Short answers, pulled from the story.
An angle is a figure formed by two lines, or rays, that meet at a common point. Each line is called a side of the angle, and the shared point is called the vertex. The term also denotes the size or magnitude of that figure.
The common angles are the zero angle at 0 degrees, the acute angle below 90 degrees, the right angle at exactly 90 degrees, the obtuse angle between 90 and 180 degrees, the straight angle at 180 degrees, the reflex angle between 180 and 360 degrees, and the full angle at 360 degrees. An angle that is not a multiple of a right angle is called oblique.
Angles are commonly measured in degrees, radians, and turns. One full turn equals 360 degrees or about 6.28 radians, a radian is roughly 57.2957 degrees, and other units include the gradian, arcminute, arcsecond, and milliradian. The degree was set historically so that a straight angle equals 180.
An angle is dimensionless because measuring it in radians divides one length, the arc, by another length, the radius, so the units of length cancel out. In the International System of Quantities the angle is defined as dimensionless, and the radian is dimensionless in the International System of Units.
Pierre Wantzel proved in 1837 that trisecting an angle with only a compass and straightedge cannot be performed for most angles. The ancient Greek mathematicians could bisect any angle and trisect only certain ones.
The vertical angle theorem states that vertically opposite angles, formed when two straight lines intersect, are always equal in measure. Eudemus of Rhodes attributed the proof to Thales of Miletus, who reportedly observed Egyptians measuring vertical angles when drawing intersecting lines.
The full Moon has an angular diameter of approximately 0.5 degrees, or 30 arcminutes, when viewed from Earth. This is roughly the same angular diameter as the Sun seen from Earth.