Advanced Geometry Theorems/Properties/Postulates Master List

Addition Property of Equality (APE) : If a = b, then a + c = b + c

Subtraction Property of Equality (SPE) : If a = b, then a - c = b - c

Transitive Property of Equality (Transitive) : If a = b, and b = c, then a = c

Substitution Property of Equality (Subst.) : If it's known that a = b, then b can replace a in any situation.

Identity Property of AdditionIdentity Property of Multiplication (Identity Prop.) : a + 0 = aa × 1 = a

Multiplicative Property of 0 (Mult. Prop 0) : a × 0 = 0 and 0 × a = 0

Rule for Subtraction (Def. Subtract.) : a-b=a+(-b)

Properties of Negative in Products : a(-b)=(-a)b=-ab, (-a)(-b)=ab

Definition of Altitude (Def Altitude) : A line which passes through a vertex of a triangle, and meets the opposite side at right angles.

Side Side Side (SSS) : If three corresponding sides of two triangles are congruent, then the triangles are congruent.

Corresponding Parts of Congruent Triangles are Congruent (CPCTC) : If given two congruent triangles, any corresponding parts are congruent.

Isosceles Triangle Theorem (ITT) : If two sides of a triangle are congruent, then the angles opposite are congruent.

Definition of a Midpoint (Def. Midpoint) : The point exactly halfway between the two endpoints of a segment.

Midpoint Theorem : If E is the midpoint of segment AC, then AE = 1/2AC

Definition of an Angle Bisector (Def ∠ Bisector) : A line or ray that divides an angle into two congruent angles.

Angle Bisector Theorem (∠ Bisector Thm) : If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Definition of Complimentary Angles (Def Comp ∠'s) : Two angles whose measures add up to 90°.

SCAC : Supplements of congruent angles or the same angle are congruent.

2 adj. ∠'s formed by ⊥ lines are ≅ : If two adjacent angles are formed by the intersection of two perpendicular lines, then the angles are congruent.

2 lines form adj. ≅ ∠'s ⟶ ⊥ lines : If two lines form adjacent congruent angles, then the lines are perpendicular.

Same Side Interior Angles Theorem (SSIA) : Two same side congruent angles are supplementary if and only if the lines cut by the transversal are parallel.

Triangle Sum Theorem (Δ Sum) : The sum of the measure of the interior angles of any triangle is 180°.

The sum of the exterior angles of a polygon : 360°

The measure of each interior angle of a regular polygon : 180(n-2) / n [n is the number of sides]

Definition of a Rectangle : A quadrilateral with four right angles.

Definition of a Rhombus : A quadrilateral with four congruent sides.

Triangle Midline/Midsegment Theorem : The midline (midsegment) of a triangle is parallel to the third side and half its length.

Midpt + || = Bisector : A line that contains the midpoint of one side of a triangle and is parallel to another side bisects the third side.

Two lines perpendicular to the same line are perpendicular to each other. : Two lines perpendicular to the same line are perpendicular to each other.

Circle : Center isn't part of the circle.

Tangent : Line that intersects a circle at one point.

Point of Tangency : Where the tangent intersects the circle.

Arc : Subset of the circumference.

Minor Arc : An arc with a measure less than 180°.

Circumscribed Polygon : All are sides are tangent to the circle.

All radii ≅ : All radii are congruent.

In a circle, congruent arcs have congruent chords. : In a circle, congruent arcs have congruent chords.

Inscribed Angles Theorem : The measure of an inscribed angle equals half the measure of its intercepted arc.

A tangent to a circle is perpendicular to the radius drawn to the point of tangency. : A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

A line that is perpendicular to a radius at a point of a circle is tangent to the circle. : A line that is perpendicular to a radius at a point of a circle is tangent to the circle.

In a circle, the perpendicular bisector of a chord contains the center of the circle. : In a circle, the perpendicular bisector of a chord contains the center of the circle.

In a circle, congruent chords are equidistant from the center. : In a circle, congruent chords are equidistant from the center.

Angle Bisector Theorem : If AD bisects <BAC, then AB/BD = AC/DC

AA~ Theorem : If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Chord Theorem : When two chords intersect in a circle, the product of the divided parts of one chord equals the product of the divided parts of the other.ab = cd

Secant Theorem : When two secants intersect at a common exterior point, the product of the external part of one secant and the whole secant equals the product of the other.a(a+b) = c(c+d)