Admittedly, parallel lines and transversal problems are very contrived in most cases and are not very real-world relevant. But, I still like those problems because they give the kids some basic algebra practice, while forcing them to think about the meanings of their equations.
In the last two years, I have consistently introduced the series of parallel-lines-and-transversal theorems using the same worksheets, and they have worked very well for my kids. My theory behind these worksheets is that: A.) In order to learn the angles vocabulary, kids need to be actively engaged in visualizing the relationships. So, why not start with making them visualize the relationships before introducing the terms? B.) Once they learn the terms, kids can discover all angle relationships via a protractor. C.) At the end, you give them some memory tools or some color-coding shortcuts to quickly figure out angle relationships for setting up angle equations. That way, even if a kid can't remember the name of an angle relationship, they can still have enough geometric knowledge to move through the algebra part. D.) In the end, as a quick check-in, kids should be able to quickly pick out the correct equations corresponding to different diagrams.
So, here we go.
1. Getting kids to visualize angle location relationships before introducing the terms.
2. Getting kids to conjecture about angle measurement relationships on their own.
3. Using colors to re-inforce angle relationships. (And I give them the memory tool that only "Same-side interior" and "Same-side exterior" angles are "Supplementary." Every other pair of recognizable relationship is congruent.)
4. Final check-in. Kids should all be able to pick out the right equations.
These are worksheets and not activities, but they are very effective in teaching the basics of angle relationships. Follow it up with a day of textbook algebra problems practice, and your kids are golden on this often-tested concept! (My Holt Geometry textbook also has an interesting angles word problems activity that I have adopted the last couple of years, which works well as an extension to ask kids to look at the application of angles in a slightly more realistic situation.)
And of course, you can always defer to Erastothenes to show kids some gee-whiz angles math.
By the way, stay tuned for my kids' straight-edge and compass construction projects. Neatest pure-geometry thing we've done in a while.