!

The properties of an isosceles triangle express the following theorems.

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

Theorem 2. In an isosceles triangle, the bisector drawn to the base is the median and height.

Theorem 3. In an isosceles triangle, the median drawn to the base is the bisector and height.

Theorem 4. In an isosceles triangle, the height drawn to the base is the bisector and the median.

Let us prove one of them, for example, Theorem 2.5.

Proof. Consider an isosceles triangle ABC with base BC and prove that ∠ B = ∠ C. Let AD be the bisector of triangle ABC (Fig. 1). Triangles ABD and ACD are equal according to the first sign of equality of triangles (AB = AC by condition, AD is the common side, ∠ 1 = ∠ 2, since AD ​​is the bisector). It follows from the equality of these triangles that ∠ B = ∠ C. The theorem is proved.

Using Theorem 1, we establish the following theorem.

Theorem 5. The third criterion for the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal (Fig. 2).

Comment. The sentences established in examples 1 and 2 express the properties of the perpendicular bisector to the segment. It follows from these proposals that the perpendicular bisectors of the sides of a triangle intersect at one point.

Example 1 Prove that the point of the plane equidistant from the ends of the segment lies on the perpendicular bisector to this segment.

Decision. Let the point M be equidistant from the ends of the segment AB (Fig. 3), i.e. AM = VM.

Then ΔAMV is isosceles. Let us draw a line p through the point M and the midpoint O of the segment AB. By construction, the segment MO is the median of the isosceles triangle AMB, and therefore (Theorem 3), and the height, i.e., the straight line MO, is the perpendicular bisector to the segment AB.

Example 2 Prove that each point of the perpendicular bisector of a segment is equidistant from its ends.

Decision. Let p be the perpendicular bisector to segment AB and point O be the midpoint of segment AB (see Fig. 3).

Consider an arbitrary point M lying on the line p. Let's draw segments AM and VM. Triangles AOM and VOM are equal, since their angles at the vertex O are straight, the leg OM is common, and the leg OA is equal to the leg OB by condition. From the equality of triangles AOM and BOM it follows that AM = BM.

Example 3 In the triangle ABC (see Fig. 4) AB \u003d 10 cm, BC \u003d 9 cm, AC \u003d 7 cm; in triangle DEF DE = 7 cm, EF = 10 cm, FD = 9 cm.

Compare triangles ABC and DEF. Find correspondingly equal angles.

Decision. These triangles are equal in the third criterion. Accordingly, equal angles: A and E (they lie opposite the equal sides BC and FD), B and F (they lie opposite the equal sides AC and DE), C and D (they lie opposite the equal sides AB and EF).

Example 4 In figure 5 AB = DC, BC = AD, ∠B = 100°.

Find angle D.

Decision. Consider triangles ABC and ADC. They are equal in the third feature (AB = DC, BC = AD by condition and side AC is common). From the equality of these triangles it follows that ∠ B = ∠ D, but the angle B is 100°, hence the angle D is 100°.

Example 5 In an isosceles triangle ABC with base AC, the exterior angle at vertex C is 123°. Find the angle ABC. Give your answer in degrees.

Video solution.

In this lesson, the topic "Isosceles triangle and its properties" will be considered. You will learn how the isosceles and equilateral triangles look and how they are characterized. Prove the theorem on the equality of angles at the base of an isosceles triangle. Consider also the bisector theorem (median and height) drawn to the base of an isosceles triangle. At the end of the lesson, you will go over two problems using the definition and properties of an isosceles triangle.

Definition:Isosceles A triangle is called which has two equal sides.

Rice. 1. Isosceles triangle

AB = AC - sides. BC - base.

The area of ​​an isosceles triangle is half the product of its base times its height.

Definition:equilateral A triangle is called in which all three sides are equal.

Rice. 2. Equilateral triangle

AB = BC = SA.

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

Given: AB = AC.

Prove:∠B = ∠C.

Rice. 3. Drawing to the theorem

Proof: triangle ABC \u003d triangle DIA according to the first sign (on two equal sides and the angle between them). From the equality of triangles follows the equality of all corresponding elements. Hence, ∠B = ∠C, which was to be proved.

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

Given: AB = AC, ∠1 = ∠2.

Prove: BD = DC, AD perpendicular to BC.

Rice. 4. Drawing for Theorem 2

Proof: triangle ADB = triangle ADC by the first feature (AD - common, AB = AC by condition, ∠BAD = ∠DAC). From the equality of triangles follows the equality of all corresponding elements. BD = DC since they lie 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 = . Hence, AD is the height of the triangle, which was to be proved.

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

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

To find: BC, AC, AB.

Decision:

Rice. 5. Drawing for example 1

We denote the base BC as a, then AB \u003d AC \u003d 2a.

2a + 2a + a = 50.

5a = 50, a = 10.

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

Example 2: Prove that all angles in an equilateral triangle 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 was to be proved.

Answer: Proven.

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

1. Alexandrov A.D., Werner A.L., Ryzhik V.I. etc. Geometry 7. - M.: Enlightenment.
2. Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. et al. Geometry 7. 5th ed. - M.: Enlightenment.
3. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichy V.A. - M.: Education, 2010.

1. Dictionaries and encyclopedias on "Akademik" ().
2. Festival of pedagogical ideas "Open Lesson" ().
3. 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. Sadovnichy 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 the other. Find the sides of the triangle. How many solutions does the problem have?