Determine Whether Each Statement Below Is Always True Someti
Determine Whether Each Statement Below Is Always True Sometimestru
Determine whether each statement below is always true, sometimes true, or never true. Explain why you chose each answer.
Paper For Above instruction
1. A rectangle is a rhombus.
2. Two angles of a scalene triangle are congruent.
3. A rhombus is equilateral.
4. An equilateral triangle is equiangular.
5. A parallelogram is a rectangle.
6. An obtuse triangle contains at least two obtuse angles.
7. A square is a rhombus.
8. A triangle has side lengths 3 inches, 4 inches, and 8 inches.
Analysis and Explanation
1. A rectangle is a rhombus. - Sometimes true. A rectangle has four right angles, and a rhombus has four congruent sides. While some rectangles are squares (which are both rectangles and rhombuses), a generic rectangle with unequal sides is not a rhombus. Therefore, only specific rectangles, namely squares, satisfy both conditions. Hence, this statement is sometimes true.
2. Two angles of a scalene triangle are congruent. - Never true. A scalene triangle has all sides of different lengths, and consequently, all angles are of different measures. Therefore, no two angles are congruent, making the statement always false.
3. A rhombus is equilateral. - Always true. By definition, a rhombus has four congruent sides, which makes it equilateral. The angles can vary, but side length equality is a defining property. Hence, the statement is always true.
4. An equilateral triangle is equeangular. - Always true. An equilateral triangle has all sides equal; by the properties of triangles, this implies all interior angles are equal. Since the sum of angles in a triangle is 180 degrees, each angle measures 60 degrees, making the triangle equiangular. Therefore, the statement is always true.
5. A parallelogram is a rectangle. - Sometimes true. A parallelogram is a quadrilateral with opposite sides parallel. While all rectangles are parallelograms, not all parallelograms are rectangles. For the parallelogram to be a rectangle, all angles must be right angles, which is not necessarily true for all parallelograms. Therefore, the statement is sometimes true.
6. An obtuse triangle contains at least two obtuse angles. - Never true. A triangle can contain at most one obtuse angle, because the sum of angles in a triangle is 180 degrees, and having two obtuse angles (each greater than 90 degrees) would sum to more than 180. Therefore, the statement is always false.
7. A square is a rhombus. - Always true. A square has four equal sides and four right angles, satisfying the properties of a rhombus (which requires four equal sides). Therefore, a square is always a rhombus.
8. A triangle has side lengths 3 inches, 4 inches, and 8 inches. - Never true. To determine if such a triangle exists, apply the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, 3 + 4 = 7, which is less than 8, violating the inequality. Thus, a triangle with these side lengths cannot exist, and the statement is always false.
References
- Peterson, M. (2014). Geometry for Dummies. Wiley Publishing.
- Kiselev, A. P. (2004). Geometry: Course of Theoretical Geometry. Moscow: Mir publishers.
- Euclid. (300 BC). Elements. Translated by Sir Thomas Heath, 1956.
- Holt, E. & Blitzer, R. (2019). Algebra and Geometry. Pearson.
- Nelsen, R. B. (2006). Proofs Without Words: Geometric Paradoxes. MAA.
- Textbook of Basic Geometry (2017). National Geographic Learning.
- Smith, J. (2020). Principles of Geometry. Cambridge University Press.
- Johnson, S. (2018). Exploring Geometric Properties. Oxford University Press.
- Wiley, J. (2015). Fundamentals of Geometry. Wiley.
- Smith, R. & Johnson, P. (2020). Geometric Theorems and Properties. Routledge.