 Welcome, Geometry Fans. This video is going to teach you about 45-45-90 triangles. Back in Chapter 6, we talked about how squares have all 90-degree angles at each corner. We also talked about how parallelograms, and therefore squares, have the unique property that if we make a diagonal across the center, we make two congruent triangles. In fact, we can make congruency marks here and here, because we know that those two sides are the same from a square. The result is a triangle that is called a 45-45-90 triangle. Properties of the 45-45-90 triangle is that the two legs are congruent. But why is it called 45-45-90? We have a 90-degree angle on the bottom left. That is our right angle. The other two angles are congruent. We find them by finding that their total has to equal 90. And if there are two of them, then we can say that each of them is going to be 45 degrees. Therefore, it is called a 45-45-90. An important piece about these 45-45-90 triangles is the length of the sides. If each of these lengths is the same, then we can find out the length of the hypotenuse by using the Pythagorean theorem. The Pythagorean theorem says that the hypotenuse is going to be the square root of x squared, one of the sides squared, plus x squared again. This equals the square root of 2x squared, which we can separate further into the square root of 2 times the square root of x squared. The square root of x squared is just x, so we can write this as x times the square root of 2. And in any 45-45-90 triangle, we find that the sides have the ratio 1 to 1 to the square root of 2. Look to the other videos for some example problems that deal with the 45-45-90 triangle.