# MATH 4345/5345 Lab #1 - Exploring the Angle Bisectors of a Parallelogram

Author:
Todd Abel
We discussed in class the angle bisectors of the interior angles of a parallelogram, and determined that: Angle bisectors of opposite interior angles are either parallel or the same line. We guessed that the angle bisectors, except in certain circumstances, will actually intersect to form a rectangle. To prove that, though, we need to prove that the angle bisectors of adjacent angles are perpendicular. Use the applet below to try and justify why that might be true.

## Angles of a Parallelogram

What do you notice about the measures of adjacent angles in a parallelogram? What does this imply about the triangle formed by angle bisectors of two adjacent angles?

## Now Let's Make Some Conjectures

So the angle bisectors intersect to form a rectangle. Drag points in the applet below in order to make conjectures about the following questions: 1) In what case(s), if any, will the central rectangle formed by angle bisectors be a square? 2) In what case(s), if any, will the central rectangle not exist? 3) In what case(s), if any, will the triangles formed by the angle bisectors and the included sides be isosceles?

## Now work to prove it

Create a sketch or series of sketches in Geogebra to confirm your conjectures. You'll do this in your own Geogebra environment. Create a sketch that illustrates whether your conjecture was correct. For instance, if I were to conjecture that the central rectangle is a square when the parallelogram is a rhombus, I would construct a rhombus (one that is always a rhombus no matter how I drag it) and the angle bisectors to show it is true. You can do all three in one sketch (add some text to explain each one) or create a separate sketch for each. When you're done, visit the menu in the upper right to save the sketch, then share it using the link. Copy and paste the links below.