# 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: For a given right-angled triangle with the legs and and the hypotenuse, form the square of the edge length and cut off the triangle four times as in the figure on the left. What remains is a square of the area. If you cut off the same four triangles in the way indicated on the right, the smaller squares with the areas and remain. Since the remainders must be the same, the assertion follows.

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

and

Proof: From follows with the height theorem and analogously