Angles
Angle. Degree and radian measures of an angle.
Right (direct), acute and obtuse angle. Mutually
perpendicular straight lines. Signs of angles.
Supplementary (adjacent) angles. Vertically
opposite (vertical) angles. Bisector of an angle.
Angle is a geometric figure ( Fig.1 ), formed by two rays OA and OB ( sides of an angle ), going out of the same point O (a vertex of an angle).
An angle is signed by the symbol  and three letters, marking ends of rays and
a vertex of an angle: AOB (moreover, a vertex letter is placed in the middle).
A measure of an angle is a value of a turn around a vertex O, that transfers a ray OA to the position OB. Two units of angles measures are
widely used: a radian and a degree. About a radian measure see below in the point “A length of arc” and also in the section
“Trigonometry”.
A degree measure. Here a unit of measurement is a degree ( its designation is ° or deg ) – a turn of a
ray by the 1/360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg. One degree is divided by
60 minutes ( a designation is ‘ or min ); one minute – correspondingly by 60 seconds ( a designation is “ or sec ).
An angle of 90 deg ( Fig.2 ) is called a right or direct angle; an angle lesser than 90 deg ( Fig.3 ), is called an acute
angle; an angle greater than 90 deg ( Fig.4 ), is called an obtuse angle.
Straight lines, forming a right angle, are called mutually perpendicular lines. If the straight lines AB and MK are perpendicular, this is signed as:
AB  MK.
Signs of angles. An angle is considered as positive, if a rotation is executed
opposite a clockwise , and negative – otherwise. For example, if the ray OA displaces to the ray OB as shown on Fig.2, then
AOB = + 90 deg; but on Fig.5
AOB = – 90 deg.
Supplementary (adjacent) angles ( Fig.6 ) – angles AOB
and COB, having the common vertex O and the common side OB;
other two sides
OA
and
OC
form
a
continuation
one
to
another.
So,
a
sum of supplementary (adjacent) angles is equal to 180 deg.
Vertically opposite (vertical) angles ( Fig.7) – such two angles with a common vertex, that sides of one angle are continuations
of the other:
AOB and COD
( and also AOC and
DOB ) are vertical angles.
A bisector of an angle is a ray, dividing the angle in two ( Fig.8 ). Bisectors of vertical angles (OM and ON, Fig.9) are
continuations one of the other. Bisectors of supplementary angles (OM and ON, Fig.10) are mutually perpendicular lines.
The
property of an angle bisector:
any point of an angle bisector is placed by the same distance from the angle
sides.
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