Regents Prep explains that the quickest method for proving that a quadrilateral is a parallelogram is if one pair of opposite sides of the quadrilateral are both parallel and congruent. However, several other forms of proof exist as well.
According to Wikipedia, in Euclidean geometry, a parallelogram is a simple quadrilateral, with four edges and four corners, characterized by its two pairs of parallel sides. Because of its parallel orientation, the opposite sides of a parallelogram are equal in length. The opposite angles of a parallelogram are also equal in measure. This congruence is proven by Euclid's parallel postulate, also known as Euclid's fifth postulate; it is part of Euclid's "Elements," which serves as the basis for the shortest method of proof previously described.
The basic definition of a parallelogram lends itself to several other methods of proof. For instance, one can prove that a quadrilateral is a parallelogram if the two pairs of opposite angles in the quadrilateral are congruent. Further, if the consecutive angles in the quadrilateral are supplementary, or equal to 180 degrees, then it is a parallelogram. Finally, one can also prove a quadrilateral is a parallelogram if the two diagonals of the quadrilateral can bisect each other, or divide each other equally in half.Learn More
A counterexample, in geometry as in other areas of mathematics and logic, is an example that one uses to prove that a particular statement is false. A simple example from primary mathematics uses the statement "the inverse of a number is never an integer," and its counterexample would be 1/4. The inverse of 1/4 is 4, which is an integer. For geometry, finding counterexamples involves a few more calculations.Full Answer >
The Barbados Slave Codes were laws set up by the British to justify the practice of slavery and legalize the planters' inhumane treatment of their slaves. Under these codes, the slaves had the status of farm animals or chattel and had no human rights.Full Answer >
The Declaration of Independence was written to justify the American colonies' break from Great Britain and to make it easier to gain international support from potential allies. It enumerated the rights of man and the grievances the colonists had against British rule.Full Answer >
The incenter of a triangle is defined as the point where all three angle bisectors intersect. It can be found using a compass and a straight edge by constructing the angle bisectors of any two vertices of the triangle and marking the point where they cross. Two angle bisectors are sufficient to define the point where all three intersect, so the third angle bisector need not be drawn.Full Answer >