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Is there a way to change the vertical spacing between lines for only one paragraph? Suppose, for example, I want everything in my document to be single-spaced, except for one paragraph that I want to be double spaced (or custom spaced to any length in general). Are there any simple suggestions for how to do this?

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You can place the adjustment to the line spacing inside a group (meaning it'll reset afterward). You have to make sure that there's an empty line before closing the group, or an explicit \par to ensure that the paragraph building feature can be performed using the newly-specified parameters. So something like

% ...stuff before

\begingroup
\setstretch{1.5}% Adjust line spacing
% ...your paragraph here
\par
\endgroup

% ...more stuff here

But setspace conveniently provides the spacing environment that does all that:

enter image description here

\documentclass{article}

\usepackage{setspace,lipsum}

\begin{document}

\lipsum[1]

\begin{spacing}{1.5}%
\lipsum[1]
\end{spacing}

\lipsum[1]

\end{document}

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