Geometry Theorems
Midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
Definition of Supplementary Angles
2 angles whose sum is 180
Between any two points there is exactly one line
A and B lie in plane JOG; therefore A and B are collinear
Definition of Midpoint
A point that divides a segment into two congruent segments
Reason sides are congruent
Given, def of midpoint, reflexive property
Reasons angles are congruent
Given, vertical angles, AI AE corresponding angles (must have parallel lines), def of angle bisector
Definition of congruency
IF segment AB is congruent to segment PQ, then AB=PQ
Definition of Segment Bisector
If A bisects B, then CA=AT
Definition of Equilateral Triangle
If ABC is equilateral, AB=BC=AC
Segment Addition Postulate
If B is between A and C, then AB + BC = AC
Definition of Isosceles Triangle
If DEF is isosceles with vertex angle E, then DE=EF
Definition of rift triangles
If R and T are right angles, then R=T
Transitive (syllogism)
If a=b and b=c then a=c
Addition Property of Equality
If a=b, then a+c=b+c
Subtraction Property of Equality
If a=b, then a-c=b-c
Multiplication Property of Equality
If a=b, then ac=bc
Substitution Property of Equality
If a=b, then b can be substituted for a in any expression
Law of Detachment
If p->q is true and p is true, then q is true
right angle
If two angles are complementary and supplementary then each is a
Congruent Complements Theorem
If two angles are complementary to the same angle, then they are congruent.
Congruent Supplements Theorem
If two angles are supplementary to the same angle, then they are congruent.
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then pairs of alternate exterior angles are congruent
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
consecutive interior angles (aka same side interior angles)
If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary.
Definition of Complementary Angles
Two angles whose sum is 90 degrees
Vertical Angles Theorem
Vertical angles are congruent
Distributive property (sum)
a(b + c) = ab + ac
Distributive property (difference)
a(b - c) = ab - ac
Reflexive Property
a=a
AAS
angle angle side
ASA
angle side angle
CPCTC
corresponding parts of congruent triangles are congruent
Distance formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Angle Addition Postulate
for any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts
HL
hypotenuse leg
Division Property of Equality
if a = b and c is not equal to 0, then a/c = b/c
Symmetric Property
if a=b, then b=a
SSS~
if all corresponding sides are proportional, then the triangles are similar
AA~
if two corresponding angles are congruent, then triangles are similar
SAS~
if two corresponding sides are proportional and the individual angles are congruent, then the triangles are similar
SAS
side angle side
SSS
side side side
