Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
Does an isosceles trapezoid have two sets of parallel sides? Why or why not? Explain.
Does a rhombus have two sets of congruent sides? Why or why not? Explain.
Answer:
Yes, because the diagonals of a rhombus, which is a parallelogram, are perpendicular. In a parallelogram diagonals bisect each other.
No, because in an isosceles trapezoid the sides that are not parallel are equal.
Yes, because a rhombus has two sets of parallel sides and all sides are congruent.
To answer the question of whether the diagonals of a parallelogram are perpendicular, we need to understand the properties of a parallelogram.
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are parallel and congruent, meaning they have the same length. Additionally, the opposite angles of a parallelogram are congruent.
Now, let's consider the diagonals of a parallelogram. A diagonal is a line segment that connects two non-adjacent vertices of a polygon. In a parallelogram, the diagonals bisect each other, meaning they intersect at their midpoints.
To determine if the diagonals of a parallelogram are perpendicular, we need to examine the angles that are formed where the diagonals intersect.
If the diagonals of a parallelogram are perpendicular, then the angles formed at their intersection would each measure 90 degrees. That is, the diagonals would form right angles at their intersection point.
In a rhombus, which is a type of parallelogram, the diagonals are indeed perpendicular. This can be proven using the properties of a rhombus. A rhombus is a parallelogram in which all sides are congruent. Additionally, the diagonals of a rhombus bisect each other at right angles, making them perpendicular.
Therefore, in summary, the diagonals of a parallelogram (specifically a rhombus) are perpendicular because they form right angles at their intersection point.
Moving on to the question of whether an isosceles trapezoid has two sets of parallel sides, we need to understand the properties of an isosceles trapezoid.
An isosceles trapezoid is a quadrilateral with one pair of parallel sides and two non-parallel sides that are congruent. The non-parallel sides are often referred to as the legs of the trapezoid, while the parallel sides are called the bases.
Now, let's consider the parallel sides of an isosceles trapezoid. Since it is an isosceles trapezoid, only one pair of sides is parallel. The other pair of sides, which are the legs, are not parallel.
The legs of an isosceles trapezoid are congruent, meaning they have the same length. However, they are not parallel to each other. Therefore, an isosceles trapezoid does not have two sets of parallel sides.
Finally, let's discuss whether a rhombus has two sets of congruent sides. A rhombus is a special type of parallelogram in which all sides are congruent.
In a rhombus, all four sides have the same length, which means they are congruent. Therefore, a rhombus does have two sets of congruent sides - the opposite sides are congruent to each other.
To summarize, while the diagonals of a parallelogram, specifically a rhombus, are perpendicular, an isosceles trapezoid does not have two sets of parallel sides, and a rhombus does have two sets of congruent sides.