Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
4. Does an isosceles trapezoid have two sets of parallel sides? Why or why not? Explain.
5. Does a rhombus have two sets of congruent sides? Why or why not? Explain.
Answer: 3. Yes, because the diagonals of a rhombus, which is a parallelogram, are perpendicular. In a parallelogram diagonals bisect each other.
4. No, because in an isosceles trapezoid the sides that are not parallel are equal.
5. Yes, because a rhombus has two sets of parallel sides and all sides are congruent.
To answer these questions, it helps to understand the properties and definitions of each shape.
1. Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
A parallelogram is a four-sided polygon with two pairs of parallel sides. The opposite angles in a parallelogram are congruent, meaning they have equal measures. However, the diagonals of a parallelogram are not necessarily perpendicular.
To explain why, we can use a counterexample. Imagine a parallelogram that is a rectangle. In a rectangle, the opposite angles are all right angles, which means the diagonals are perpendicular. However, not all parallelograms are rectangles. For example, a parallelogram could be a rhombus, with equal sides but non-right angles.
In general, the diagonals of a parallelogram only bisect each other, meaning they divide each other into two equal segments. This means that the diagonals cross each other at a 90-degree angle only in specific cases, such as when the parallelogram is a rectangle.
2. Does an isosceles trapezoid have two sets of parallel sides? Why or why not? Explain.
An isosceles trapezoid is a quadrilateral with two parallel sides and two non-parallel sides of equal length. The non-parallel sides are known as legs, while the parallel sides are called bases. To determine whether an isosceles trapezoid has two sets of parallel sides, we need to consider the definition and properties of the shape.
In an isosceles trapezoid, the bases are parallel to each other. This is one set of parallel sides. However, the legs of an isosceles trapezoid are not parallel to each other. The legs meet at a vertex, which means they converge. Therefore, an isosceles trapezoid does not have two sets of parallel sides.
3. Does a rhombus have two sets of congruent sides? Why or why not? Explain.
A rhombus is a quadrilateral with four sides of equal length. Each side of a rhombus is congruent to the others, making it a special type of parallelogram.
To determine whether a rhombus has two sets of congruent sides, we need to consider the definition and properties of the shape. In a rhombus, all sides are equal in length, which means it has a single set of congruent sides. Unlike a rectangle, which also has equal sides, a rhombus does not have 90-degree angles. Thus, the diagonals of a rhombus bisect each other, but they are not necessarily perpendicular.
In summary, a rhombus has one set of congruent sides, while an isosceles trapezoid has one set of parallel sides. The diagonals of a parallelogram are not necessarily perpendicular, but in a rectangle (a specific type of parallelogram), the diagonals are perpendicular.