Chapter 4: Chapter 4

Problem 38

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

Short Answer

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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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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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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Chapter 4: Chapter 4

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

Short Answer

Step by step solution

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Step 1: Sum of angles in a triangle

Step 2: Assume a triangle has two obtuse angles

Step 3: Add the two obtuse angles

Step 4: Add the third angle

Step 5: Sum of angles in triangle contradiction

Step 6: Conclusion

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