Geometry How to Draw a Plane
P l a northward e Chiliad eastward o m e t r y
An Adventure in Language and Logic
based on
Dwelling house
Introduction. Geometry: The study of figures
Introduction to Logic.
Hypothesis and conclusion
Necessary and sufficient
Valid arguments
Volume I
First Principles.
Definitions
Postulates
Axioms or Mutual Notions
CONSTRUCTIONS
Proposition one
On a given direct line to construct an equilateral triangle.
Proposition 2
From a given point to draw a straight line equal to a given direct line.
Proffer 3
Given two diff straight lines, to cut off from the longer line
a straight line equal to the shorter line.
CONGRUENT TRIANGLES
Proposition 4 (Side-Angle-Side)
If 2 triangles take two sides equal to two sides respectively, and if the angles contained past those sides are also equal, so the remaining side will equal the remaining side, the triangles themselves will be equal areas, and the remaining angles will be equal, namely those that are opposite the equal sides.
THE ISOSCELES TRIANGLE
Proffer v
In an isosceles triangle the angles at the base are equal.
PROOF Past CONTRADICTION
Proffer vi
If 2 angles of a triangle are equal, and then the sides opposite them volition be equal.
Coinciding TRIANGLES 2
Proposition 8 (Side-Side-Side)
If 2 triangles have two sides equal to two sides respectively, and if the bases are also equal, and then the angles will be equal that are contained by the two equal sides.
BISECTIONS
Proposition nine
To bisect a given bending.
Proposition 10
To bisect a given directly line.
PERPENDICULARS
Proposition 11
To depict a straight line at right angles to a given direct line from a given betoken on it.
Suggestion 12
To a given straight line that may be made as long as we please, and from a given point not on it, to draw a perpendicular line.
Correct ANGLES, VERTICAL ANGLES
Proposition thirteen
When a straight line that stands on another straight line makes ii angles, either it makes two correct angles, or information technology makes angles that together are equal to two correct angles.
Proposition 14
If two directly lines are on contrary sides of a given straight line, and, coming together at one point of that line they brand the adjacent angles equal to two right angles, then the 2 direct lines are in a direct line with 1 another.
Proposition 15
When two directly lines intersect 1 another, the vertical angles are equal.
THE SIDES AND ANGLES OF A TRIANGLE
Proffer 16
If ane side of a triangle is extended, and then the outside angle is greater than either of the reverse interior angles.
Proposition 17
Whatever two angles of a triangle are together less than two right angles.
Suggestion 18
A greater side of a triangle is opposite a greater bending.
Proposition nineteen
A greater angle of a triangle is reverse a greater side.
Suggestion 20
Any two sides of a triangle are together greater than the third side
Proposition 22
To construct a triangle whose sides are equal to three given straight lines: thus whatsoever ii of them taken together must be greater than the third.
Proposition 23
On a given straight line and at a given point on it, to construct an bending equal to a given bending.
Coinciding TRIANGLES 3
Proposition 26 (Angle-Side-Angle)
If 2 triangles take 2 angles equal to two angles respectively, and one side equal to ane side, which may be either the sides between the equal angles or the sides reverse i of them, and then the remaining sides will equal the remaining sides (those that are opposite the equal angles), and the remaining bending will equal the remaining angle.
THE THEORY OF PARALLEL LINES
Proposition 27
If a straight line that meets 2 straight lines makes the alternate angles equal, and so the two straight lines are parallel.
Proffer 28
If a direct line that meets two straight lines makes an exterior angle equal to the opposite interior angle on the aforementioned side, or if it makes the interior angles on the same side equal to 2 right angles, and then the two straight lines are parallel.
Proposition 29
If two direct lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle equal to the contrary interior angle on the same side, and it makes the interior angles on the same side equal to ii right angles.
Proposition thirty
Directly lines that are parallel to the same straight line are parallel
to each other.
THE THREE ANGLES OF A TRIANGLE
Suggestion 31
Through a given bespeak to depict a direct line parallel to a given straight line.
Suggestion 32
If i side of a triangle is extended, then the outside angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to 2 right angles.
PARALLELOGRAMS
Proposition 33
The straight lines which join the extremities on the same side of two equal and parallel straight lines, are themselves equal and parallel.
Proposition 34
In a parallelogram the opposite sides and angles are equal, and the diagonal bisects the surface area.
EQUALITY OF NON-Coinciding FIGURES
Proposition 35
Parallelograms on the same base of operations and in the same parallels are equal.
Proposition 36
Parallelograms on equal bases and in the aforementioned parallels are equal.
Proposition 37
Triangles on the same base and in the same parallels are equal.
Proposition 38
Triangles on equal bases and in the same parallels are equal.
Proposition 39
Equal triangles that are on the same base and on the aforementioned side of it, are in the same parallels.
Suggestion 41
If a parallelogram and a triangle are on the same base of operations and in the same parallels, the parallelogram is double the triangle.
Construction OF A SQUARE
Proposition 46
On a given direct line to describe a square.
THE PYTHAGOREAN THEOREM
Proposition 47
In a right triangle the foursquare drawn on the side opposite the right angle
is equal to the squares drawn on the sides that make the right angle.
Proffer 48
If the square drawn on ane side of a triangle is equal to the squares drawn on the other 2 sides, then the bending contained by those 2 sides is a correct angle.
Additional exercises
champagnecionse86.blogspot.com
Source: https://themathpage.com/aBookI/plane-geometry.htm
0 Response to "Geometry How to Draw a Plane"
Postar um comentário