# All triangles have a hypotenuse

### Right triangle

**Definition:** Under a *right triangle* one understands a triangle that has a right angle.

Because of the theorem about the sum of the angles in the triangle, the other two angles are then acute angles and complement each other, i.e. *Complement angle*. The side opposite the right angle is called the *hypotenuse* of the triangle and the other two sides the *Catheters*. Usually one chooses the designations in the right triangle in such a way that the right angle is, i.e. the hypotenuse and and the cathetus.

The shape of a right triangle is clearly determined by the specification of the angle. In particular, the shape of a right-angled isosceles triangle (or) is clearly defined. It is then half of a square that has been halved along its diagonal.

**Area of a right triangle:** If you turn a right-angled triangle around the center of its hypotenuse and join the two triangles, you get a rectangle with edge lengths and, which therefore has the area. So applies to the area of the right triangle

**Pythagorean theorem:** In every right-angled triangle, the terms selected above apply

**Proof:**

**Comment:** The reverse of the Pythagorean theorem also applies. *Applies to the three sides * The proof results from the congruence theorem **sss**, according to which two triangles that match on all three sides are congruent to one another, and according to the Pythagorean theorem there is a right-angled triangle with the three sides indicated.

**Altitude rate**: The height of the right triangle through the point divides the hypotenuse into sections that abut the side and that abut the side. With these designations applies

**Proof:**According to the Pythagorean theorem, the following applies in the given right-angled triangle and the two right-angled partial triangles

as well as and with are obtained and hence from which the assertion follows.

**Catheter set:** With the same designations as in the height theorem applies

**Proof:** From follows with the height theorem and analogously

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