$a)$
$\dfrac{a+b}{b}$$=$ $\dfrac{c+d}{d}$
$=$ $\dfrac{a}{b}$ $+$ $\dfrac{b}{b}$$=$ $\dfrac{c}{d}$$+$ $\dfrac{d}{d}$
$=$ $\dfrac{a}{b}$ $+1=$ $\dfrac{c}{d}$$+1$
$=$ $\dfrac{a}{b}$ $=$ $\dfrac{c}{d}$
$b)$
$\dfrac{a-b}{b}$$=$ $\dfrac{c-d}{d}$
$=$ $\dfrac{a}{b}$$-$ $\dfrac{b}{b}$$=$ $\dfrac{c}{d}$$-$ $\dfrac{d}{d}$
$\dfrac{a}{b}$ $-1=$ $\dfrac{c}{d}-1$
$= $ $\dfrac{a}{b}=$$\dfrac{c}{d}$
$c)$
$\dfrac{a+c}{c}=$$\dfrac{b+d}{d}$
$=$ $\dfrac{a}{c}+$$\dfrac{c}{c}=$$\dfrac{b}{d}+$$\dfrac{d}{d}$
$\dfrac{a}{c}+1=$$\dfrac{b}{d}+1$
$=$ $\dfrac{a}{c}=$ $\dfrac{b}{d}$
$d)$
$\dfrac{a+c}{b+d}=$$\dfrac{a-c}{b-d}$
$=$ $\dfrac{a}{b+d}+$$\dfrac{c}{b+d}$$=$ $\dfrac{a}{b-d}-$$\dfrac{-c}{b-d}$
$=$ $\dfrac{a}{b+d}-$$\dfrac{a}{b-d}=$$\dfrac{-c}{b-d}-$$\dfrac{c}{b-d}$
$=$ $\dfrac{a(b-d)-a(b+d)}{(b+d)(b-d)}$
$=$ $\dfrac{a}{b}=$$\dfrac{c}{d}$
