Diagonals in a Quadrilateral
A diagonal of a polygon is created when two non-consecutive vertices are connected by a segment. Place the mouse over the picture below for an illustration.
In convex quadrilaterals, the diagonals can:
* be non-congruent
* be congruent
* bisect each other
* meet at right angles
Find all of the possible combinations of diagonals that can occur and construct the quadrilateral that they define.
First, I drew a trapezoid and created the diagonals from each vertex. I found the measure of each angle to use for a comparison. I noticed that the opposite interior angles were congruent. However, the segments formed by the intersection of the diagonals was not congruent.
Next I created a square and its diagonals. I looked to see what relationships there were. The diagonals bisected each other and formed perpendicular lines. The interior angles were all 90 degrees.
The next figure was a kite. I drew the diagonals inside the kite. I determined that one diagonal was a perpendicular bisector of the other diagonal. Also two sets of congruent triangles were formed. The interior angles all equal 90 degrees.
The next figure that I drew was a rectangle. Once I created the diagonals, I measured the segments created by the intersection to see if they were congruent. Also the opposite interior angles were congruent and that was proved by measurements. All triangles were also congruent.