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Unique Line Postulate
 Through any two points there is exactly one line
 (a line is determined by two points)

Postulate: If two lines intersect, then they intersect...
...in exactly one point

Unique Plane Postulate
 Through any three noncollinear points there is exactly one plane
 (three noncollinear points determine a plane)

Postulate: If two planes intersect, then they intersect...
...in exactly one line.

Angle Addition Postulate

Linear Pair Postulate

Linear Pair Theorem

Vertical Angles Theorem
If two angles are vertical, then they are congruent.

Parallel Postulate (Playfair's Axiom)
 Through a point not on a given line, there is exactly one line parallel to the given line.
 (Allows you to add a parallel line to a diagram)

Perpendicular Postulate
 Through a point not on a given line, there is exactly one line perpendicular to the given line.
 (Allows you to add a perpendicular line to a diagram)

Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then corresponding angles are congruent.

Converse of the Corresponding Angles Postulate
If two lines are cut by a transversal, and corresponding angles are congruent, then the lines are parallel.

Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Converse of the Alternate Interior Angles Theorem
If two lines are cut by a transversal, and alternate interior angles are congruent, then the lines are parallel.

Same Side Interior Angles Theorem
If two parallel lines are cut by a transversal, then same side interior angles are supplementary angles.

Converse of the Same Side Interior Angles Theorem
If two lines are cut by a transversal, and same side interior angles are supplementary, then the lines are parallel.

Triangle AngleSum Theorem
The sum of the measures of the interior angles of a triangle is 180 degrees.

No Choice Theorem
If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent.


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

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

Theorem: The bisector of the vertex angle of an isosceles triangle is...
 ... the perpendicular bisector of the base.

Theorem: If a triangle is equilateral, then the triangle is...
 ...equiangular.

Congruent Supplements Theorem
If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Right Angle Congruence Theorem
If two angles are right angles, then they are congruent.

Angle Bisector Theorem

Midpoint Theorem

Segment Division Theorem
The like divisions of congruent segments are congruent.

Angle Division Theorem
The like divisions of congruent angles are congruent.

Segment Multiplication Theorem
The like multiples of congruent segments are congruent.

Angle Multiplication Theorem
The like multiples of congruent angles are congruent.

Theorem: If the exterior sides of two adjacent acute angles are perpendicular then...
... the angles are complementary

Theorem: If two lines are perpendicular then they form...
...congruent adjacent angles.

Theorem: If two lines form congruent adjacent angles, then they are...
...perpendicular.

Polygon AngleSum Theorem
The sum of the measures of the interior angles of an ngon is (n  2)180

Polygon Exterior AngleSum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

Theorem: The number of diagonals in an ngon is...

SideSideSide Theorem (SSS)
If the three sides of one triangle are congruent to the three sides of another triangle,then the two triangles are congruent.

SideAngleSide Theorem (SAS)
 If two sides and the included angle of one
 triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

AngleSideAngle Postulate (ASA)
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Theorem: All radii of a circle are...
...congruent.

HypotenuseLeg Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Angle AngleSide Theorem (AAS)
 If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle, then the two triangles are congruent.

Corresponding parts of congruent triangles are congruent (CPCTC)
What it does...
CPCTC allow us to prove that corresponding parts of congruent triangles are congruent.

Corresponding parts of congruent triangles are congruent (CPCTC)
What it does NOT do...
CPCTC does NOT prove that triangles are congruent.

Perpendicular Bisector Theorem
 If a point is on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem
 If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.

Theorem: If two points are equidistant from the endpoints of a segment, then the two points...
 determine the perpendicular bisector of the segment.

Angle Bisector Theorem II
 If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

Converse of the Angle Bisector Theorem II
 If a point in the interior of an angle is equidistant from the sides of an angle, then the point is on the angle bisector.

Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the...
vertices of the triangle.

Circumcenter
The point of concurrency of the perpendicular bisectors of a triangle.

Theorem: The angle bisectors of a triangle are concurrent at a point equidistant from the...
...sides of the triangle.

Incenter
The point of concurrency of the angle bisectors of a triangle.

3 Circumcenter Facts
 Found using perpendicular bisectors
 Equidistant from vertices of a triangle.
 Is the center of circle that passes through vertices.

3 Incenter Facts
 Found using angle bisectors
 Equidistant from sides of a triangle
 Is the center of circle that shares one point with each side of a triangle

The circumcenter of a right triangle is always located on...
...the midpoint of the hypotenuse

Median (of a triangle)
A segment whose endpoints are a vertex and the midpoint of the opposite side.

Centroid
The point of concurrency of the medians of a triangle.

Centroid Theorem
The medians of a triangle are concurrent at a point that is twothirds the distance from each vertex to the midpoint of the opposite side.

Altitude (of a triangle)
The perpendicular segment from a vertex of a triangle to the line containing the opposite side.

Theorem: The lines that contain the altitudes of a triangle...
...are concurrent.

Orthocenter
The point of concurrency of the altitudes of a triangle.

4 Centroid Facts
 Found using medians

