True Or False?

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Identify each statement as true or false:

• An equilateral triangle always has 3 acute angles.
• A scalene triangle can also be an isosceles triangle.
• A trapezoid is a quadrilateral.
• All quadrilaterals are parallelograms.
• A rectangle is a square.
• An equilateral triangle can never have a right angle.
• Every rhombus has two sets of parallel sides.
• All quadrilaterals with 4 right angles are squares and rhombi.
• Can a square also be categorized as a parallelogram? Why or why not?
• Sophia used a protractor and measured the angles of her triangle. She found that one angle was 90°. Could this triangle be equilateral? Why or why not?
• Trevor recorded the sides of his triangle as 3m, 5m, and 5m. He said the triangle could be categorized as acute, right, or obtuse. Is he correct? Justify your reasoning.
• If Zoe colors in a quadrilateral with only one pair of parallel sides, what is the name of her polygon?
• Kirah gives clues about her polygon: it's a quadrilateral with opposite sides parallel, four right angles, and one set of sides is parallel. Who is she?
• Diego claims a rhombus and a square are the same shape, just different names. Tanya disagrees. Who is correct? Explain.
• Melissa classifies a triangle as both isosceles and scalene. Is this possible? Why?
• Korena gives clues: the shape has less than four sides, two angles less than 90°, and one angle measuring 121°. Who is the shape?

Paper For Above instruction

Triangles and quadrilaterals are fundamental shapes in geometry that exhibit diverse properties based on their sides and angles. Understanding the classifications and characteristics of these shapes is essential for developing geometric reasoning and problem-solving skills.

Firstly, consider the statement that an equilateral triangle always has three acute angles. An equilateral triangle has three equal sides and three equal angles. Since the sum of interior angles in any triangle is 180°, and all three are equal in an equilateral triangle, each angle measures 60°, which is less than 90°, making all three angles acute. Therefore, the statement is true.

Next, a scalene triangle is defined as having all sides of different lengths. An isosceles triangle has at least two sides equal. It is possible for a triangle to be both scalene and isosceles if "scalene" only refers to no sides equal, which contradicts the definition. However, more precisely, a triangle cannot be both scalene and isosceles simultaneously, since they are mutually exclusive categories. Therefore, the statement that a scalene triangle can also be an isosceles triangle is false, because by definition, a scalene triangle cannot be isosceles.

Regarding quadrilaterals, a trapezoid (or trapezium in some regions) is defined as a quadrilateral with at least one pair of parallel sides. Since it has four sides, the statement that a trapezoid is a quadrilateral is true. Furthermore, not all quadrilaterals are parallelograms, which require two pairs of parallel sides. Parallelograms include rectangles, squares, rhombuses, and certain other shapes, but quadrilaterals that have only one pair of parallel sides (like trapezoids) are not parallelograms. Hence, the statement "All quadrilaterals are parallelograms" is false.

The statement that a rectangle is a square is only true if the rectangle has four sides equal, which defines a square. Therefore, in general, a rectangle (with four right angles but not necessarily all sides equal) is not a square. Thus, the statement is false.

An equilateral triangle can never have a right angle because all its angles are 60°, which are acute. If one angle is 90°, the other two must total 90°, but since the three angles sum to 180°, and two angles being 45° and 45° would make the triangle isosceles and acute, not equilateral. Therefore, an equilateral triangle cannot have a right angle, and the statement is true.

Every rhombus has two sets of parallel sides, a defining property that makes it a parallelogram, specifically a rhombus as a special case with all sides equal. This statement is true.

Quadrilaterals with four right angles are rectangles. If all sides are also equal, the shape is a square. Rhombi have four equal sides but do not necessarily have right angles. Thus, the statement that all quadrilaterals with four right angles are squares and rhombi is false because it conflates these different shapes.

A square can be categorized as a parallelogram because it has two pairs of parallel sides, four right angles, and four equal sides. Therefore, a square is a special type of parallelogram, confirming that the answer to whether a square can also be a parallelogram is yes, because of its properties.

Considering Sophia's triangle with a 90° angle, and questioning if it could be equilateral: since all angles in an equilateral triangle are 60°, and having a 90° angle contradicts this, the triangle cannot be both equilateral and have a 90° angle. Therefore, it is not equilateral.

Trevor's recorded sides of 3m, 5m, and 5m suggest the triangle is isosceles, with two sides equal. Since two sides are 5m, the angles opposite these sides are equal, and the triangle could be acute (all angles less than 90°), right (if the Pythagorean theorem is satisfied), or obtuse (if one angle exceeds 90°). He is correct, because knowing the sides alone allows classification into these types based on the angles, which depend on the specific measurements.

Zoe coloring a quadrilateral with only one pair of parallel sides indicates it is a trapezoid, as that is the defining property of a trapezoid (or trapezium).

Kirah's clues fit a rectangle: a quadrilateral with opposite sides parallel and four right angles, indicating a rectangle.

Diego's statement that a rhombus and a square are the same shape, just different names, is incorrect. While both are parallelograms with four sides, a square has all sides equal and four right angles, whereas a rhombus doesn't necessarily have right angles. Thus, Diego is incorrect; they are different shapes.

Melissa's classification of a triangle as both isosceles and scalene is impossible because these are mutually exclusive: scalene triangles have all sides of different lengths, while isosceles triangles must have at least two equal sides. Therefore, her classification is inconsistent.

Korena's clues describe a shape with less than four sides, two angles less than 90°, and an angle measuring 121°, which is obtuse. Since the shape has fewer than four sides, it must be a triangle, specifically an obtuse triangle with an additional angle less than 90°, and one angle greater than 90°, satisfying the given clues.

References

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