\documentclass{article}
\usepackage{amsthm}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\begin{document}
\section{Theorems}
\begin{theorem}
Let A be the hypotenuse, B be the base, and C be the height of the triangle.
According to the Pythagorean theorem, the square of the hypotenuse of a right
angled triangle is equal to the sum of the squares of the sides of the triangle.
\end{theorem}
\section{Definition}
\begin{definition}
Pythagoras theorem states that, in a right triangle, the square of the hypotenuse
is equal to the sum of the square of the other two sides.
\end{definition}
\section{Corollary}
\begin{corollary}
if a triangle has side lengths a, b, c, with, then: If \[c^2 > a^2 + b^2\], then the
triangle is obtuse. If \[c^2 = a^2 + b^2\], then the triangle is right. If \[c^2 < a^2
+ b^2\], then the triangle is acute.
\end{corollary}
\section{Lemma}
\begin{lemma}
Triangles with the same base and height have the same area. A triangle which
has the same base and height as a side of a square has the same area as a half of
the square.
\end{lemma}
\end{document}