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Chapter 5: Problem 59

Determine whether the statement is true or false. Justify your answer. If three sides or three angles of an oblique triangle are known, then the triangle can be solved.

Short Answer

Expert verified
The statement is partially true. If all three sides are known, an oblique triangle can be completely solved using the Law of Cosines. But if only the three angles are known, we cannot solve for the sides, thus the triangle cannot be solved entirely.

Step by step solution

01

Comprehension of the Statement

The statement is: 'If three sides or three angles of an oblique triangle are known, then the triangle can be solved.' This implies that knowing the measure of three sides or three angles of an oblique triangle would allow us to completely solve the triangle, meaning determining all measures, including all sides and angles.
02

Analyzing the Sides Scenario

If three sides of a triangle are known, then by the Law of Cosines, it is possible to solve the oblique triangle. This law allows for calculating the angle measures of the triangle when all side measures are known using the formula: \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \), where \( a, b, \) and \( c \) represent the sides of the triangle and \( C \) is the angle opposite side \( c \). Hence, if the three sides of a triangle are known, the triangle can indeed be solved.
03

Analyzing the Angles Scenario

If only the three angles of an oblique triangle are known, it is not possible to solve the triangle completely. Although you can accurately calculate the value of each angle (as the internal angles of any triangle sum up to 180 degrees), you will not be able to determine the lengths of the sides, since having the angles alone does not provide enough information to solve for the sides. Therefore, if only the three angles of a triangle are known, the statement is false, as the triangle cannot be solved completely.

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