# About Seifert-Van Kampen Theorem

I have the following problem of the book "Massey, Algebraic Topology an Introduction" (Chapter 4, section 4, problem 1):

Let $$X$$ arcwise connected and $$U,V$$ open arcwise connected subsets of X. Suppose that $$U\cup V=X$$, $$U\cap V$$ non-empty and arcwise connected.

Consider $$\varphi_1:\pi(U\cap V)\to\pi(U)$$, $$\varphi_2:\pi(U\cap V)\to\pi(V)$$, $$\psi_1:\pi(U)\to\pi(X)$$, $$\psi_2:\pi(V)\to\pi(X)$$, all morphism induced by the inclusions.

Prove that if $$\varphi_2$$ is isomorphism onto, then so is $$\psi_1$$.

I tried to use the first version of Seifert-Van Kampen Theorem but I have no result. Maybe is a wrong way to do this problem.

Any solution or hint would be appreciated.