Geometry Postulates and Properties for Proofs 

Reflexive Property 

  • A segment or angle is congruent to itself.

    • Ex A=A

Substitution Postulate 

  • DO NOT CONFUSE WITH SUBTRACTION POSTULATE

  • If two things are congruent to the same thing then they are congruent to each other. 

    • Ex. If AB≅DC, then AB≅EF, EF≅DC

    • Ex:

x=y

x+5=7

y+5=7

Addition Postulate 

  •  If you add the same thing to two equal things then the result is equal. 

    • Ex. If A=B, then A+3=B+3 or If A=B, then A+C=B+C.

    • Ex. for fig. 1:  If AB≅CD, then AD-CD=CB-CD

fig 1.

Subtraction Postulate 

  •  If you subtract the same thing from two equal things then the result is equal. 

    • Ex. If A=B, then A-3=B-3 or If A=B, then  A-C=B-C.

    • Ex. for fig. 2: If AD≅CB then AD-CD≅CB-CD


Segment Bisector

  • A line that intersects a segment and cuts it into two congruent parts.

    • Ex. Line segment AB intersects Line segment CD at point E. CE≅ED and CE+ED=CD

Angle Bisector

  • A line (or part of a line) that divides an angle into two congruent parts.

    • Ex. Ray BD bisects ∠ABC. ∠ABD≅DBC.



Median

  • A line segment that goes from the vertex of a triangle to the MIDPOINT of the opposite side.

    • Ex. AD is the median of BC. D is the midpoint of BC


Altitude

  • A segment that goes from the vertex of a triangle and is PERPENDICULAR to the opposite side.

    • Ex. BD is an altitude drawn to AC. ∠BDA and ∠BDC are right angles. ∠BDA≅∠BDC

Isosceles Triangle

  • A triangle with exactly two congruent sides and two congruent angles.

    • Ex. AB≅BC and ∠A≅∠C

Right Triangle

  • A triangle with a right angle

    • Ex. △ABC is a right △  because  it has a right angle.

Equilateral Triangle

  • A triangle with three congruent sides and three congruent angles.

    • Ex. AB≅BC≅AC and ∠A≅∠B≅∠C

Symmetric Property

  • Quantities are equal forwards and backwards.

    • Ex. AB≅BA and ∠ABC≅∠CBA

Transitive Property

  • Two quantities equal to the same quantity are equal to each other.

  • There has to be a diagonal between the first and second line otherwise it is Substitution Postulate.

    • Ex. A=B

       B=C

   ∴A=C

  • Not transitive:

    A=B

    C=B

∴A=C

Segment Partition

  • Part+Part=Whole

    • AB+BC=AC

Angle Partition

  • Smaller angle + Smaller angle = Whole angle

    • ∠ABD +∠DBC ≅ ∠ABC