Download Plane Geometry

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The Building Bloks of Geometry
In the history of architecture geometric rules based on the ideas of proportions
and symmetries formed fixed tools for architectural design
Sacred geometry
St. Vitus Cathedral
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Golden ratio (divine proportion)
ab a
 
a
b

1 5
 1.618
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Architecture was profoundly influenced by the golden ratio. The Egyptian
Pyramids, Stonehenge, the Acropolis, all gothic cathedrals,… these
architectural wonders and many more are said to have been designed and
built using the golden ratio.
Euclid: Elements, book II
Proposition 11
To cut a given straight line so that the rectangle
contained by the whole and one of the segments equals
the square on the remaining segment.
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Here are some definitions you will need to remember.
• Point – names an exact location on a plane.
A point is that which has no part.
Euclid: Elements, book I
• Line – a collection of points forming a straight path that extends
infinitely in opposite directions.
A line is breadthless length.
A straight line is a line which lies evenly with the points on itself.
Euclid: Elements, book I
• Plane – a perfectly flat surface that extends forever in all
directions.
• Segment – part of a line between two endpoints.
• Angle – formed by 2 rays with a common endpoint called a vertex.
Pleural of vertex is vertices.
Right triangle
Parallelogram
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Congruent - figures that have the same size and shape.
Segments that have the same length are congruent.
Angles that have the same measure are congruent.
The symbol for congruence is  , which is read “
congruent to”.
Types of angles
Acute angle - any angle which measures less
than 90°
Right angle - any angle which measures
exactly 90°
Obtuse angle - any angle which measures
>90°, but <180°
Straight angle - any angle which measures
exactly 180°
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Vertical angles – 2 angles formed by
intersecting lines. They are always equal
in measure. They are across from one
another.
Angle 1 , 3 are vertical angles.
Angle 2 , 4 are vertical angles.
Angle 1 and angle 2 are not vertical.
Supplementary angles – 2 angles whose
measures add up to 180°. Supplementary
angles can be placed so that they form a
straight line.
Angle 1 and angle 2 are supplementary.
The line passing through points A, B, and C
is a straight line.
True or False
• If 2 distinct lines do not intersect, then they are
parallel.
• If 2 lines are parallel, then a single plane contains them.
• If 2 lines intersect, then a single plane contains them.
• If a line is perpendicular to a plane, then it is
perpendicular to all lines in that plane.
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Solids
vertex
edge
face
cylinder
pyramid
cone
prism
cube
tetrahedron
sphere
octahedron
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Angle between line and plane
Dihedral angle (Двугранный угол)
-angle between two intersecting planes
Angle between two planes (α, β, green)
in a third plane (pink) which cuts the
line of intersection at right angles.
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Euclid: Elements, book I
Proposition 1:
To construct an equilateral triangle on a given finite
straight line.
Euclid: Elements, book I
Proposition 46
To describe a square on a given straight line.
Let AB be the given straight line.
It is required to describe a square on the straight line
AB.
Construction:
Draw AC at right angles to the straight line AB from
the point A on it. Make AD equal to AB. Draw DE
through the point D parallel to AB, and draw BE through
the point B parallel to AD
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Euclid: Elements, book II
Proposition 14
To construct a square equal to a given rectilinear figure.
Let A be the given rectilinear figure
Construct the rectangular parallelogram BCDE equal to the rectilinear figure A
(Book I, prop 45).
To construct a square equal to a given paralellogram.
Geometric mean theorem
Geometric mean of the two
segments equals the altitude.
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http://geotest.geometry.cz/2016/?lang=EN
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