Download Proof statements review

General Statements for
Geometry Proofs
Math B
Reflexive Property - A quantity is equal to itself.
Symmetric Property of equality – Equality can be expressed in either order
Transitive Property – If a = b and b = c, then a = c.
Substitution Postulate - A quantity can be substituted for an equal quantity.
The whole is equal to the sum of its parts
Addition Postulate - If two equal quantities are added together with 2 other
equal quantities, the sums are equal.
Subtraction Postulate - If two equal quantities are subtracted from 2 other
equal quantities, the differences are equal.
Multiplication Postulate - If two equal quantities are multiplied together
with 2 other equal quantities, the products are equal.
Doubles of equal quantities are equal
Division Postulate - If two equal quantities are divided by 2 other equal
quantities, the quotients are equal. (As long as the quantity you are dividing
by is not zero!)
Halves of equal quantities are equal.
Powers Postulate - The squares of equal quantities are also equal.
Roots Postulate - The positive root of equal quantities are also equal.
Definition of Midpoint - The point that divides a segment into two equal
segments.
A line segment had one and only one midpoint
Definition of angle Bisector - The line segment that cuts an angle into 2
equal angles
An angle has one and only one bisector
All right angles are congruent
All straight angles are congruent
If 2 angles are congruent then their complements are congruent
If 2 angles are congruent then their Supplements are congruent
If 2 angles form a linear pair then their supplementary
Vertical angles are congruent
___________________________________________________________
Corresponding parts - Parts of congruent triangles are congruent
Reflexive Property (geo shapes) – Any geometric shape is congruent to itself
Symmetric Property (geo shapes) – Any congruence can be expressed in any
order.
Transitive Property (geo shapes) – 2 Geometric shapes congruent to a third
shape are congruent to each other.
SAS - Two triangles are congruent if two sides and the included angle are
congruent to two sides and the included angle of another triangle.
ASA - Two triangles are congruent if two angles and the included side are
congruent to two angles and the included side of another triangle.
SSS - Two triangles are congruent if all three sides are congruent to the
three sides of another triangle.
AAA - Two triangles are similar if all three angles are congruent to the
three angles of another triangle
Altitude of triangle – A segment from the vertex of a triangle and
perpendicular to the other side.
Median of a triangle - A segment from the vertex of a triangle to the
midpoint of the opposite side.
The base angles of an isosceles triangle are congruent.
If the base angles of a triangle are congruent then the 2 opposite sides are
congruent
The bisector of the vertex angle of an isosceles triangle is the perpendicular
bisector of the base of the triangle.
_____________________________________________________________
Transitive Property of inequality - If a > b and b > c, then a > c.
Substitution Property of inequality – If a > b and c = b, then a > c.
Addition Postulate of inequality – If a > b and c = d, then a + c > b + d
Subtraction Postulate of inequality – If a > b and c = d, then a – c > b – d
The sum of the lengths of 2 sides of a triangle is greater than the length of
the third side.
The measure of an exterior angle of a triangle is greater than the measure of
either of the nonadjacent interior angles.
If the lengths of 2 sides of a triangle are unequal, the measure of the angles
opposite these sides are unequal and the greater angle is across from the
greatest side.
_____________________________________________________________
If 2 points are each equidistant from the endpoints of a line segment, the
points determine the perpendicular bisector.
Any point that lies on a perpendicular bisector of a line segment is
equidistant from the endpoints
Two lines that are parallel to a third line, are parallel to each other
Two lines that are perpendicular to a third line, are parallel to each other
If 2 angles of one triangle are congruent to 2 angles of another triangle, the
third angles of each triangle have to be equal.
AAS - If 2 angles and the opposite side are congruent in 2 different triangles
then the triangles are congruent.
HL – Two right triangles are congruent if the hypotenuse and a leg of the 2
triangles are congruent to the hypotenuse and leg of the other triangle.
The exterior angle of a triangle is equal to the sum of the nonadjacent
interior angles
_____________________________________________________________
If both pairs of opposite sides are congruent then the quadrilateral is a
parallelogram
If one pair of sides is congruent and parallel then the quadrilateral is a
parallelogram
If both pairs of opposite angles are congruent then the quadrilateral is a
parallelogram
The diagonals of a quadrilateral bisect each other then the quadrilateral is a
parallelogram
If you have a parallelogram with one right angle, it is a rectangle
If the diagonals of a quadrilateral are congruent then the quadrilateral is a
rectangle
If the sides of a quadrilateral are all congruent then the quadrilateral is a
rhombus
If the diagonals of a quadrilateral are perpendicular then the quadrilateral is
a rhombus
If one angle of a rhombus is a right angle then the quadrilateral is a square
If a trapezoid is isosceles then the base angles of the trapezoid are equal
_____________________________________________________________
Prove similar triangles with AA
If a line that is parallel to one side of a triangle and intersects the other sides
in different points cuts off a triangle that is similar to the original triangle
Product of the means equals the product of the extremes
If an altitude is drawn to the hypotenuse of a right triangle, the two smaller
triangles are similar to the original triangle