1
Geometry – Proofs Reference Sheet
Here are some of the properties that we might use in our proofs today:
#1. Definition of Isosceles Triangle – says that “If a triangle is isosceles then TWO or more sides are
congruent.”
#2. Isosceles Triangle Theorem – says that “If a triangle is isosceles, then its BASE ANGLES are
congruent.”
#3. Definition of Acute Triangle/Definition of Obtuse Triangle – says that
 “If a triangle is an acute triangle, then all of its angles are less than 90 degrees.”
 “If a triangle is an obtuse triangle, then one of its angles is greater than 180 degrees.”
#4. Definition of Perpendicular – says that “If two lines, rays, segments or planes are perpendicular,
then they form right angles (as many as four of them).”
#5. Definition of Right Angle/Definition of Acute Angle/Definition of Obtuse Angle – says that:
 “If an angle is a right angle, then the angle must EQUAL 90 degrees.”
 “If an angle is an acute angle, then the angle must be less than 90 degrees.”
 “If an angle is an obtuse angle, then the angle must be greater than 90 degrees.”
#6. ASA, SSS, SAS, AAS – Proves that two triangles are congruent.
 ASA: says that “If two angles and an included side of one triangle are congruent to two
corresponding angles and an included side of another triangle, then the triangles are
congruent.”
2
 SSS: says that “If all three sides of one triangle are congruent to all three corresponding sides of
a another triangle, then the triangles are congruent.”
 SAS: says that “If two sides and an included angle of one triangle are congruent to two
corresponding sides and an included angle of another triangle, then the triangles are
congruent.”
 AAS: says that “If two angles and a non-included sides of one triangle are congruent to two
angles and a non-included side of another triangle, then the triangles are congruent.”
#7. CPCTC – says that “If you use ASA, SSS, SAS, or AAS to prove that two triangles are congruent, then
all other corresponding parts (sides & angles) of the congruent triangles are going to be congruent.”
#8. Reflexive Property – says that “something is congruent to itself.”
#9. Segment Addition Postulate/ Angle Addition Postulate – used when we do part + part = whole (for
either sides or angles).
#10. Definition of Vertical Angles – says that “If two non-adjacent angles are created by intersecting
lines, then those angles are known as vertical angles.”
#11. Vertical Angle Theorem – says that “If two angles are vertical angles, then their measures are
going to be congruent to one another.”
3
#12. Definition of Linear Pair – says that “If two angles are adjacent and form a line, then they form
what’s known as a linear pair.
#13. Linear Pair Postulate – says that “If two angles form a linear pair, then those angles are also going
to be supplementary.”
#14. Definition of Supplementary/Definition of Complementary – says that
 “If two angles are supplementary, then their measures add up to 180 degrees.”
 “If two angles are complementary, then their measures add up to 90 degrees.”
#15. Definition of Parallel Lines – says that “If lines in the same plane do not intersect, then the lines are
parallel.”
Angle relationships due to parallel lines
#16. Definition of alternate interior angles – says that “If two angles are alternate interior, then they
are on opposite sides of a transversal and are both on the interior to two lines (whether parallel or
not).”
#17. Alternate interior angle theorem – says that “If two lines are parallel and alternate interior angles
are formed, then the angles will be congruent to one another.”
#18. Converse of alternate interior angle theorem – says that “If alternate interior angles are
congruent, then the lines that form them will be parallel to one another.”
#19. Definition of alternate exterior angles – says that “If two angles are alternate exterior, then they
are on opposite sides of a transversal and are both on the exterior to two lines (whether parallel or
not).”
#20. Alternate exterior angle theorem – says that “If two lines are parallel and alternate exterior angles
are formed, then the angles will be congruent to one another.”
#21. Converse of alternate exterior angle theorem – says that “If alternate exterior angles are
congruent, then the lines that form them will be parallel to one another.”
4
#22. Definition of corresponding angles – says that “If two angles are corresponding, then they are on
same side of a transversal and are both on corresponding sides (one interior/one exterior) to two lines
(whether parallel or not).”
#23. Corresponding Angle Postulate – says that “If two lines are parallel and corresponding angles are
formed, then the angles will be congruent to one another.”
#24. Converse of corresponding angle postulate – says that “If corresponding angles are congruent,
then the lines that form them will be parallel to one another.”
#25. Definition of same side interior angles – says that “If two angles are same side interior, then they
are on the same side of a transversal and are both on the interior to two lines (whether parallel or not).”
#26. Same side interior angle theorem – says that “If two lines are parallel and same side interior
angles are formed, then the angles will be supplementary to one another.”
#27. Converse of same side interior angle theorem – says that “If same side interior angles are
supplementary, then the lines that form them will be parallel to one another.”
#27. Definition of a midpoint – says that “If a point is a midpoint, then the point divides a segment into
TWO equal parts.”
#28. Midpoint Theorem – says that “If a point is a midpoint, then the point divides a segment so that
each part of the segment is equal to ONE HALF of the whole segment.”
#29. Definition of a Median – “If a segment is a median, then it is a segment whose endpoints are the
vertex of a triangle and the midpoint of the opposite side of the triangle.”
#30. Definition of centroid – says that “If a point is a centroid, then it is a point of concurrency of the
medians inside of a triangle.”
#31. Centroid Theorem – says that “If a point of concurrency of a triangle is a centroid, then the point
that they are concurrent at is two thirds the distance from each vertex to the midpoint of the opposite
side.”
5
#32. Definition of Angle Bisector – says that “If a segment, ray, line or plane is an angle bisector, then it
divides an angle into TWO equal parts.”
#33. Angle Bisector Theorem – says that “If a segment, ray, line or plane is an angle bisector, then it
divides an angle so that each part of the angle is equal to ONE HALF of the whole angle.”
It also says: “If a point is on the bisector of an angle, then the point is equidistance from the sides of the
angle.”
#34. Hinge Theorem (SAS Inequality Theorem)
#35. Converse of Hinge Theorem (SSS Inequality Theorem)
#36. Isosceles Triangles Property – Remember that the following things happen.
The following will be the same segment: (Median; Altitude; Angle bisector)
Vertex Angle Bisector Conjecture
“If a triangle is an isosceles triangle, then the median, angle bisector, and altitude will be the same
segment.”
#37. Triangle Sum Theorem – says that “If a polygon is a triangle, then its interior angles will measure a
sum of 180 degrees.”
#38. Definition of equidistant – says that “If a point is equidistant from two other points (or objects),
then it is the same distance from the other two points (or objects).”
#39. Definition of Segment Bisector – says that “If a segment, ray, line or plane is a segment bisector,
then it divides an segment into TWO equal parts.”
6
#40. Transitive Property – says that “If one expression is equal/congruent to a second expression, and
that second expression is equal/congruent to a third expression, then the first and third expressions are
also equal/congruent.”
#41. Substitution Property – says that “If we insert an expression into an equation in place of another
expression, then we have used substitution.”
#42. Addition/Subtraction/Division/Multiplication – says that “If we add, subtract, multiply, or divide
an number on BOTH sides of the equal sign in an equation, then we have carried out one of those basic
arithmetic operations.”
#43. Distributive Property – says that “If we multiply items on the same side of the equal sign of an
equation in which parentheses are involved, then we have used the distributive property.”
#44. Combining Like Terms – says that “If we add or subtract expressions on the same side of the equal
sign, then we have combined like terms.”
#45. Exterior Angle Theorem – says that “If we have an exterior angle of a triangle, then its measure
will equal the sum of its two remote interior angles.”
#46. Definition of Altitude – says that “If a segment is an altitude, then it is a segment originating from
one of the vertices of a triangle and its perpendicular to an opposite side.”