High School: Geometry
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Whether students think you're acute or obtuse, the important thing is that they all know you're right. Not only can you prove it to them, you'll get them to prove it for themselves. Yep, it's time to talk about theorems about lines and angles, which all point to the same end goal: proving congruence.
For many students, proofs are the most grueling part of geometry. It's best to ease them in by reminding them that proofs are just logical statements that flow from one to the next. They aren't a secret alien language, no more so than algebra or trigonometry. Honest.
Students should know that a theorem is different from a postulate because it is a formula or statement deduced from a chain of reasoning, or a series of other proofs already accepted. A postulate, on the other hand, is an assumed truth based on rational geometric principles.
Different students prefer different types of proofs. Some take a liking to paragraph proofs so they can provide a stream-of-consciousness rant and have it be an acceptable answer. Others prefer the two-column proof so that all their arguments are clearly laid out in front of them. Be lenient as to the format of the proof at first, but make sure your students know what the point of a proof actually is and what their goal is in writing one.
Knowing basic information and definitions about lines and angles is crucial, since they'll be the building blocks of your students' arguments. For instance, many theorems concerning lines and angles will make use of the fact that a straight line is 180°. Vertical angles come to mind.
You can let students know that once one proof is worked out, they can use the result in future proofs. Much like Pokémon, proofs are meant to be collected and used when tackling difficult problems. So we can use the vertical angle proof to prove another theorem about alternate interior angles.
Can Pikachu do that? We didn't think so.
Here is recap video of vertical angle theorem.
- ACT Math 1.2 Plane Geometry
- ACT Math 1.3 Plane Geometry
- ACT Math 1.4 Plane Geometry
- Parallel Lines and Transversals
- Proving Lines are Parallel
- Proving Lines are Perpendicular
- All You Need to Know about Slopes of Parallel and Perpendicular Lines
- Vertical Angle Theorem