We've been focusing on segments for a while, and we think angles deserve some TLC, too. Aside from being the inspiration for several games and puzzles (like tangrams) and the backbone of many ancient proofs, this idea has led to some of the most breathtaking paradoxes in mathematics. This idea of showing objects are congruent by splitting them into smaller pieces and matching them up is one of the most fascinating ideas in geometry. Why are we adding this random BC to both sides of an equation? This is just a way to formally write out our intuitive argument about chopping big segments into smaller segments. It might seem like step 3 comes out of nowhere. If we write it out as a proof, this is what we have. Whaddayaknow? Those are exactly the segments we're given information about. Right away we see that BC shows up in both of them, so we can match those parts up. AC can be split into AB and BC, while BD can be split into BC and CD. Instead, let's split them into more manageable pieces. We can't jump straight to working with AC and BD. This problem is quite a bit trickier than what we've covered so far, since we aren't explicitly given any information about the segments we want to prove something about. Given the diagram below, show that AC ≅ BD.įrom the diagram, we see that AB ≅ CD by the tick marks, so that's our given information. Here, B may not be assumed between A and C because you have to go out of your way to pass through it (like passing through Des Moines on your way from L.A. Seems obvious, right? This fact is often called the Segment Addition Postulate and frequently comes up in geometrical proofs.

Remember that in this situation, AB + BC = AC. We don't really have a choice, but we might as well see Portland and Crater Lake. to Seattle, we need to pass through Oregon. We can assume B is between A and C, since the segment from A to C passes through B. Like vertical angles, you can read 'betweenness' directly from a diagram. Thus, we can use those two facts as given statements. Looking at the figure, we see that ∠1 and ∠2 are vertical angles, and ∠2 is congruent to ∠3 (by the tick marks). Given the following figure, show that ∠1 is congruent to ∠3. So how can we take these congruent line segments and angles and convert them into proofs? Well, we'll show you. In general, objects satisfying these three properties are called equivalence relations, since they behave a lot like actual equality. Since segments and angles are congruent when they have equal measures, it makes sense that congruence also has the reflexive, symmetric, and transitive properties.

One of the nice things about congruence is that it has a lot in common with equality.

Geometry proofs