A triangle is isosceles if it has. Isosceles triangle. Detailed theory with examples

The properties of an isosceles triangle are expressed by 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 altitude.

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

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

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

Proof. Let us 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 a common side, ∠ 1 = ∠ 2, since AD ​​is a bisector). From the equality of these triangles it follows that ∠ B = ∠ C. The theorem is proven.

Using Theorem 1, the following theorem is established.

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 congruent (Fig. 2).

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

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

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

Then Δ AMV is isosceles. Let us draw a straight 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 to a segment is equidistant from its ends.

Solution. 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 straight line p. Let's draw segments AM and BM. Triangles AOM and BOM are equal, since their angles at vertex O are right, leg OM is common, and leg OA is equal to leg OB by condition. From the equality of triangles AOM and BOM it follows that AM = BM.

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

Compare triangles ABC and DEF. Find the corresponding equal angles.

Solution. These triangles are equal according to the third criterion. Correspondingly, equal angles: A and E (lie opposite equal sides BC and FD), B and F (lie opposite equal sides AC and DE), C and D (lie opposite equal sides AB and EF).

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

Find angle D.

Solution. Consider triangles ABC and ADC. They are equal according to the third criterion (AB = DC, BC = AD by condition and side AC is common). From the equality of these triangles it follows that ∠ B = ∠ D, but angle B is equal to 100°, which means angle D is equal to 100°.

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

Video solution.

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 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 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 = . 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.

1. Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Education.
2. Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. 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. Sadovnichego V.A. - M.: Education, 2010.

1. Dictionaries and encyclopedias on Academician ().
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. 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- This triangle, in which the two sides are equal in length. Equal sides are called lateral, and the last one is called the base. A-priory, 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. Also equal bisectors , medians And heights drawn from these angles.
• The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. Centers inscribed And described circles lie on this line.
• Angles opposite to equal sides are always spicy(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 circumcircle , r- radius inscribed.

The sides can be found as follows:

Angles can be expressed in the following ways:

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

Square 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).

see also

Wikimedia Foundation. 2010.

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

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