Geometry Proofs Cheat Sheet: All theorems, postulates, etc

Angles : 1. Supplementary: Add to 1802. Complementary: Add to 903. Vertical angles: angles opposite each other when two lines cross4. Adjacent angles: angles that come out of the same vertex

Basic Properties for proofs : 1. Reflexive: a=a2. Symmetric: if a=b then b=a3. Transitive: if a=b and b=c then a=c4. Substitution: if a=b then both can be substituted for each other in any equation5. Addition: if a=b then a+c=b+c6. Subtraction: if a=b then a-c=b-c7. Multiplication: if a=b then ac=bc8. Division: if a=b and c≠0, then a/c=b/c9. Commutative: a+b=b+a ; ab=ba10. Associative: (a+b)+c=a+(b+c) ; (ab)c=a(bc)11. Distributive: a(bc)=(ab)+(ac)

Quadrilaterals : 1. Trapezoid:a. 1 pair of parallel sides (bases)b. angles on the same side of bases are supplementary2. Isosceles trapezoid:a. 1 pair of parallel sidesb. congruent legsc. 2 pairs of congruent base anglesd. diagonals are congruent3. Kite:a. 2 consecutive pairs of congruent sidesb. 1 pair of congruent opposite anglesc. diagonals perpendicular4. parallelograma. Opposite sides are congruent b. Opposite angels are congruentc. Consecutive angles are supplementaryd. If one angle is right, then all angles are right.e. The diagonals of a parallelogram bisect each other.f. Each diagonal of a parallelogram separates it into two congruent triangles.5. Rectanglea. parallelogram with all angles congruentb. diagonals congruent6. Rhombusa. parallelogram with all sides congruentb. diagonals perpendicularc. each diagonal bisects a pair of opposite angles6. squarea. Both a rhombus and rectangleb. all angles congruentc. all sides congruent

Proving a quad is a parallelogram : To prove a quadrilateral is a parallelogram, prove any of the following conditions:1. Both pairs of opposite sides are parallel. (note: this is the definition of a parallelogram)2. Both pairs of opposite sides are congruent.3. Both pairs of opposite angles are congruent.4. An interior angle is supplementary to both of its consecutive angles.5. Its diagonals bisect each other.6. A pair of opposite sides is both parallel and congruent.

Proving a quad is an Isosceles Trapezoid : 1. It is a trapezoid and has a pair of congruent legs. (definition of isosceles trapezoid)2. It is a trapezoid and has a pair of congruent base angles.3. It is a trapezoid and its diagonals are congruent.

Proving similar triangles : AA congruent to AASSS (Sides in proportion)SAS (2 sides in proportion, angle in between is congruent)