High School: Geometry
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
If your students have gotten tired of lines and angles, they're in luck. Tell them that triangles are nothing like lines and angles. Instead, tell them triangles are new, edgy, and they've got some good points…three of them, actually. Conveniently leave out the fact that triangles are just three line segments joined together.
Students should know the basic definitions that come with triangles and how to classify them based on angles and sides. When they can use the words "equiangular" and "isosceles" in everyday conversation, you'll know you've done a good job.
Students should also be comfortable with the angles of a triangle, both interior and exterior. They should know that all the interior angles of a triangle add up to 180°, and they should know how to prove it. It's better to introduce these concepts to them by using concepts they should already know, like parallel lines and transversals.
But that's just the tip of the triangular iceberg. There's way more inside triangles than just three interior angles. For instance, we can fill a triangle with medians, line segments that join the vertices of a triangle to the midpoints of its opposite sides. We can also connect the midpoints of each side in the triangle to form a similar triangle that's half the size of the original one.
Finally, students shouldn't get lost with all the theorems and postulates. They all build on each other, and it's best to keep track of these proofs and postulates so that students don't get confused. Also, students should know that using proofs and theorems they've already learned isn't cheating; it's applying the skills they've learned, and it's highly encouraged.
If studying triangles still seems more confusing than the Bermuda Triangle, give plenty of examples and draw on knowledge that's already been introduced. It's difficult to learn something new without understanding the basics, so go back and re-derive some proofs if needed. You'll get grumbles in the moment, but they'll thank you for it later. Better than being stuck in the Bermuda Triangle, anyway.
- Exterior and Remote Interior Angles
- Formal Proofs
- Midsegment Theorem
- Proving Triangles Congruent
- Centers of a Triangle