A kosarad üres
A few basic geometric concepts
(reminder)
- A straight line is divided into two rays by a point lying on the straight line. (Figure 8)
- Two points of a straight line define a line segment. (Figure 9)
Figure 8
\latex{ P }
Figure 9
\latex{ A }
\latex{ B }
Figure 10
\latex{ e }
- A plane is divided into two half-planes by a straight line lying on the plane. (Figure 10)
- The space is divided into two half-spaces by a plane. (Figure 11)
- Two rays starting at a given point divide the plane into two parts. These plane parts are called angular domain or shortly angle. (Figure 12)
If we rotate a ray in the plane about its starting point in any direction, then the angle defined by the starting and the ending position of the ray as arms is called the rotation angle. (Figure 13)
The rotation angle is positive, if we rotate in the counter-clockwise or anti-clockwise direction, it is negative, if we rotate in the clockwise direction. (Figure 14)
The rotation angle is positive, if we rotate in the counter-clockwise or anti-clockwise direction, it is negative, if we rotate in the clockwise direction. (Figure 14)
Figure 11
\latex{ S }
 \latex{\alpha} arm of an angle arm of an angle angular domain vertex (plural:vertices) Figure 12 \latex{ O } |  \latex{\alpha} Figure 13 \latex{ O } |  \latex{\alpha} \latex{\beta} rotating in the positive direction rotating in the negative direction \latex{\beta\lt0} \latex{\alpha\gt0} counter-clockwise or anti-clockwise clockwise Figure 14 \latex{ O } \latex{ O } |
Types of angles
\latex{0^{\circ}\lt\alpha\lt 90^{\circ}}
\latex{\alpha=90^{\circ}}
\latex{90^{\circ}\lt\alpha\lt 180^{\circ}}
\latex{\alpha=180^{\circ}}
\latex{180^{\circ}\lt\alpha\lt 360^{\circ}}
\latex{\alpha=360^{\circ}}
acute angle
right angle
obtuse angle
straight angle
concave (reflex) angle
complete angle
convex angles
\latex{0^{\circ}\lt\alpha\leq180}
\latex{0^{\circ}\lt\alpha\leq180}
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
\latex{\alpha}
Figure 15
Special angle pairs
- If two convex angles have a common vertex, and their arms are mutually each other's extensions, then these are called vertical angles. (Figure 16) Vertical angles are of equal size.
- If two convex angles have one arm in common and the other two arms make up a straight line, then these are called adjacent angles. (Figure 17) The sum of adjacent angles is \latex{180^{\circ}}.
- If the sum of two angles is \latex{180^{\circ}}, then these are called supplementary angles.
\latex{\alpha}
\latex{\alpha'}
\latex{\beta}
\latex{\beta'}
Figure 16
\latex{\alpha=\alpha'};
\latex{\beta=\beta'}
- If the sum of two angles is \latex{90^{\circ}}, then these are called complementary angles.
- If the arms of two convex or two concave angles are mutually in the same direction, then these are called corresponding angles, if the arms are mutually in the opposite direction, then these are called alternate angles (Figure 18). (\latex{\alpha} and \latex{\beta} are corresponding angles, \latex{\beta} and \latex{\gamma} are alternate angles.) Corresponding angles and alternate angles are of equal size.
- Another type of special angle pairs is made of two angles the arms of which are mutually perpendicular, but these do not have an own name in English.
\latex{\alpha+\beta=180^{\circ}}
\latex{\alpha}
\latex{\beta}
Figure 17
If both of the angles with mutually perpendicular arms are acute angles or both are obtuse angles, then they are of equal size; if one of them is an acute angle, the other one is an obtuse angle, then these are supplementary angles. (Figure 19)
\latex{\alpha}
\latex{\beta}
\latex{\gamma}
\latex{\delta}
Figure 19
\latex{\beta}
\latex{\alpha}
\latex{\alpha=\beta}
\latex{\delta=\gamma}
\latex{\alpha+\beta=180^{\circ}}
\latex{\alpha}
\latex{\beta}
\latex{\gamma}
\latex{\alpha=\beta}
\latex{\beta=\gamma}
Figure 18
Distance
- The distance between two points is the length of the line segment connecting the points. (Figure 20) Possible notations: \latex{AB, \;d(A; B); \;d_{AB}}
Figure 20
\latex{ A }
\latex{ B }
Note: \latex{ AB } can denote line segment \latex{ AB } itself, but it can also denote the distance between point \latex{ A } and point \latex{ B }. It should be unambiguously clear from the context in which sense it is used.
- The distance from a point to a straight line is the distance between the given point and the base point of the line segment through the point and perpendicular to the given straight line (Figure 21). (If we take the distances between each point of the straight line and point \latex{ P }, then distance \latex{ PT } is the shortest among these.)
If \latex{P\in e}, then \latex{d(P; e)=0}.
\latex{d(P; e)= d(P; T)= PT}
\latex{ P }
\latex{ T }
\latex{ e }
Figure 21
- The distance of two parallel straight lines is the distance from an arbitrary point of one of the straight lines to the other straight line. (Figure 22)
- The distance of two intersecting straight lines is \latex{ 0 }.
- The distance from a point to a plane is the distance between the given point and the base point of the line segment through the point and perpendicular to the plane.
If \latex{P\in S}, then \latex{d(P; S)=0}.
So that this definition does have a meaning we have to determine what it means that a straight line is perpendicular to a plane.
Figure 22
\latex{d(e; f)= d(P; f)= d(Q; e)= PQ}
\latex{ P }
\latex{ Q }
\latex{ f }
\latex{ e }
DEFINITION: A straight line is perpendicular to a plane, if it is perpendicular to every straight line of the plane.
Since there are infinitely many straight lines among the straight lines of the plane which make a pair of skew lines with the given straight line, we also have to define the perpendicularity of two skew lines.
Figure 23
\latex{g\cap e=M}
\latex{f\parallel g, \; g\bot e}
\latex{f\parallel g, \; g\bot e}
\latex{f\bot e}
\latex{\Downarrow}
\latex{ M }
\latex{ e }
\latex{ g }
\latex{ f }
DEFINITION:Two skew lines are perpendicular to each other if one of the straight lines and a straight line through an arbitrary point of the first straight line and parallel with the second straight line are perpendicular to each other. (Figure 23)
Figure 24
\latex{d(P; S)= d(P; T)= PT}
\latex{ P }
\latex{ T }
\latex{ S }
The following can be proven:
THEROEM: If a straight line is perpendicular to two nonparallel coplanar straight lines, then the straight line is perpendicular to the plane. (Figure 24)
- The distance of two parallel planes is the distance from an arbitrary point of one of the planes to the other plane. (Figure 25)
- The distance of two intersecting planes is \latex{ 0 }.
Note: From the above examples it can be seen that we chose the shortest from the distances between the points of the given point sets as the distance of the two point sets. It happens in the same way when defining the distance between less special point sets, except for the cases when there is no shortest distance. Later on we are going to see an example of such a case, then we are going to come back to the definition of distance.
S1
Figure 25
S2
\latex{d(S_1; S_2)=d(P; S_2)=\\d(Q; S_1)=PQ}
\latex{ P }
\latex{ Q }
Exercises
{{exercise_number}}. A straight line and points \latex{ A }, \latex{ B }, \latex{ C }, \latex{ D } and \latex{ E } lying on the straight line in this order from left to right are given. Let us mark the following point sets on the straight line. (When we say line segment we mean a closed line segment, i.e. the two end-points of the line segment are included.)
a) \latex{AB\cup BC}
b) \latex{BC\cap AD}
c) \latex{AC\cup BE}
d) \latex{(AC\cup BD)\cap CE}
{{exercise_number}}. Into how many parts is a straight line divided by
a) \latex{3;}
b) \latex{5;}
c) \latex{10;}
d) \latex{n}
distinct points of its own? How many line segments and how many rays do we get in each of the cases?
{{exercise_number}}. How many points did we give on the straight line, if these points divide the straight line so that there would be
a) \latex{5;}
b) \latex{9;}
c) \latex{20;}
d) \latex{n}
pieces of line segments among the parts which can have at most a common end-point?
{{exercise_number}}. How many points did we give on the straight line, if these points define
a) \latex{1;}
b) \latex{3;}
c) \latex{6;}
d) \latex{15;}
e) \latex{55}
line segments? (The above numbers give the number of all possible line segments.)
{{exercise_number}}. How many straight lines are defined in the plane by
a) \latex{2;}
b) \latex{5;}
c) \latex{7;}
d) \latex{10;}
e) \latex{n}
points no three of which are collinear? Create a table and formulate your assumption.
{{exercise_number}}. How many intersection points are defined in the plane by
a) \latex{2;}
b) \latex{4;}
c) \latex{6;}
d) \latex{10;}
e) \latex{n}
straight lines, if no two of them are parallel, and there is no point lying on the plane through which more than two straight lines pass?
{{exercise_number}}. Points \latex{ A }, \latex{ B }, \latex{ C } and \latex{ D } in this order are points of a straight line. Fill in the missing cells in the table.
\latex{AB}
\latex{BC}
\latex{CD}
\latex{AC}
\latex{BD}
\latex{AD}
\latex{3 \;m}
\latex{40 \;cm}
\latex{5 \;m}
\latex{8\;m}
\latex{1\;cm}
\latex{3\;cm}
\latex{7\;cm}
\latex{10\;cm}
\latex{60\;cm}
\latex{2\;mm}
\latex{11\;km}
\latex{2\;cm}
\latex{13\;km}
\latex{18\;km}
\latex{3.3\;cm}
{{exercise_number}}. The ratio of two angles with a common arm is \latex{2 : 7}. One of the angles is \latex{100^{\circ}} greater than the other one. Give the sum of the two angles.
{{exercise_number}}. Straight lines \latex{ a } and \latex{ b } in the figure are parallel. Give the size of the angles denoted by the arcs, if we know that angle \latex{\alpha} is
- \latex{ 30º };
- \latex{ 60º11’ };
- \latex{ 48º };
- \latex{ 53.2º }.
\latex{\alpha}
\latex{ b }
\latex{ a }
{{exercise_number}}. \latex{\alpha } and \latex{\beta} are two angles with one common arm. Their sum is 250º, and we know that the extension of the arm of angle a not common with angle b divides angle b into two parts in the ratio of
a) \latex{1: 2;}
b) \latex{1:3;}
c) \latex{2:3}.
Give the two angles.
{{exercise_number}}. An aeroplane takes off from the airport in the direction of west, then it turns to north-west. After flying for one and a half hours it turns by \latex{ –45º }, then after half an hour it turns by \latex{ +60º }. In what direction is it heading now compared to the original direction (west)?
