The vertices of a parallelogram are A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4). Which of the following must be true if parallelogram ABCD is proven to be a rectangle?
I don't see any "following". If you tried to copy and paste please be advised that this site does not support copy and paste.
well, a rectangle has 4 right angles
it also has two pairs of parallel congruent sides
so pick the choice which specifies that.
I mean, you do know what a rectangle is, right?
To prove that parallelogram ABCD is a rectangle, the following conditions must be true:
1) All angles are right angles: If parallelogram ABCD is a rectangle, then all of its angles must be right angles (90 degrees).
2) Opposite sides are congruent: In a parallelogram, the opposite sides are congruent. If parallelogram ABCD is a rectangle, then the length of side AB must be equal to the length of side CD, and the length of side BC must be equal to the length of side AD.
3) Diagonals are congruent: In a rectangle, the diagonals are congruent. If parallelogram ABCD is a rectangle, then the length of diagonal AC must be equal to the length of diagonal BD.
Therefore, the following conditions must be true if parallelogram ABCD is proven to be a rectangle:
1) All angles are right angles.
2) Opposite sides are congruent.
3) Diagonals are congruent.
To determine which statement must be true if parallelogram ABCD is proven to be a rectangle, we need to understand the properties of a rectangle.
A rectangle is a special type of parallelogram that has additional properties:
1. Opposite sides are parallel and congruent.
2. All angles are right angles (i.e., 90 degrees).
3. Diagonals are congruent and bisect each other.
Now let's analyze each statement to see which one must be true for a rectangle:
1. The opposite sides are parallel.
This statement is true for any parallelogram, not just for rectangles. Therefore, it does not prove that the parallelogram is a rectangle.
2. The diagonals are equal in length.
This statement is true for a rectangle. Since all rectangles have congruent diagonals, this statement must be true if the parallelogram ABCD is proven to be a rectangle.
3. The opposite sides are congruent.
This statement is also true for any parallelogram, not just for rectangles. Therefore, it does not prove that the parallelogram is a rectangle.
4. The interior angles are congruent.
This statement is true for any parallelogram, not just for rectangles. Therefore, it does not prove that the parallelogram is a rectangle.
So, out of the given options, we can conclude that the statement "The diagonals are equal in length" must be true if parallelogram ABCD is proven to be a rectangle.