Answer :
To determine which sentence is always true for a parallelogram, it is important to understand the properties of a parallelogram:
1. Definition of a Parallelogram:
A parallelogram is a four-sided figure with opposite sides that are parallel.
2. Properties of Parallelograms:
- Opposite sides are congruent (equal in length).
- Opposite angles are congruent (equal in measure).
- Adjacent angles are supplementary (i.e., their measures add up to 180 degrees).
- The diagonals bisect each other.
Now let's analyze each option given in the question:
Option A: All sides are congruent.
This is not always true for a parallelogram. This property holds true for a rhombus, which is a special type of parallelogram where all sides are equal, but it is not a general property of all parallelograms.
Option B: All angles are congruent.
This is not always true for a parallelogram. This property is true for a rectangle, which is a special type of parallelogram where all angles are 90 degrees, but it is not a general property of all parallelograms.
Option C: The diagonals are congruent.
This is not always true for a parallelogram. In a rectangle and specifically in a square, the diagonals are congruent, but in a general parallelogram, the diagonals only bisect each other, not necessarily being congruent in length.
Option D: Opposite angles are congruent.
This is always true for all parallelograms. Regardless of the specific shape or the particular types of the parallelogram (whether it's a rectangle, rhombus, or another type), the property that opposite angles are congruent always holds true.
Given our analysis, the correct choice for a property that always holds true for a parallelogram is:
D. Opposite angles are congruent.
1. Definition of a Parallelogram:
A parallelogram is a four-sided figure with opposite sides that are parallel.
2. Properties of Parallelograms:
- Opposite sides are congruent (equal in length).
- Opposite angles are congruent (equal in measure).
- Adjacent angles are supplementary (i.e., their measures add up to 180 degrees).
- The diagonals bisect each other.
Now let's analyze each option given in the question:
Option A: All sides are congruent.
This is not always true for a parallelogram. This property holds true for a rhombus, which is a special type of parallelogram where all sides are equal, but it is not a general property of all parallelograms.
Option B: All angles are congruent.
This is not always true for a parallelogram. This property is true for a rectangle, which is a special type of parallelogram where all angles are 90 degrees, but it is not a general property of all parallelograms.
Option C: The diagonals are congruent.
This is not always true for a parallelogram. In a rectangle and specifically in a square, the diagonals are congruent, but in a general parallelogram, the diagonals only bisect each other, not necessarily being congruent in length.
Option D: Opposite angles are congruent.
This is always true for all parallelograms. Regardless of the specific shape or the particular types of the parallelogram (whether it's a rectangle, rhombus, or another type), the property that opposite angles are congruent always holds true.
Given our analysis, the correct choice for a property that always holds true for a parallelogram is:
D. Opposite angles are congruent.
