Triangle Inequality
The sum of the lengths of two sides of a triangle is always greater than the remaining side
Parallel Postulate
Given two lines l and m and a third line t intersecting both l and m forming interior angles a and b on the same side of t, if a + b < 180, then l and m intersect at some point on the same side of t as a and b
Playfair's Parallel Axiom
Given a line l and a point P in the plane, there is EXACTLY ONE line through P parallel to l
Congruent Figures
- Two angles congruent if they have the same measure- Two line segments congruent if they have the same length- Two circles are congruent if they have the same radius- Two triangles congruent if all corresponding sides and angles are the same size- All rays- All lines
Exterior Angle Theorem
An exterior angle is equal to the sum of the opposite interior angles
Exterior Angle Inequality
An exterior angle is greater than either of its opposite interior angles
Thales' Theorem
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Simple Cyclic Quads
A simple quad is cyclic if and only if its opposite angle pairs sum to 180
Non-Simple Cyclic Quads/ Bow Tie Theorem
A non-simple quad can be inscribed in a circle if and only if the opposite angles are equal
Characterization of the Perpendicular Bisector
Given a line segment AB, a point X is on the perpendicular bisector of AB if and only if X is equidistant to A and B
Characterization of the Angle Bisector
Given a non-reflex angle theta, a point X is on the angle bisector of theta if and only if it is equidistant to the arms of the angle
Open Jaw Equality
Given triangle ABC and triangle DEF with AB=DE and BC=EF, then angle ABC is LESS than angle DEF if and only if AC is LESS than DF
Midline Theorem
If P and Q are the respective midpoints of sides AB and AC of triangle ABC, then PQ is parallel to BC and the lengths of these segments satisfy PQ=BC/2
Converse of the Midline Theorem
Let P be the midpoint of side AB of triangle ABC, and let Q be a point on AC such that PQ is parallel to BC, then Q is the midpoint of AC
Converse of a P implies Q statement
Q implies P
Contrapositive of a P implies Q statement
Not Q implies Not P
Negation or denial of a P implies Q statement
No if and thenP and Not QIf P implies Q is true then P and Not Q is false
Opposite angles
Opposite angles in an X are equal
SAS
two sides and the included angle of one are congruent to two sides and the included angle of the other
SSS
three sides of one triangle are congruent to the three sides of the other
ASA
Two angles and the included side are congruent to two angles and the included side of the other
SAA
Two angles and a side of one are congruent to two angles and a side of the other
HSR
if the hypotenuse and one side of a right triangle is congruent to the hypotenuse and one side of another triangle, then the two triangles are congruent
angle-side inequality
In triangle ABC, angle ABC is larger than angle ACB if and only if AC is longer than AB
Parallel lines
- Adjacent angles total 180- alternate angles are equal- Alternate exterior angles are equal- corresponding angles are equal- Two lines are parallel
Cyclic
A polygon that can be inscribed in a circle is cyclic
Perpendicular bisectors
a triangles perpendicular bisectors are concurrent
How do we construct a circumcircle?
Make 2 of the perpendicular bisectors of AB, AC, BCLet P be where these 2 lines intersect, make C(p, PA)