Download PLANE GEOMETRY 1 (In the line of Elements, ignoring definitions

PLANE GEOMETRY 1
(In the line of Elements, ignoring definitions and theorems that lack modern rigour and adding
things that are needed for completion. Book 1)
Undefinables Point, Any one of those 3 things
Definitions
The remaining two of those three things, Equal lengths, Smaller length, Greater
length, Angle, Equal angles, Smaller angle, Larger angle, Triangle, Equilateral,
Isosceles, Congruent triangles, Plane, Intersecting lines, Perpendicular lines,
Right angle, Straight angle, Adjacent angles, Opposite angles, Exterior angle,
Transversal, Alternate angles, Parallelogram, Rectangle, Square
Axiom 1
Axiom 2
Axiom 3
Axiom 4
: Two distinct points determine a unique straight-line
: SAS congruence
: The fifth postulate
: There exists a unique circle with centre as any given point and radius as any given line
Theorem 1
Theorem 2
: In an isosceles traingle, angles at the ‘base’ are equal.
: In a triangle, the sides opposite to equal angles are equal.
Theorem 3
: SSS congruence
Construction 1 : Bisect a given angle.
Construction 2 : Bisect a given line-segment.
Construction 3 : Construct a line through a given point and perpendicular to a given line.
Lemma 1
: If adjacent angles form two right angles, the not common arms form a straight line.
Theorem 5
Theorem 6
Theorem 7
Theorem 8
: Exterior angle of a triangle is greater than opposite interior angle.
: In a triangle, the angle opposite to greater side is greater
: In a triangle, the side opposite to greater angle is greater
: Two sides in a triangle is greater than the third.
Lemma 2
: Two lines on a side of a traingle meeting within the triangle are lesser than the other two
sides of the triangle but form a greater angle than these sides of triangle.
Construction 4 : Construct a triangle whose sides are given.
Construction 5 : Construct a given angle on a given line and at a given point on the line.
Lemma 3
: If in a triangle an angle/side is increased, the side/angle opposite to it is increased.
Theorem 9
: ASA congruence
Theorem 10
: If, and only if, alternate angles are equal or corresponding angles are equal or interior
angles on the same side of the transversal form two right angles then the lines are parallel.
Construction 6 : Construct a line parallel to a given line and passing through a given point
Theorem 11
Theorem 12
: Exterior angle of a triangle is opposite interior angles.
: Angles in a triangle are two right angles.
Theorem 13
: In a parallelogram, opposite sides and angles are equal and diagonals bisect each other.
Lemma 4
: Diagonal of a parallelogram divides it into congruent triangles.
Theorem 14
: Parallelograms on the same base and between the same parallels are equal and are double
the triangle on the same base and between the same parallels
Theorem 15
Theorem 16
: Pythagoras theorem
: Converse of the Pythagoras theorem