(a) Which sets of measurements could be the interior angle measures of a a triangle? Select each correct answer A \( 20^{\circ}, 80^{\circ}, 100^{\circ} \) B \( 35^{\circ}, 35^{\circ}, 105^{\circ} \) C \( 15^{\circ}, 75^{\circ}, 90^{\circ} \) D \( 60^{\circ}, 60^{\circ}, 60^{\circ} \) E \( 10^{\circ}, 10^{\circ}, 160^{\circ} \)
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To determine which sets of measurements could be the interior angle measures of a triangle, we remember that the sum of the interior angles in a triangle must always equal \(180^{\circ}\), and all angles must be positive and less than \(180^{\circ}\). Now, let’s break down each option: - A: \(20^{\circ} + 80^{\circ} + 100^{\circ} = 200^{\circ}\) (not valid) - B: \(35^{\circ} + 35^{\circ} + 105^{\circ} = 175^{\circ}\) (not valid; one angle too large) - C: \(15^{\circ} + 75^{\circ} + 90^{\circ} = 180^{\circ}\) (valid!) - D: \(60^{\circ} + 60^{\circ} + 60^{\circ} = 180^{\circ}\) (valid!) - E: \(10^{\circ} + 10^{\circ} + 160^{\circ} = 180^{\circ}\) (valid!) Thus, the valid angle sets are C, D, and E. Now let’s add some fun facts! Did you know that triangles have been studied for thousands of years? The ancient Greeks, particularly mathematicians like Euclid, were fascinated by triangles and their properties, laying down rigorous foundations for geometry! In the real world, triangles aren’t just theoretical shapes; they are used heavily in architecture and engineering! Triangles provide stability to structures—think of a bridge or a roof truss—because their shape distributes weight evenly. Next time you see a triangular structure, you’ll appreciate the geometric genius behind it!
