proof of parallelogram theorems
This was proved in the parent (http://planetmath.org/ParallelogramTheorems) article. ∎
If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Suppose is the given parallelogram, and draw .
If one pair of opposite sides of a quadrilateral are both parallel and congruent, the quadrilateral is a parallelogram.
Again let be the given parallelogram. Assume and that and are parallel, and draw .
Since and are parallel, it follows that the alternate interior angles are equal: . Then by SAS, since they share a side.
Again by CPCTC we have that , so both pairs of sides of the quadrilateral are congruent, so by Theorem 2, the quadrilateral is a parallelogram. ∎
The diagonals of a parallelogram bisect each other.
Let be the given parallelogram, and draw the diagonals and , intersecting at .
Since is a parallelogram, we have that . In addition, and are parallel, so the alternate interior angles are equal: and . Then by ASA, .
By CPCTC we see that and , proving the theorem. ∎
If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
Let be the given quadrilateral, and let its diagonals intersect in .
Then by assumption, and . But also vertical angles are equal, so and . Thus, by SAS we have that and .
By CPCTC it follows that and that . By Theorem 1, is a parallelogram.
|Title||proof of parallelogram theorems|
|Date of creation||2013-03-22 17:15:48|
|Last modified on||2013-03-22 17:15:48|
|Last modified by||rm50 (10146)|