Cumulative List
Assumptions, Theorems, Definitions, Properties, & Postulates
Assumptions (from a diagram)
 Straight lines (angles)
 Intersection of lines
Theorems
 If two angles are supp. (or comp.) to  angles (or the same angle), then they are  .
 If two angles are right (or straight) angles, then they are congruent.
 If two lines intersect, then opposite angles are  .
 If two angles are both supp. and  , then they are right angles.
 If two sides of a triangle are  , then their opposite angles are  .
(note: converse true as well)
 If two pts. are equidistant from the endpoints of a segment, then they determine the
 bis. of the segment.
 If two coplanar lines are  to the same line, then they are .
 If alt. int. angles are  , then
lines. (note: converse true as well)
 If alt. ext. angles are  , then
lines. (note: converse true as well)
 If corresp. angles are  , then
lines. (note: converse true as well)
 If given a triangle, then the sum of the measures of its angles is 180.
 If two angles of a triangle are congruent to two angles in another triangle, then the
remaining angles are congruent.
 AAS (congruency method for proving congruent triangles)
 If given a triangle, the measure of an exterior angle is equal to the sum of the
measures of its two remote interior angles.
 Area of a Triangle = (1/2) x (product of any two sides) x sin(included angle)
Cumulative List
Assumptions, Theorems, Definitions, Properties, & Postulates
 Law of Sines (use when given ASA, AAS, or SSA – the ambiguous case)
 Law of Cosines (use when given SSS or SAS)
 AA / SSS (proportional) / SAS (proportional) – methods for proving similar
triangles
 If a parallel line is constructed to any side of a triangle, then it divides the other two
sides (of the triangle) proportionally.
 If a series of parallel lines cut several transversals into parts, then these parts are
proportional.
 If a ray bisects an angle of a triangle, then it divides the side opposite the angle into
parts with the same ratio as the one between the two adjacent corresponding sides
which make up the angle.
 If a radius is perpendicular to a chord of a circle, then it bisects the chord.
 If two tangent segments share a common endpoint outside the circle, then they are
congruent.
 If an angle is an incribed angle, then its measure is one-half the measure of its
intercepted arc.
 If a triangle is inscribed with one side a diameter, then it is a right triangle.
 If a quad is inscribed in a circle, then its opposite angles are supplementary.
 If two chords intersect and both sets of endpoints are connected (creating a
“butterfly” shape), then the created triangles are similar.
 If an angle is an tangent-chord angle, then its measure is one-half the measure of its
intercepted arc.
 If an angle is an chord-chord angle, then its measure is one-half the sum of the
measures of its intercepted arcs.
 If an angle has its vertex outside a circle, then its measure is one-half the difference
of the measures of its intercepted arcs.
 If two chords intersect inside a circle then the product of the segments of one chord
equals the product of the segments of the other.
Cumulative List
Assumptions, Theorems, Definitions, Properties, & Postulates
 If two secants share a common endpoint outside a circle then the product of one
whole secant and its exterior part equals the product of the other whole secant and
its exterior part.
 If a secant and a tangent share a common endpoint outside a circle then the product
of the whole secant and its exterior part equals the square of the tangent.
Definitions
Please note that the converse of a definition is always true. Be sure you are using the
appropriate form of the conditional statement when constructing an argument.
 If two angles are supp., then they form a straight angle (line).
 If two angles are comp., then they form a right angle.
 If a point is a midpoint of a segment, then it divides the segment into two congruent
parts.
 If two segments (or lines/rays) are perpendicular, then they form a right angle(s).
 If a segment (or line/ray) bisects an angle, then it divides it into two congruent parts.
 If a segment connects the vertex of a triangle to the midpt. of the opposite side, then
it is a median.
 If a segment connects the vertex of a triangle to the opposite side such that it is
perpendicular to this side, then it is an altitude.
 If a line divides an angle into two congruent parts, then it is an angle bisector.
 If a line is perpendicular to a segment and passes through its midpt., then it is the
perpendicular bisector of that segment.
 If a triangle has at least two congruent sides, then it is isosceles.
 If two congruent segments share an endpt, then this point is equidistant from the
other endpts of these segments.
 Circle: the set of all points in a plane equidistant from a common point (the center
of the circle)
 Radius: a line segment connecting the center point to a point on the circle
 Chord: a line segment connecting two points on the circle
Cumulative List
Assumptions, Theorems, Definitions, Properties, & Postulates
 Tangent: a line that intersects a circle at only one point.
 Central Angle: an angle whose vertex is the center of a circle.
 Inscribed Angle: an angle whose vertex is on the circle.
 Secant: a segment (line) that intersects a circle at two points.
Properties
These surface quite frequently so it is important to be able to anticipate their use so as to
make efficient use of these properties.





Reflexive Property
Substitution Property
Transitive Property
Addition/Subtraction Property
Multiplication/Division Property
Postulates
Please note the number postulates are few.
 Side-Angle-Side (SAS) – method used to prove triangles are congruent.
 Angle-Side-Angle (ASA) – method used to prove triangles are congruent.
 Side-Side-Side (SSS) -- method used to prove triangles are congruent.
 Two points determine a line.
 Perpendicular Postulate
 Parallel Postulate
 All radii of a circle are congruent.
 A tangent is perpendicular to a radius drawn to the point of tangency.