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Answer :
More than one triangle could be created having sides measuring 8cm and 15cm and an included angle measuring [tex]60\textdegree[/tex]. The correct answer is option C. Sides measuring 8 cm and 15 cm and an included angle measuring 60°
To determine which measurements could create more than one triangle, let's analyze each option:
- A. Angles measuring 85°, 45°, and 60°: The sum of the angles is 190°, which exceeds 180°, so this is not a valid triangle.
- B. Angles measuring 95°, 35°, and 50°: The sum of the angles is 180°, forming a unique triangle since the angles fully determine the shape.
- C. Sides measuring 8 cm and 15 cm and an included angle measuring 60°: This is a classic case of SAS similarity criteria in a triangle that can create more than 1 similar triangle.
- D. Sides measuring 5 inches, 7 inches, and 13 inches: According to the triangle inequality theorem, this cannot form any triangle because the sum of any two sides must be greater than the third side (5 + 7 = 12, which is less than 13).
Multiple triangles can be constructed having sides measuring 8 cm and 15 cm and an included angle measuring 60°.
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The correct option is C. Sides measuring 8 cm and 15 cm and an included angle measuring 60°
To determine which measurements could create more than one triangle, let's apply the triangle inequality theorem.
A. Angles measuring 85°, 45°, and 60°:
- Since the sum of angles in a triangle is always 180°, these angles can form a triangle. However, only one triangle can be formed with these angles.
B. Angles measuring 95°, 35°, and 50°:
- Again, the sum of angles in a triangle is 180°, so these angles can form a triangle. However, only one triangle can be formed with these angles.
C. Sides measuring 8 cm and 15 cm and an included angle measuring 60°:
- This option describes the side-side-angle (SSA) case, which may or may not form a unique triangle depending on the lengths of the sides. There can be two different triangles, one if the angle is acute and one if it's obtuse.
D. Sides measuring 5 inches, 7 inches, and 13 inches:
- These side lengths satisfy the triangle inequality theorem (the sum of the lengths of any two sides is greater than the length of the third side). However, only one triangle can be formed with these side lengths.
So, the option that could create more than one triangle is: C. Sides measuring 8 cm and 15 cm and an included angle measuring 60°
