Q:

What must one justify to prove that a quadrilateral ABCD is a parallelogram?

A:

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.


Is this answer helpful?

Similar Questions

  • Q:

    What are the types of quadrilaterals?

    A:

    A quadrilateral is a four-sided shape, such as a rectangle or parallelogram. In order for a shape to be considered a quadrilateral, it has to have four straight sides and be two-dimensional.

    Full Answer >
    Filed Under:
  • Q:

    How many quadrilaterals are there?

    A:

    A quadrilateral is a four-sided polygon. There are many different possible shapes and variations a quadrilateral could take, but there are six main types according to the BBC.

    Full Answer >
    Filed Under:
  • Q:

    How do you prove the complementary angle theorem?

    A:

    The first theorem used to prove that two angles are complementary states that if two angles are complementary to a third angle, then they are typically congruent to each other. The second theorem states that complements of congruent angles are congruent, which means that if two angles are complements of congruent angles, they are congruent to each other. Complementary angles are those that have a sum of 90 degrees.

    Full Answer >
    Filed Under:
  • Q:

    What is a counterexample in geometry?

    A:

    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 >
    Filed Under:

Explore