Common Core Standards: Math
5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
If you've worked as a babysitter (or ever interacted with a little kid), you've certainly had a conversation very similar to this one at some point in your past:
You: It's time for bed!
You: Because it's 10 pm.
You: Because I'm the babysitter, and I promised your mom you'd be asleep by now.
You: Because little kids need their sleep.
You: Because you need sleep to be healthy.
And so it goes, long into the night, until you fall asleep while the kid continues to watch TV.
Little kids have an inborn need to have things proven to them. (Need another example? Try telling a toddler that it's a bad idea to put things up her nose. Then get ready to call 911.)
Mathematicians are the same way. They would never accept a rule like, "Never put things up your nose," just because someone says it's true. Even after you tell them it's a bad idea, they'd need to prove it. (Now you know that if mathematicians don't stick pens up their noses, they've proven it to be a bad idea.)
Your students should step up to the plate and prove some geometric rules on their own. Before they do, though, they should know that theorems are proven statements of fact and postulates are logical assumptions that are basic enough not to be proven. They should also understand the properties of lines, angles, and triangles.
Once they've grasped these concepts well enough, they can prove theorems about parallel lines and transversals and their angles. Then, students can take this information and apply it to triangles.
It's important that students know they can apply theorems and concepts from one area of geometry to another. For instance, seeing a triangle as two parallel lines cut by two transversals will help them understand connections between angles in both those situations.
Here's an example of the exterior angle theorem in action that can be shown to students.