Corollary 26.7:

Suppose that $U$, $V$ are open path-connected subsets of a topological space such that $U$ is simply connected and $i:U\cap V\longrightarrow V$ is the inclusion map then, taking a base point $x_0\in U\cap V$,

$$\pi_1(U\cup V, x_0) = \pi_1(V, x_0)/\langle$$   Im$$\; i_*\rangle,$$

where $\langle$   Im$\; i_*\rangle$ denotes the normal subgroup generated by the image of $i_*$.