MATH 4345/5345 Lab #1 - Exploring the Angle Bisectors of a Parallelogram
- 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?
Angle Bisectors of a Parallelogram
Conjecture 1 (In what case(s), if any, will the central rectangle formed by angle bisectors be a square?)
Conjecture 2 (In what case(s), if any, will the central rectangle not exist?)
Conjecture 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.