Geometry axioms used in proofs
Last update current as of January 4th, 2003
developed by K. Steinman – mathematics graduate student L.I.U
The reason for the creation of this website is because of a lack of a comprehensive lists of “reasons” used in common geometry proofs, such as the triangle proofs in High Schools. There has not been any textbook I have come across that attempts to list all of the axioms. In my quest for a list I have developed the following list which will be updated. This list also includes axioms for circle and parallelogram proofs.
1)reflexive property a=a
2)symmetric property a=b then b=a
3)transitive property [ a=b b=c then a-c ]
4)Substitution postulate (x=4y y=k) --> x=4k
5)Partition postulate: a whole = sum of all its parts
6)Addition postulate (a=b and c=d) then (a+c =b+d)
7)Subtraction postulate (a=b and c=d) then (a-c=a-d)
8)multiplication postulate: (a=b and c=d) then ac=bd
9)division postulate (a=b and c=d) then a/c = b/d
10)powers postulate (a=b) then aⁿ=bⁿ
11)roots postulate (a=b) then a^(1/n)=b^(1/n)
12)Motion postulate: a geometric figure can be moved (copied) without change
13)Extension postulate: line segments can be extended
14)2 points determine a line
15) 2 lines intersect in at most 1 point
16) At any given point only one perpendicular to a line can be drawn
17)The sum of t measures of angles with 1 vertex – 360
18)The sum of angles, with a common vertex, on one side of a line =180
Some simple angle Theorems
19)All right angles are congruent
20)All straight angles are congruent
21)If 2 angles are supplements of the same angle, the are congruent
22)If 2 angles are complements of the same angle, the are congruent
23)If 2 angles form a liner pair, they are supplementary
24)Vertical angles are congruent
25)If 2 angles are congruent, their complements and supplements are congruent
26)Corresponding parts of congruent triangles are congruent (CPCTC)
S.A.S is congruent to S.A.S
A.S.A is congruent to A.S.A
S.S.S is congruent to S.S.S
Base angles of an isosceles triangles are = (if 2 sides are congruent, the sides opposite are congruent)
The bisector of the vertex angle of isosceles triangle is a median to the base (bisects the base)
The bisector of the vertex angle of isosceles triangle is perpendicular to the base
Every equilateral triangle is equiangular
If intersecting lines form congruent adjacent angles, the are perpendicular
Any point on the perpendicular bisector of a segment is equidistant from its endpoints
if 2 points are equidistant from the endpoints of a segment, the pts. Determine the perpendicular bisector.
Coplaner line are either parallel or intersecting
Reflexive symmetric and transitive properties apply to parallel
There is a unique line through a five point not on another line, parallel to any other line
If a line intersects 1 of 2 parallel lines, it intersects the other
Iff alternate interior angles are congruent, lines are parallel
Iff corresponding angles are congruent, lines are parallel
iff interior angles on the same side of a transversal are supplementary, lines are parallel
Sum of degree measures of angles of a triangle is 180º
If 2 angles of triangles are congruent, the third angle is congruent
Each angle of an equilateral triangle is 60º
Each acute angle of an isos. Rt. Triangle =45º
Sum of all angles of a quadrilateral is 360º
A.A . S = A.A . S
2 rt. Triangles are congruent if hypotenuse and acute angle congruent hypotenuse. And acute angle [by a.a.s]
H.L = H.L
The exterior of a triangle = sum of measure of remote interior angles
Sum of interior angles of n-gon = 180(n-2)
Sum of exterior angles of a polygon = 180
A diagonal divides a parallelogram into two equal triangles
Two consecutive angles of a parallelogram are supplementary
opposite
angles of a quadrilateral are equal iff it is a parallelogram
