Angles formed by crossing lines

author: Петя
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Българска версия:

§   When lines а, b and  с are intercepting each other then 8 angles are formed which have following designations:

а) alternate angles: angle 3 and angle 5; angle 4 and angle 6 ; angle 1 and angle 7; angle 2 and angle 8

b) corresponding angles: angle 1 and angle 5; angle 2 and angle 6; angle 3 and angle 7; angle 4 and angle 8;

c) adjacent angles: angle3 and angle 6; angle 4 and angle 5; angle 1 and angle 8; angle 2 and angle 7 

 

§ Axiom for parallel lines: Trough a point which doesn't lie on a given line, not more than one line which is parallel to the given line can go trough it.

§ Theorems - corollaries from the axiom

• If two lines are parallel separately to another line then the first two lines are parallel to each other.

• If two lines are parallel and there is another line that crosses one of those lines, then the third line crosses the other given line, too.

§Theorems - signs for parallelism of two lines

• If two lines cross a third line and one pair alternate angles are equal then the lines are parallel.

• If two lines cross a third line and one pair corresponding angles are equal then the lines are parallel.

• If two lines cross a third line and one pair adjacent angles have a sum 180° then the lines are parallel.

• If two lines are perpendicular to another line then these lines are parallel.

§Theorems - characteristic of two parallel lines:

If we cross two parallel lines with a third one, then:

1.      each pear alternate angles are equal;

2.      each pear corresponding angles are equal;

3.      each pear adjacent angles have a sum of 180°.

 

§ Theorems - corollaries:

• If one line is perpendicular to one of two parallel lines then it is also perpendicular to the other line.

•   Trough a point which does not lie on a given line, only one line can go which is perpendicular to the given line.

§   A length from point М to line а is the length of a segment ММ₁ with end point М and heed М₁ on the perpendicular which is let down from point М to line а.

§   Two angles which rays lies respectively on parallel lines, are called angles with mutually parallel rays.

§ If two angles are with mutually parallel rays and: 

а) the angles are from one kind (acute or obtuse) then they are equal;

b) the angles are from different kinds then their sum is equal on 180°;