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Chapter 1: Problem 22

Can a triangle contain more than one obtuse angle? Explain.

Short Answer

Expert verified
A triangle cannot contain more than one obtuse angle. Having two obtuse angles in a triangle would cause the sum of the angles to be greater than 180 degrees, contradicting the property of a triangle where the sum of its angles must always be 180 degrees. If a triangle has one obtuse angle, the sum of the other two angles must be less than 90 degrees.

Step by step solution

01

Recall the properties of a triangle

A triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always 180 degrees.
02

Define an obtuse angle

An obtuse angle is an angle with a measure greater than 90 degrees but less than 180 degrees.
03

Consider a triangle with a single obtuse angle

Let's assume a triangle has one obtuse angle, which means it has a measure greater than 90 degrees. Since the sum of the angles in a triangle is 180 degrees, the other two angles must have a sum less than 90 degrees (because 180 - 90 = 90).
04

Analyze the possibility of a triangle with more than one obtuse angle

Now, let's assume a triangle has two obtuse angles. This would mean that the measures of these two angles would both be greater than 90 degrees. However, the sum of two such angles would be greater than 180 degrees (since 90 + 90 = 180), which contradicts the property of a triangle where the sum of the angles must always be 180 degrees.
05

Conclusion

A triangle cannot contain more than one obtuse angle. If a triangle has one obtuse angle, the sum of the other two angles must be less than 90 degrees. If a triangle were to have two obtuse angles, then the sum of the angles in the triangle would be more than 180 degrees, which contradicts the property of a triangle.

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