synthetic proof

proof built using a system of postulates & theorems in which the prop's of figures, but not their actual measurements, r studied

justiifications of synthetic proof

given statements

definitions

postulates

previously proved theorems

Bisecting Diagonals Th

If the 2 diagonals of a quadrilateral bisect each other, then the quadirlateral is a parallelogram.

implications

if-then statements that can be represented by symbols

Ex: p→q reads If p, then q

logically equivalent

either both truth or both false

original & contrapositive

not logically equivalent

just cuz the original = true, doesn't mean converse & inverse r

contrapositive

~q → ~p

If not q, then not p

converse

q → p

If q, then p

inverse

~p → ~q

If not p, then not q

median (of triangle)

a segment joining a vertex to the midpoint of the opp. side

coordinate proof

a proof based on a coord. system in which all points r represented by ordered pairs of #'s

justificiations for coordinate proof

distance & midpoint formulas

parallel lines have the same slope

perp. lines have slopes that are negative reciprocals of each other

a geometric figure may be placed anywhere in the coordinate plane

distance formula

midpoint formula

Isosceles Median Theorem

In an isosceles triangle, the medians drawn to the legs are equal in measure.

isosceles trapezoid

a trapezoid w/ a line of symmetry that passes through the midpoints of the bases

Isosceles Trapezoid Theorem

In an isosceles trapezoid:

- the legs r equal in measure
- the diagonals r equal in measure
- the 2 angles @ each base r equal in measure

inclusive definition

a definition that includes all possibilities

exclusive definition

a definition that excludes some possibilities

quadrilateral chart

past postulates & theorems

Addition Prop. of Equality Post.

Subtraction Prop. of Eq. Post

Mult Prop of Eq Post

Div Prop of Eq Prop Post

Reflexive Prop of Eq Prop Post

Substitution Property Post

Distributive Prop Post

If 2 angles r supplements of the same angle, then they r equal in measure.

If 2 angles r complements of the same angle, then they r equal in measure

Straight Angle Post - If the sides of an angle form a straight line, then the angle is a straight angle with measure 180^{o}

Angle/Segment Addition Post - For any segment or angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts.

Vertical angles are equal in measure

The sum of the measures of the angles of a triangle is 180^{o}

An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote interior angles

If 2 sides of a triangle r equal in measure, then the angles opp those sides r equal in measure

If 2 angles of a triangle r equal in measure, then the sides opp those angles r equal in measure

If a triangle is equilateral, then it is also equiangular, with three 60^{o} angles.

If a triangle is equiangular, then it's also equilateral

The sum of the angle measures of an n-gon is given by the formula S(n) = (n -2)180^{o}

If 2 parallel lines r intersected by a transversal, then corresponding angles r equal in measure

If 2 parallel lines r intersected by a transversal, then alternate int angles r = in measure

If 2 parallel lines r intersceted by a trnsvrsl, then co-interior angles r supp

If 2 lines r intersected by a transversal & corresponding angles r = in measure, then the lines r parallel

If 2 lines r intersected by a transversal & alt int angles r = in measure, then the lines r parallel

If 2 liens r intersected by a transversal & co-interior angles r supp, then the lines r parallel

If 2 lines r perp to the same transversal, then they r parallel

If a transversal = perp to one of 2 parallel lines , then its is perp to the other one also

Thru a pt not on a given line, there is 1 and only 1 line parallel to the given line

If a pt is the same distance from both endpts of a segment, then it lies on the perp bisector of the segment

A segment can be drawn perp to a given line from a pt not on the line. The length of this segment is the shortest distance from the pt to the line.

AA Similarity - If 2 angles of 1 tirangle r equal in measure to 2 angles of another triangle, then the 2 triangles r similar

If a line is drawn from a pt on one side of a tirangle parallel to another side, then it forms a tirangle similar to the original triangle

In a tirangle, a segment that connects the midpts of 2 sides is parallel to the third side & half as long

ASA Th

AAS Th

SAS Post

SSS Post

If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed r similar to the original triangle & to each other

Pythagorean Th

If the altitude is drawn to the hypotenuse of a right triangle, then the measure of the altitude is the geometric mean b/w the measures of the parts of the hypotenuse

The sum of the lengths of any 2 sides of a tirangle is greater than the length of the 3rd side

In an isosceles triangle, the medians drawn to the legs r equal in measure

In a parallelogram, the diagonals have the same midpt

In a rectangle, the diagonals r equal in measure

In a kit, the diagonals r perp to each other

In a parallelogram, opp sides r equal in measure

If a quadrilateral is a parallelogram, then consecutive angles r supp

If a quad is a parallelogram, then opp angles r equal in measrue

The sum of the measures of the angles of a quad is 360^{o}

If both pairs of opp angles of a quad r equal in measreu, then the quad = a parallelogram