The Lesson
A right triangle (or a rightangled triangle) is a
triangle containing an angle of 90° (called a
right angle).
The Oxford English Dictionary defines a right triangle as "a triangle containing a right angle."
The Parts of a Right Triangle

A right triangle has one right angle (an angle of 90°). It is shown by drawing a small square.

The longest side (opposite the right angle) is called the hypotenuse.

The other two shorter sides are called legs or catheti (singular cathetus). They are perpendicular to each other.

If another angle is labelled (such as the angle labelled θ in the image below) then we can name the two shorter sides.
The side next to the angle is called the adjacent. The side opposite the angle is called the opposite.Note: The adjacent and opposite only exist in relation to an angle. The same side can be an adjacent or an opposite depending on which angle they refer to.
What's in a Name?
A right triangle gets its name from the fact it has a right angle  an angle of 90°.
The right triangle is called a rightangled triangle in British English.
The Other Angles in a Right Triangle
The 3 angles in a triangle
must add to 180°.
If one angle is 90°, the other two angles must add to 90°.