Here’s how each set of conditions relates to forming triangles:
| Triangle Conditions | Unique, More Than One, or No Triangle | |-------------------------------------------------------------------------|--------------------------------------| | Side lengths of 5 inches, 12 inches, and 20 inches | No Triangle | | Angle measures of 42°, 53°, and 85° | Unique Triangle | | Side lengths of 8 centimeters, 17 centimeters, and 24 centimeters | Unique Triangle | | Angle measures of 36°, 36°, and 108° | Unique Triangle | | Angle measures of 39° and 74°, and a side length of 9 centimeters between those angles | More Than One Triangle | | Side lengths of 14 inches and 16 inches, and an angle measure of 27° between those sides | Unique Triangle |
Explanation:
- For the side lengths of 5, 12, and 20 inches, they do not satisfy the triangle inequality: 5 + 12 is not greater than 20.
- The angles of 42°, 53°, and 85° add up to 180° and form a valid triangle.
- The side lengths 8, 17, and 24 satisfy the triangle inequality.
- The angles of 36°, 36°, and 108° also sum to 180° and form a valid triangle.
- The angle measures of 39° and 74° with a side in between can form more than one triangle if you consider the triangle's congruence and various arrangements.
- The side lengths of 14 inches and 16 inches with an angle between them follow the SSA condition which creates a unique triangle by the law of sines.
Make sure to cross-check the conditions if needed for thorough understanding!
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