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If $|\mathbf{a}|+|\mathbf{b}|=|\mathbf{c}|$ and $\mathbf{a}+\mathbf{b}=\mathbf{c}$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is
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$\begin{aligned}
& \mathbf{a}+\mathbf{b}=\mathbf{c} \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2 \mathbf{a} \cdot \mathbf{b}=|\mathbf{c}|^2 \\
& \text { and }|\mathbf{a}|+|\mathbf{b}|=|\mathbf{c}| \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2|\mathbf{a}||\mathbf{b}|=|\mathbf{c}|^2 \\
& \therefore \mathbf{a} \cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}| \Rightarrow \cos \theta=1 \Rightarrow \theta=0 .
\end{aligned}$
& \mathbf{a}+\mathbf{b}=\mathbf{c} \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2 \mathbf{a} \cdot \mathbf{b}=|\mathbf{c}|^2 \\
& \text { and }|\mathbf{a}|+|\mathbf{b}|=|\mathbf{c}| \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2|\mathbf{a}||\mathbf{b}|=|\mathbf{c}|^2 \\
& \therefore \mathbf{a} \cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}| \Rightarrow \cos \theta=1 \Rightarrow \theta=0 .
\end{aligned}$
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