Yes, they do exist.
A triangle that has one right angle and is isosceles is called an isosceles right triangle.
Isosceles
An isosceles triangle can have a 90 degree angle, but doesn't necessarily. Similarly, a right triangle can be an isosceles triangle, but isn't necessarily.
It is the shape formed by three non-collinear points on the surface of a sphere which are joined together by straight lines in which two of the sides are of the same length. An example might be a triangle from the earth's North pole to two points on the same latitude - specially the equator.
There can be only one pair of perpendicular lines in an isosceles triangle when it is a right angled isosceles triangle. A non right angled isosceles triangle has no pairs of perpendicular lines.
equilateral triangle (and so, if it is in Euclidean (plane) space, it has 3 angles all of which are 60°) otherwise (in Hyperbolic or Spherical space) it is an isosceles triangle.
An acute triangle can be an isosceles triangle, but it doesn't have to be. An isosceles triangle can be an acute triangle, but it doesn't have to be.
A right triangle that is also isosceles cannot have an angle of 57 deg 24 sec unless it is in spherical - as opposed to plane, or flat - space.
isosceles triangle
A right triangle and an isosceles triangle have the fact that they are both triangles in common. A right triangle can also be an isosceles triangle.
An isosceles triangle is a triangle that has two equal sides.
Are isosceles triangle sometimes an equilateral triangle
You cannot. An isosceles triangle cannot be scalene and a scalene triangle cannot be isosceles. So an isosceles scalene triangle cannot exist.
how is an isoscelels trapezoid related to a isosceles triangle
An obtuse triangle is not necessarily isosceles, but it can be.
In an isosceles triangle and an isosceles trapezoid, both base angles are congruent
A triangle that has one right angle and is isosceles is called an isosceles right triangle.