# Thoughts on teaching geometry

For the theorem involving the radius and tangent of a circle, the students will be given a diagram without the angle being measured. An interactive is unnecessary in this case, because it's obvious that any tangent is just a rotation of the diagram. First, students will be asked what they notice about the angle between the radius and tangent of a circle. It won't take them long to conjecture that it's always a \(90^\circ.\) The proof by contradiction will simply be given to them, because a proof by contradiction is likely too difficult for the vast majority of students.

Each geometry theorem may have one or two Geogebra files. One if the measurement can be made precise, and two if the measurement cannot. For example, the exterior angle theorem has one Geogebra file, because all measurements can be shown without giving away the theorem, whereas the theorem about angles in semicircles will have two, because we cannot label the angle without giving away the theorem. What follows is a description of the process for each case.

### Exterior angle theorem

For the exterior angle theorem, the students begin with the interactive and are asked what they notice. It shouldn't take long before they notice that the exterior angle is the sum of the two interior angles. Their next task is to prove the theorem if they can. Then they will be shown a proof, in case their proof is fallacious or they couldn't come up with a proof.

### Thales' theorem

The theorem about angles in semicircles will have two Geogebra files. The first is to allow the students to experiment and make a conjecture. Surely, after a bit of experimentation, they will conjecture the angle to be \(90^\circ.\) Then they will be given an interactive with the angle measured. This second interactive should bolster their confidence in the conjecture, but it's not a proof, because they haven't verified their conjecture for the infinitely many configurations. Their next task is to prove the theorem if they can. Then they will be shown a proof, in case their proof is fallacious or they couldn't come up with a proof.

### Side splitter theorem

The student is unlikely to discover the side splitter theorem on their own. Simply tell them the theorem, then have them play with the interactive to give them evidence.

### Alternate interior angles

Once students can measure angles, this theorem can be discovered. Have the student draw various parallel lines cut by a transversal. Have them measure the alternate interior angles, then have the student make a conjecture.

### Vertical angles

Again, once students can measure angles, this theorem can be discovered. Have the student draw various pairs of intersecting lines. Have them measure the vertical angles, then have the student make a conjecture.