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If $$\frac{a+b}{b+c} = \frac{c+d}{d+a}$$, which of the following statements is always true?
Given, $$\dfrac{a+b}{b+c} = \dfrac{c+d}{d+a}$$
From, the properties of ratio, $$\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}=\dfrac{a+b+c+d}{a+b+c+d}$$
Now there are two possibilities,
i.) $$a+b+c+d\ne\ 0$$
When, $$a+b+c+d\ne\ 0$$, $$\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}=\dfrac{a+b+c+d}{a+b+c+d}=1$$
So, $$\dfrac{a+b}{b+c}=1$$
or, $$a+b=b+c$$
or, $$a=c$$
ii.) $$a+b+c+d=\ 0$$
In this case $$\dfrac{a+b+c+d}{a+b+c+d}$$ won't give a finite value.
So, either $$a=c$$ or $$a+b+c+d=0$$ is always true.
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