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Problem 38

# Prove that a triangle can have, at most, one obtuse angle.

Expert verified
To prove that a triangle can have, at most, one obtuse angle, we assume that a triangle has two obtuse angles, say angle $$A$$ and angle $$B$$. Since both angle $$A$$ and angle $$B$$ are obtuse, their sum will be greater than 180 degrees. Adding angle $$C$$ to the sum of angle $$A$$ and angle $$B$$, we have: $A + B + C > 180° + C$. However, we know that the sum of the angles in a triangle is equal to 180 degrees, which means $A + B + C = 180°$. But from our calculation, we concluded that $$A + B + C > 180° + C$$, which is a contradiction. Therefore, our assumption that a triangle can have two obtuse angles is false, and a triangle can have, at most, one obtuse angle.
See the step by step solution

## Step 1: Sum of angles in a triangle

Recall that the sum of the interior angles of a triangle is always equal to 180 degrees. This will be an important fact to help us prove that a triangle cannot have more than one obtuse angle.

## Step 2: Assume a triangle has two obtuse angles

We will use proof by contradiction. Let's assume that a triangle has two obtuse angles, say angle $$A$$ and angle $$B$$.

## Step 3: Add the two obtuse angles

Now, let's add the two obtuse angles together. An obtuse angle is an angle greater than 90 degrees. Since we have assumed that both angle $$A$$ and angle $$B$$ are obtuse, their sum will be greater than 180 degrees. Mathematically, this can be represented as: $A + B > 180°$

## Step 4: Add the third angle

Now, let's consider the third angle of the triangle, angle $$C$$. It must also be greater than 0 degrees, as it is an interior angle of a triangle. Adding angle $$C$$ to the sum of angle $$A$$ and angle $$B$$, we have: $A + B + C > 180° + C$

## Step 5: Sum of angles in triangle contradiction

However, we know that the sum of the angles in a triangle is equal to 180 degrees. This means that: $A + B + C = 180°$ But from step 4, we concluded that $$A + B + C > 180° + C$$, which is a contradiction. Therefore, our assumption that a triangle can have two obtuse angles is false.

## Step 6: Conclusion

After proving that a triangle cannot have two obtuse angles, it directly follows that a triangle can have, at most, one obtuse angle.

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