3070.12 – A Supplement/Complement Conjecture

Suppose the acute angle ∠B\angle B has nn degrees. True or false: if ∠B\angle B is subtracted from its supplement the result is twice ∠B\angle B’s complement.

  • Give nn several values and test this statement.

  • Show algebraically that the statement is either true or false.

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Solution

If you try a few examples, you'll soon believe the statement is true.

Let's try a general argument. Let ∠S\angle S be the supplement of ∠B\angle B, and ∠C\angle C be the complement.

Then:

S−B=180−B−B=180−2B=2(90−B)=2C\begin{align*}S - B &= 180 - B - B \\&= 180 - 2B \\&= 2(90 - B) \\&= 2C\end{align*}