Congruence

By the Side-Angle-Side (SAS) congruence theorem, if two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent.

By the Angle-Side-Angle (ASA) congruence theorem, if two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent.

By the Side-Side-Side (SSS) congruence theorem, if three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent.

Two triangles are congruent if they have congruent sides and angles, similar sides and angles, or equal areas.

If two triangles are congruent, then their corresponding sides, angles, and areas are all congruent.

By the Hypotenuse-Leg (HL) congruence theorem, if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

A triangle with two congruent sides and one non-congruent side is called an isosceles triangle.

A triangle with all three sides congruent is called an equilateral triangle.

Two triangles are similar if they have similar sides and angles, but not necessarily congruent sides and angles.

If two triangles are similar, then their corresponding sides are proportional, their corresponding angles are congruent, and their corresponding areas are proportional.

By the Hypotenuse-Hypotenuse (HH) similarity theorem, if the hypotenuse of one right triangle is congruent to the hypotenuse of another right triangle, then the triangles are similar.

A triangle with two congruent angles and one non-congruent angle is called an isosceles triangle.

A triangle with one angle measuring 90 degrees is called a right triangle.

By the Pythagorean theorem, in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

By the converse of the Pythagorean theorem, if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.