No, but a square pyramid does.
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Yes its base contains 4 corner right angles
Shadow of a building have right angles at the base of the building. Braces of buildings have right angles.
Sum of exterior angles: 360 degrees Sum of interior angles: 180 degrees The square of its hypotenuse is equal to the sum of its base squared plus its height squared.
A lot of things. For example, a cube has 8 vertices, the square pyramid has 5. A cube also has more edges and faces than the pyramid. All the faces of the cube are squares, while the pyramid only has one square face. Also, all the angles in the cube are right angles, while the angles in a square pyramid (in the usual sense of a pyramid where the apex is directly above the center of the base) are acute. In any case, they are not all right angles.