We finish by noting, viewing the top line, that + + = 180°, however these are    also the interior angles of the triangle formed. Q.E.D. (Latin for erat “quod demonstrandum” meaning "what was required to be proved") Similar Triangles & Theorem of Pythagoras There are many proofs of this theorem I’ll present the most obvious proof. We – start with a right triangle, that is, a triangle with one right angle and refer to the following diagram. We label the vertices of the triangle, that is, points where the sides of the triangle intersect, using lowercase letters, the sides of the triangle using uppercase letters, and angles using Greek letters. Note that the vertex opposite the side uses the same letter and the angle at the vertex uses the corresponding Greek letter. Then we drop a perpendicular to the hypotenuse (longest side, line segment ab) and intersecting Point c. Note that refers to the length of the line segment from Point c to Point d. Also note “D” that is the length of the hypotenuse, C1 is the length from Point d to Point a, C2 the “C” length from Point d to Point b, and therefore C = C1 + C2. Now, to proceed, we must discuss Any similar figure, including Similar Triangles. triangles, has the same shape as another, but a different size. For triangles this means angles are preserved, that is, the angles are the same in similar triangles. Using what we’ve reviewed previously, that is, the sum of all angles in a triangle equals 180°, we assign symbols to all the angles in the previous figure. After doing this, therefore: is similar to and is similar to . Also This is the meaning of in similar figures, all lengths scale the same. “same shape, different size.” This also means that lengths are in proportion in similar figures and, now, we can write a mathematical equation, a proportion, to describe this: and We will use these equations in just a little bit to derive a second proof of the Pythagorean Theorem. Before we do that, note: of + of = of .” “area ” “area ” “area And areas are proportional to the length of one side squared, therefore the is immediately proven: Pythagorean Theorem Let’s now revisit the equations above applying cross-multiplication: and
We finish by noting, viewing the top line, that  +  +  = 180°, however these are also the interior angles of the triangle formed. Q.E.D. (Latin for “quod erat demonstrandum” meaning "what was required to be proved") Similar Triangles & Theorem of Pythagoras There are many proofs of this theorem – I’ll present the most obvious proof. We start with a right triangle, that is, a triangle with one right angle and refer to the following diagram. We label the vertices of the triangle, that is, points where the sides of the triangle intersect, using lowercase letters, the sides of the triangle using uppercase letters, and angles using Greek letters. Note that the vertex opposite the side uses the same letter and the angle at the vertex uses the corresponding Greek letter. Then we drop a perpendicular to the hypotenuse (longest side, line segment ab) and intersecting Point c. Note that “D” refers to the length of the line segment from Point c to Point d. Also note that “C” is the length of the hypotenuse, C1 is the length from Point d to Point a, C2 the length from Point d to Point b, and therefore C = C1 + C2. Now, to proceed, we must discuss Similar Triangles. Any similar figure, including triangles, has the same shape as another, but a different size. For triangles this means angles are preserved, that is, the angles are the same in similar triangles. Using what we’ve reviewed previously, that is, the sum of all angles in a triangle equals 180°, we assign symbols to all the angles in the previous figure. After doing this, therefore: is similar to and is similar to . Also in similar figures, all lengths scale the same. This is the meaning of “same shape, different size.” This also means that lengths are in proportion in similar figures and, now, we can write a mathematical equation, a proportion, to describe this: and We will use these equations in just a little bit to derive a second proof of the Pythagorean Theorem. Before we do that, note: “area of ” + “area of ” = “area of .” And areas are proportional to the length of one side squared, therefore the Pythagorean Theorem is immediately proven: Let’s now revisit the equations above applying cross-multiplication: and