Write what you would do to prove indirectly that a triangle cannot have two obtuse angles.

Solution

Verified$\text{\color{#c34632}Step 1:}$ Assume that a triangle can have two obtuse angles. In particular, in $\triangle ABC$, assume that $\angle A$ and $\angle B$ are obtuse angles.

$\text{\color{#c34632}Step 2:}$ By the definition of a obtuse angle $m\angle A > 90\text{\textdegree}$ and $m\angle B > 90\text{\textdegree}$.

By the addition property of equality, $m\angle A + m\angle B > 180\text{\textdegree}$. By the protractor postulate $m\angle C =n\text{\textdegree}$,where $n$ is a positive number less than or equal to 180. By the addition property of equality, $m\angle A + m\angle B + m\angle C > 180\text{\textdegree} + n\text{\textdegree}$. By the triangle-sum theorem, $m\angle A + m\angle B + m\angle C=180\text{\textdegree}$.

$\text{\color{#c34632}Step 3:}$ Note that the last two statements in Step 2 are contradictory. Therefore, the assumption that a triangle can have two obtuse angles is false. The given statement must be true.

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