 In Proclus' classification of the propositions into problems and theorems, he identifies Book 1 Proposition 4 as the first true theorem in the elements. And it's quite a mouthful. If two sides and the included angle of one triangle are equal to the two sides and the included angle of a second, the two triangles are equal, and the remaining angles and sides of one triangle are equal to the corresponding angles and sides of the other. Now Euclid's phrasing is different from our own, but don't get lost in the words. We know this as the side-angle-side congruence, giving us the corresponding parts of congruent triangles, our congruent CPCTAC. And you might ask yourself, why do we talk about congruence when Euclid talks about equal? To understand the reason, let's take a look at Euclid's proof. So let's take our two triangles with side AB equal to side DE, side AC equal to side DF, and the included angle BAC equal to the angle EDF. Now since side AB is equal to side DE, we can place it atop DE and it will coincide end to end. And that's because of our common notion, things which coincide are equal. Likewise AC can be placed atop DF and angle BAC is equal to angle EDF. And that means the angles and the sides coincide, they can be placed right on top of each other. And that means the endpoint C coincides with F and the endpoint B with E. But axiom one between any two points, a straight line may be drawn, BC coincides with EF, because they're straight lines between the same points. Since BC coincides with EF and AB with DE, that also means angle ABC coincides with angle DEF. And likewise the angle BCA has to coincide with the angle EFD, they must also be equal. And here we see an important distinction in Euclid. In the elements, equal means that we can make things coincide. And this works fine for plane figures, we can take a plane figure apart and make it coincide with another. But this doesn't work so well for solid figures, we can't make two identical solid figures coincide. And so instead we might say things like the lengths of the side and the measures of the angles are equal. And because the lengths and measures are not the sides of the angles themselves, we use the term congruent. And so we say that the sides of the angles are congruent. So without going into the details, Euclid then proves in a similar fashion a number of very important propositions. Book 1, Proposition 5, base angles of isosceles triangles are equal. Proposition 8, what we now know as side, side, side congruence. Proposition 15, vertical angles are equal. And Proposition 26, what we now call angle, side, angle congruence. How about parallelograms? Euclid categorizes quadrilaterals in the following way. A square is an equilateral right-angled quadrilateral. An oblong is right-angled but not equilateral. A rhombus is equilateral but not right-angled. A rhomboid has opposite sides and angles equal but is neither right-angled nor equilateral. And all other quadrilaterals are trapezius. And there are a couple things worth pointing out about Euclid's definitions. First, we might note that the definitions are exclusive. Every quadrilateral fits exactly one of the categories. And we also note Euclid's definition of a parallelogram. Specifically, there isn't one. And similarly, while trapezius fall into the all other quadrilaterals, there's no specific definition for a trapezoid. However, Euclid does know that parallelogramic regions have certain useful properties. And so we'll call them parallelograms even though Euclid doesn't have a specific term for them beyond parallelogramic regions. So Euclid proves an important property of parallelograms. Book 1, Proposition 29. If a transversal cuts to parallel lines, the alternate interior angles are equal. And this allows him to prove opposite sides and opposite angles of parallelogramic regions are equal. What about areas? We know many area formulas in geometry, and here's what Euclid has to say about areas. Um, not a lot. In fact, there isn't any area formula as we would recognize it in Euclid. Later on, we'll see that there are propositions in Euclid that we can interpret as area formulas, but the idea of calculating the area of a figure is nowhere in Euclid. The closest you get in book 1 is the following. Parallelograms, which are on the same base, edit the same parallels, equal one another. Well, let's take A, B, C, D, and B, C, F, E to be parallelogramic regions, both on base B, C, and between the parallels, A, F, and B, C. Since we know that opposite sides of a parallelogram are equal, we know that A, B is equal to D, C. B, C is equal to A, D, and B, C is equal to E, F, because of their opposite sides of a parallelogramic region. And things equal to the same thing are equal to each other, and so we know that A, D is equal to E, F. Now, suppose we add D, E to both. Since we're adding equals to equals, the results are equal, and so A, E is equal to D, F. And by the properties of parallel lines, angle F, D, C is equal to angle F, A, B. And so side angle side congruence means that triangle A, E, B is equal to triangle D, F, C. At this point, we introduce what I sometimes call scotch tape and scissors geometry. A lot of geometry can be understood by the proper application of scotch tape and scissors. So in this case, we're going to apply those scissors, and we're going to remove triangle D, G, E from both of these triangles. And since equals subtracted from equals leaves the result equal, then we know that the remaining regions have to be equal. So I'm going to add triangle B, C, G to both. And if equals are added to equals, the results are equal. And notice that when we tape these two pieces together, we get parallelogramic region A, B, C, D, which is equal to parallelogramic region B, C, F, E. This is an example of a dissection proof, a very important type of proof in the Euclidean geometry. Two figures are shown to be equal because we can dissect and reconstitute one from the other.