Geometry: Triangle Congruence, Properties, and Theorems

Definition of Congruent Triangles

If there exists a 1-1 correspondence between the vertices of two triangles such that the corresponding sides and angle are congruent, then the triangles are congruent.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

SAS Postulate

If there exists a 1-1 correspondence between the vertices of two triangles such that two sides and their included angle of 1 triangle are congruent to the second, then the triangles are congruent.

ASA

If there exists a 1-1 correspondence between the vertices of two triangles such that two angles and their included side of 1 triangle are congruent to the second, then the triangles are congruent.

Definition of Isosceles

A triangle is isosceles iff two sides are congruent.

ITT

If two sides of a triangle are congruent then the angles opposite them are congruent.

CITT

If two angles of a triangle are congruent, then the sides opposite them are congruent.

SSS

If there exists a 1-1 correspondence between the vertices of two triangles such that the corresponding sides are congruent, then the triangles are congruent.

Concurrence

Three lines are concurrent iff they intersect at the same point.

Median of a triangle

The segment from the MP of one side of a triangle to the vertex opposite it.

Angle Bisector of a Triangle

A segment from the vertex of a triangle to the side opposite it, such that the angle is bisected.

Altitude

A segment from the vertex of a triangle to the side opposite it, such that the angle formed by the intersection is a right angle.

H.L. Theorem

If in two right triangles, there exists congruent hypotenuses and corresponding congruent legs, then the triangles are congruent.

1/3 enuf

In an isosceles triangle, if there exists a median, altitude, or angle bisector, then it is also the other two.

2/3 enuf

In a triangle, if one segment is any two (median, altitude, angle bisector), then the triangle is isosceles.

Equilateral

A triangle is equilateral iff all sides are congruent.

Equangular

A triangle is equilateral iff all angles are congruent.

Scalene

A triangle with no congruent sides or angles.

Obtuse triangle

A triangle with an obtuse angle.

Acute triangle

A triangle with all angles acute.

Equidistant

Points are equidistant to one another if the segments created between the points are congruent.

Perpendicular Bisector Th 1

In a plane, if 2 pts are equidistant from the endpoints of a given segment, then the line containing those points is the perpendicular bisector.

Perpendicular Bisector Th 2

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Perpendicular Bisector Th 3

In a plane, if a point is equidistant from the endpoints of a given segment, then it lies on the perpendicular bisector of the segment.