There are four combinations of signs for a and b where the magnitudes are the same but the signs are + or -: a and b both positive; a positive and b negative; a negative and b positive; a and b both negative.
Case 1: a=1, b=2, |a|+|b|=3=|a+b|
Case 2: a=1, b=-2, |a|+|b|=3, |a+b|=1
Case 3: a=2, b=-1, |a|+|b|=3, |a+b|=1
Case 4: a=-1, b=2, |a|+|b|=3, |a+b|=1
Case 5: a=-2, b=1, |a|+|b|=3, |a+b|=1
Case 6: a=-1, b=-2, |a|+|b|=3=|a+b|
So the proposition isn't true.