Math Unit 2 (Proofs)
Reflexive Postulate of Equality/Congruence
A number/geometric figure is eual to itself
Definition of Square Root
A quantity that when multiplied by itself yields a given number (EX: a^2 = b then square root of b is a or -a
Theorems
A true statement that needs to be proven
Distributive Postulate of Equality
An expression in the form a(b+c) = ab+ac
Auxiliary Line
An extra line you can draw to help prove
Postulates
Concepts that are so basic you cannot prove them
Synthetic Proofs
Constructing a proof without having any givens or a diagram. 1. Draw Diagram and Label 2. Write the Given and the Prove 3. Mark up the diagram 4. Make your plan. 5. Write two-column proof
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.
Straight Angle Postulate
If the sides of an angle form a straight line, the measure is 180 degrees
Converse of Corresponding Angles Postulate
If two coplanar lines are intersected by a transversal such that two corresponding angles are congruent, then the lines are parallel.
Converse of AIA Theorem
If two coplanar lines are intersected by third line such that the AIA's are congruent, than the lines are parallel. **Need to prove
Linear Pair Theorem
If two lines intersect to form adjacent angles, called a linear pair, then the angles are supplementary **Need to prove
Vertical Angles Theorem
If two lines intersect to form vertical angles, the vertical angles are congruent. **Need to prove
Alternate Interior Angles (AIA) Theorem
If two parallel lines are intersected by a transversal, than the alternative interior angles are congruent. **Need to prove
Alternate Exterior Angles (AIA) Theorem
If two parallel lines are intersected by a transversal, than the two angles on alternate sides of the transversal outside of the parallel lines are congruent. **Need to prove
Co-Interior Angles Theorem
If two parallel lines are intersected by a transversal, than the two angles on the same side of the transversal between the parallel lines are supplementary. **Need to prove
Co-Exterior Angles Theorem
If two parallel lines are intersected by a transversal, than the two angles on the same side of the transversal outside of the parallel lines are supplementary. **Need to prove
Corresponding Angles Postulate
If two parallel lines are intersected by a transversal, then all corresponding angles are congruent.
Postulates of Equality
If two quantities are equal, then adding, subtracting, multiplying, and dividing BOTH sides of the equations by the same amount will keep the quantity. (EX: solving for x algebraically)
Substitution Property
If values are equal than one may be substituted for the other (EX: A=B, B=C: A=C)
Diagonals
Parallelogram: bisect each other, perpendicular. Rhombus: bisect each other. Rectangle: bisect each other, congruent diagonals. Square: bisect each other, perpendicular, congruent. Isosceles Trapezoid: congruent diagonals.
Indirect Proofs
Proofs by contradiction, proving the the alternate is impossible, therefore it is true. 1. State what you are trying to prove 2. Negate statement from step 1 and assume that the negation is true 3. Work through logical consequences (the beef) 4. Show a clear contradiction 5. Statement from 2 must be false 6. Statement from 1 must be true
Pythagorean Triple
Proved using an indirect proof A number of a pythagorean triple can be 2x of any number in the triple is found to be FALSE A number of a pythagorean triple cannot be 2x of any number in the triple is found to be TRUE
Two Column Proofs
Statement Side and Justifications Side
Postulates of Triangle Congruence
The basic concepts that prove triangles are congruent SSS SAS ASA AAS
Whole and Parts Postulate
The measure of the whole angle or segment is equal to the sum of the parts (EX: Angle QSR + Angle RST = Angle QST)
