My mother, who reads this blog, reports that she’s lost a few nights’ sleep lately, tormented by thoughts of Knights, Knaves and Crazies. Serves her right. Once when she and I were very young, she tormented me with a geometry puzzler that I now know she must have gotten (either directly or indirectly) from Lewis Carroll; you can find it here. If she remembers the solution, she should be able to sleep tonight.
Herewith, a proof that a right angle can equal an obtuse angle. The puzzle, of course, is to figure out where I cheated.
But wait! Let’s do this as a video, since I’m starting to fool around with this technology and could use the practice. Consider this more or less a first effort. If you prefer the old ways, you can skip the video and read the (identical) step-by-step proof below the fold.
Or, if you prefer to skip the video, start here:
1. Draw a rectangle:
2. Bisect the horizontal edge of the rectangle, and continue the perpendicular bisector downward:
3, Attach, at the bottom right-hand corner, a line segment as shown, and exactly equal in length to a vertical side of the rectangle. I’ve colored the new segment and one of the vertical sides dark blue as a reminder that their lengths are equal:
4. Connect the tops of the two dark blue segments:
5. Bisect the line you just drew, and continue the perpendicular bisector downward:
6. Add the two red lines as shown. Notice that these red lines are two sides of an isoceles triangle, hence equal in length.
7. Continuing to focus on that same isoceles triangle, let’s mark the equal angles with double arcs:
8. Add the green lines as shown. Notice that these are also two sides of an isoceles triangle (because they meet on the perpendicular bisector of the triangle’s “base”), which we’ll remember because they’re the same color:
9. Now we’ve got two blue-green-red triangles, one on each side of the diagram. The two blue sides are equal, the two red sides are equal, and the two green sides are equal, so the triangles are congruent. Therefore corresponding angles are equal. Let’s mark the corresponding blue-side-to-red-side angles with single arcs to remind ourselves they’re equal:
10. Okay, the two single-arc angles are equal. So are the two double-arc angles. On the left side, if you subtract the double-arc angle from the single-arc angle, you get a right angle. On the right side, if you subtract the double-arc angle from the single-arc angle, you get an obtuse angle. Therefore a right angle and an obtuse angle are equal.