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| Geometry |
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Sets of
Points in a Plane
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Line, a half-line and line segment
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The line segment bisector
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Angle, measurement of angles
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Complementary, supplementary,
alternate and vertically opposite angles
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Angle bisector
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Construction of angles; 60°, 30°, 90°
and 45° (degrees)
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Points,
Lines and Planes in Three-Dimensional (3D) Space |
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Position
of two lines in a three-dimensional space
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Coplanar lines and skew lines
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Line and plane in a three-dimensional space
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Two planes in a
three-dimensional space
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The line of intersection of two planes
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Orthogonality of line and plane
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Orthogonality of two planes
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Orthogonal projection of a point onto a plane
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The distance from a point to a plane
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Angle between a line
and a plane, and angle between two planes |
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| Line, a half-line and line segment
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| Line is a straight one dimensional geometrical figure of infinite
length. |
| Any
pair of points uniquely determine a line, of which the segment
between given points is the shortest path between them. |
| The
intersection
point A is a common point of two lines. |
| A line divides a plane into two
half-planes. |
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| Two lines are called
parallel lines, if they lie in the same plane and don’t intersect. |
| The point
A
which lies on a line divides the line into the two half-lines (rays). |
| The part of a line between two points is called a
line segment. The line segment AB
is the part of the line
l. |
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| The line segment bisector
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| The line segment bisector crosses the segment through the
midpoint, forms a right angle and each of its points is equidistant from the segment's endpoints. |
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| Angle, measurement of angles
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| Angle is the figure formed by two half-lines or rays OA and OB
(sides of an angle) sharing a common endpoint O
called the vertex of the angle. An angle is signed AOB. |
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| The unit of angles measure is a degree,
1° (deg). The angle of a
full circle (complete revolution) equals 360 degrees. |
| One degree is divided by 60 minutes,
1° = 60', one minute by 60 seconds,
1' = 60". |
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| Complementary, supplementary,
alternate and vertically opposite angles
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| Complementary angles are two angles that have the common
vertex, one common side and whose sum is 90 degrees. |
| A sum of
supplementary (adjacent) angles is equal to 180 degrees. |
| Any
pair of angles formed between two intersecting lines and lying
on the same side of one of them are called adjacent angles.
The adjacent angles of two intersecting lines supplement each
other. |
| Two
angles having a sum of 360 degrees are called conjugate
( or explementary) angles. |
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| When two lines intersect at one point, they form four angles. The
opposite angles on the figure above are called vertical angles. Vertical angles have the same measure. |
| Angles with correspondingly
parallel sides either are equal one to another, or sum of them is 180
degrees. |
| Parallel lines intersected by a transversal
(traverse) form the angles which are equal or supplementary. |
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| The angles contained between two given lines lying on opposite
sides of the transversal are called alternate angles. |
| Two angles
with mutually perpendicular sides are equal or supplementary. |
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| Angle bisector
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| An
angle bisector is a straight line that divides an angle into two
equal angles. |
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| Construction of angles; 60°, 30°, 90°
and 45° (degrees)
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| Position
of two lines in a three-dimensional space
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| Exactly one line passes through two different points in a space. |
| Three points not on the same line
define
a plane. |
| A plane is also
defined by: |
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- a line and a point not on the line
- a point and a line, which is normal/perpendicular to the plane |
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- two lines which intersect
- two lines which are parallel. |
| Two lines are parallel if they
do not intersect and they are coplanar (lie on the same plane). |
| Two
lines which do not intersect but which are not coplanar are called
skew lines. |
| Skew lines do not lie in a single plane together. |
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| The relations between points, lines and planes are represented by sides, edges, diagonals and vertices of the rectangular
parallelepiped (cuboid) shown in the pictures below. |
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| Line and plane in a three-dimensional space
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| A line l intersects a plane
P
at a point A, as shows the left picture. |
| A line is defined to be
parallel to a plane if the line and the plane are disjoint (empty
intersection). That is, there is a line in the plane which is parallel to
the given line, as shows the right picture. |
| The line
k below lies in the plane P1
and P2 . |
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| Two planes in a
three-dimensional space
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| Two distinct planes either are disjoint (empty intersection) or
intersect in a line. |
| Two disjoint planes are called parallel. |
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| Orthogonality (or perpendicularity) of line and plane
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| A line is perpendicular to a plane if it is perpendicular to every line
in the plane that passes through its intersection point. |
| The plane
P2
is perpendicular to the plane P1
if there is a line in
P2
which is perpendicular to the plane
P1 . |
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| Orthogonal projection of a point onto a plane
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| Orthogonal projection of a point
P
onto a plane is the intersection point P' of the perpendicular drawn from
P
to the plane of projection. |
| Orthogonal projection of a line segment is a line segment or point
depending of the position of the line segment in relation to the plane
of projection. |
| The projection will be a point if the line segment is
perpendicular to the plane of projection. |
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| The distance from a point to a plane is the distance from the
point to its orthogonal projection to the plane, i.e.,
AA'
=
d(A,A'). |
| For
example, if
the surface of a triangle and the plane of projection are
mutually perpendicular, as is shown in the picture below, then
the projection of the triangle is a line segment, otherwise the
projection is a triangle. |
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| Angle between a line
and a plane, and angle between two planes |
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angle between a line and a plane is the angle between the line
and its orthogonal projection to the plane. |
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angle between two
planes is the angle between two lines, one lying in each plane, drawn perpendicular to the intersection of the planes at the same
point, as is shown below. |
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| Geometry
and use of trigonometry contents |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
