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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.

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Chapter 4

The measure of the vertex angle of an Isosceles triangle exceeds the measure of each base angle by \(30^{\circ}\). Find the value of each angle of the triangle.

Chapter 4

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Chapter 4

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