PLANE GEOMETRY

PLANE GEOMETRY

Theorems, axioms, definitions

Proof. Theorem. Axiom. Initial notions. Definitions.

Proof – a reasoning, determining some property.

Theorem – a statement, determining some property and requiring a proof. Theorems are called also as lemmas, properties, consequences, rules, criteria, propositions, statements. Proving a theorem, we are based on the earlier determined properties; some of them are also theorems. But some properties are considered in geometry as main ones and are adopted without a proof.

Axiom – a statement, determining some property and adopted without a proof. Axioms have been arisen by experience and the experience checks if they are true in totality. It is possible to build a set of axioms by different ways. But it is important that the adopted set of axioms would be sufficient to prove all other geometrical properties and minimal. Changing one axiom in this set by another we must prove the replaced axiom, because now it is not an axiom, but a theorem.

Initial notions. There are some notions in geometry  ( and in mathematics in general ), to which it is impossible to give some sensible definition. We adopt them as initial notions. The meaning of these notions can be ascertained only by experience. So, the notions of a point and a straight line are initial. Basing on initial notions we can give definitions to all other notions.

Straight line, ray, segment

In your thought you can continue a straight line infinitely in both directions.We consider a straight line as infinite. A straight line, limited from one side and infinite from another side, is called a ray. A part of a straight line, limited from both sides, is called a segment.

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 acuteangle; 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.