The big picture
Each idea is a claim about a shape. Each has an argument that proves it.
| Idea | Remember |
|---|---|
| Which claims force the name | A safe test is passed by every parallelogram. No other shape passes it. |
| One diagonal, two triangles | A landing diagonal gives alternate angles, and both side pairs come out parallel. |
| Both pairs equal | SSS on the shared diagonal, then the same alternate-angle steps. |
| Three more ways in | One pair equal and parallel, bisecting diagonals, matching opposite angles. |
| The midpoint theorem | Join two midpoints of a triangle. The segment is parallel to the third side. It is exactly half that side. |
| Medians and the centroid | The three medians meet at one point, the centroid. It splits every median 2 to 1 from the corner. |
| The midpoint 4-gon | Join the four side midpoints of any quadrilateral in order. They make a parallelogram. |
| Fold lines and half-turns | In every parallelogram the diagonals bisect each other. A half-turn about their crossing swaps each corner with the opposite one. So every parallelogram turns onto itself. A slanted one has no fold line. |
| Any 4-gon covers a floor | Four copies of one quadrilateral meet at a point. Each copy gives one corner. There is no gap and no overlap. |
You can now name the measurements that prove a parallelogram. You can give the argument for each one.
Next comes three dimensions: Surface Area and Volume. When a claim sounds true, ask what would prove it.