Something coherent is logical and clear. "I told you if you didn't do your homework you couldn't watch TV. You didn't do your homework so you can't watch TV." This is a coherent argument (and annoying as well).

The thing about such coherent arguments, as your Mom well knows, is that they are also easily and clearly understood. If someone speaks coherently, they are speaking clearly and lucidly. A slight variation on the meaning comes in the sense of something that is united clearly as a whole and a recognizable entity. You might say, for example, that New York City is a "culturally coherent region" within the United States.

Definitions of coherent
  1. adjective
    marked by an orderly, logical, and aesthetically consistent relation of parts
    “a coherent argument”
    synonyms: consistent, logical, ordered
    perfectly consistent and coherent
    capable of or reflecting the capability for correct and valid reasoning
    consistent with or based on or using reason
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    without logical or meaningful connection
    confused, disconnected, disjointed, disordered, garbled, illogical, scattered, unconnected
    lacking orderly continuity
    confused and not coherent; not clearly thought out
    illogical, unlogical
    lacking in correct logical relation
    not consistent with or using reason
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  2. adjective
    (physics) of waves having a constant phase relation
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    (physics) of waves having no stable definite or stable phase relation
  3. adjective
    capable of thinking and expressing yourself in a clear and consistent manner
    “she was more coherent than she had been just after the accident”
    synonyms: logical, lucid
    consistent with or based on or using reason
  4. adjective
    sticking together
    “two coherent sheets”
    synonyms: tenacious
    tending to adhere
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