# Properties of an isosceles triangle

A triangle whose two sides are equal to each other is called isosceles. These sides are called his side, and the third party 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 the triangle BAC. These triangles, on the first sign, are equal to each other. So it is, because BC = AC, AC = BC, angle ACB = angle ACB. From this it follows that the angle BAC = angle ABC, because these are the corresponding angles of our equal triangles. Here you have the property of angles of an isosceles triangle.

## Theorem 2

### The median in an isosceles triangle, which was conducted to its base, is also a height and a bisector

Proof of the theorem.

Suppose we have an isosceles triangle ABC, the base of which is AB, and the CD is the median, which we led 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, an isosceles triangle). Side AD = side BD, because the point D divides the segment AB into equal parts. Hence, the 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. It follows from equality 1 that a CD is a bisector. And the angle ADC and the angle BDC are adjacent angles, and it follows from equality 2 that they are both straight lines. 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 in a triangle the two angles are equal to each other, then such a triangle is isosceles

Proof of the theorem.

Suppose we have a triangle ABC, in which angle CAB = angle CBA. Triangle ABC = triangle BAC on the second sign of equality between triangles. So it is, because AB = BA; angle CBA = angle CAB, angle CAB = angle CBA. From this equality of triangles, we have the equality of the corresponding sides of the triangle - AC = BC. Then it turns out that the triangle ABC is isosceles.

## Theorem 4

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

Proof of the theorem.

In the triangle ABC we will hold the median CD. It will also be tall. Right triangle ACD = right triangle BCD, because the CD leg is common to them, and the leg AD = the 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 the three sides of a triangle are equal to the three sides of another triangle, then these triangles are equal

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.

Suppose that these triangles are not equal to each other. From here 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 in the above feature.

Assume that triangle A1B1C2 = triangle ABC. For a triangle, the vertex C2 lies with the vertex C1 relative to the straight line A1B1 in one half-plane. We assumed that the vertices C2 and C1 do not coincide. Suppose that point D is the midpoint of the segment C1C2.So we have the 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 the straight line B1D and the straight line A1D are perpendicular to the straight line C1C2.

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