Consider the following definition:
Definition: Let $(X, A)$ be a topological pair. We say $A$ has the homotopy extension property with respect to a space $Y$ if given any continuous map $f:X\longrightarrow A$ then every homotopy $F:A\times I\longrightarrow Y$ such that $F(x, 0)=f(x)$ for all $x\in A$ extends to a homotopy $G:X\times I\longrightarrow Y$ such that $G(x, 0)=f(x)$ for all $x\in X$.
Using this I'd like to show:
Theorem. Let $(X, A)$ be a topological pair where $A$ is contractibe. If $A$ has the homotopy extension property then $q:X\longrightarrow X/A$ is a homotopy equivalence.
Can anyone help me showing this?
Remark: (i) A topological space $X$ is said to be contractible if $id_X$ is null-homotopic.
(ii) I know this is proven in Hatcher, but I don't like his exposition and I didn't understand his proof.