Logic, Properties, & Theorems
If-Then Statements:
Conditional – If p, then q.
Converse – If q, then p.
Inverse – If not p, then not q.
Contrapositive – If not q, then not p.
Properties of Equality:
Reflexive
a=a
Transitive
If a=b & b=c, then a=c
Distributive
a(b+c)=ab+ac
Commutative
a+b=b+a OR ab=ba
Associative
(a+b)+c=a+(b+c) OR (ab)c=a(bc)
Addition
If a=b, then a+c=b+c
Subtraction
If a=b, then a-c=b-c
Multiplication
If a=b, then ac=bc
Division
If a=b, then a/c=b/c
Lines and Angles:
Definition of Midpoint – If B is the midpoint of AC, then AB=BC.
Definition of Bisect – Cuts exactly in half.
Definition of Perpendicular – Lines meet at a 90o angle.
Linear Pair Property – If two angles form a linear pair, then they are
supplementary (sum of their measures is 180o).
Vertical Angles Theorem – If two lines intersect, then the two pairs of vertical
angles formed are equal in measure (congruent).
Angle Addition Postulate – If P is a point in the interior of <ABC, then
m<ABP + m<PBC = m<ABC.
Segment Addition Postulate – If C is a point on line l, between points A and B,
then AC + CB = AB.
Given Parallel Lines:
If two parallel lines are cut by a
transversal, then corresponding
angles have equal measure.
(Corresponding Angles Assumption)
If two parallel lines are cut by a
transversal, then alternate interior
angles have equal measure.
Proving Parallel Lines:
If two lines are cut by a
transversal so that corresponding
angles have equal measure, then
the lines are parallel.
(Parallel Lines Property)
If two lines are cut by a
transversal so that alternate
interior angles have equal
measure, then the lines are
parallel.
If two parallel lines are cut by a
transversal, then alternate exterior
angles have equal measure.
If two lines are cut by a
transversal so that alternate
exterior angles have equal
measure, then the lines are
parallel.
If two parallel lines are cut by a
transversal, then same-side interior
angles are supplementary.
If two lines are cut by a
transversal so that same-side
interior angles are supplementary,
then the lines are parallel.
If two parallel lines are cut by a
transversal, then same-side exterior
angles are supplementary.
If two lines are cut by a
transversal so that same-side
exterior angles are supplementary,
then the lines are parallel.
Triangles:
Definition Right Triangle – Triangle with one right angle
Definition Isosceles Triangle – Triangle with two congruent sides
Definition Equilateral Triangle – Triangle with three congruent sides
Triangle Sum Theorem – The sum of the angles in any triangle is 180o.
Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles
opposite those sides are congruent and if two angles of a triangle are congruent, then
the opposite sides are congruent.
Exterior Angle Theorem – An exterior angle of a triangle has a measure equal to the
sum of the measures of the two remote interior angles.
Angle Bisector Theorem – If an angle of a triangle is bisected, then the segments it
creates on the opposite side are in proportion to the adjacent sides of the triangle.
Similarity
Side Angle Side Similarity- If the lengths of two sides of one triangle are
related by a scale factor k to two sides of a second triangle and the included
angles have equal measures, then the triangles are similar.
Side Side Side Similarity – If three sides of one triangle are related by a scale
factor k to three sides of a second triangle, then the triangles are similar.
Angle Angle Similarity – If two angles of one triangle are congruent to two
angles of a second triangle, then the triangles are similar.
Hypotenuse Leg Similarity – If the lengths of the hypotenuse and one leg of one
right triangle are related by a scale factor k to the hypotenuse and leg of a second
right triangle, then the triangles are similar.
Congruence
Side Angle Side Congruence – If two sides and the included angle of one
triangle are congruent to two sides and the included angle of another triangle,
then the triangles are congruent.
Side Side Side Congruence – If three sides of one triangle are congruent to
three sides of another triangle, then the triangles are congruent.
Angle Side Angle Congruence – If two angles and the included side of one
triangle are congruent to two angles and the included side of another triangle,
then the triangles are congruent.
Angle Angle Side Congruence – If two angles and a nonincluded side of one
triangle are congruent to corresponding parts of another triangle, then the
triangles are congruent.
Hypotenuse Leg Congruence – If the hypotenuse and leg of one right triangle
are congruent to the hypotenuse and leg of another right triangle, then the
triangles are congruent.
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
Midpoint Connector Theorem for Triangles – If a line segment joins the midpoints of
two sides of a triangle, then it is parallel to and half the length of the third side.
Parallel Bisector for Triangles – If a line goes through the midpoint of one side of a
triangle and is also parallel to a different side of the triangle, then the line intersects the
third side at its midpoint.
Quadrilaterals:
If a quadrilateral is a parallelogram, then the following properties are true:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Adjacent angles are supplementary
The diagonals bisect each other
Each diagonal divides the shape into 2 congruent triangles
The following conditions guarantee a quadrilateral to be a parallelogram:
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
The diagonals bisect each other
One pair of opposite sides is congruent and parallel
Midpoint Connector Theorem for Quadrilaterals – If the midpoints of
consecutive sides of any quadrilateral are connected, the resulting quadrilateral
is a parallelogram.