## TRIANGLES: Theorems and Postulates for Geometry

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kellrson  on May 20, 2012

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# TRIANGLES: Theorems and Postulates for Geometry

 Side-Side-Side (SSS) CongruenceIf three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
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#### Definitions

Side-Side-Side (SSS) Congruence If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS) Congruence If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA) Congruence If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Angle-Side (AAS) Congruence If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Hypotenuse-Leg (HL) Congruence (right triangle) If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
CPCTC Corresponding parts of congruent triangles are congruent.
Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS for Similarity If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
SAS for Similarity If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.
Side Proportionality If two triangles are similar, the corresponding sides are in proportion.
Mid-segment/Mid-line Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Sum of Two Sides The sum of the lengths of any two sides of a triangle must be greater than the third side
Longest Side In a triangle, the longest side is across from the largest angle.
In a triangle, the largest angle is across from the longest side.
Altitude Rule The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.
Leg Rule Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

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