Question

Use the following information to complete the problem.

Given: Quadrilateral ABCD has vertices at A(3, 6), B(2, 2), C(-5, 4), and D(-4, 8).
Prove: Quadrilateral ABCD is a parallelogram.
Match each numbered statement with the correct reason.
a) Statement 1: Opposite sides are equal in length
b) Statement 2: Opposite angles are congruent
c) Statement 3: Diagonals bisect each other
d) Statement 4: Consecutive angles are supplementary

Answer 1

To prove that quadrilateral ABCD is a parallelogram, we use the distance formula to show opposite sides are equal and that the diagonals bisect each other. Without angle measures, we cannot use statements about angles, making statements 1 and 3 concerning side lengths and diagonals the correct ones to prove the parallelogram.

Step-by-step explanation:

To prove that quadrilateral ABCD with vertices A(3, 6), B(2, 2), C(-5, 4), and D(-4, 8) is a parallelogram, we need to examine the properties of the sides, angles, and diagonals of the quadrilateral:

1. Opposite sides are equal in length: To prove this, we can use the distance formula to calculate the length of each side: AB, BC, CD, and DA. If the lengths of AB are equal to CD, and BC are equal to DA, then ABCD is a parallelogram by definition.
2. Opposite angles are congruent: This property is not directly applicable here because we do not have information about the angles.
3. Diagonals bisect each other: We can calculate the midpoint of both diagonals AC and BD. If both diagonals have the same midpoint, it implies that they bisect each other, which is characteristic of a parallelogram.
4. Consecutive angles are supplementary: This also cannot be verified without information about the angles.

Upon calculation, we can find that the lengths of AB and CD are the same, and the lengths of BC and DA are the same, which supports statement 1. Additionally, if we find the midpoint of both diagonals to be equal, it verifies statement 3, thereby proving that quadrilateral ABCD is a parallelogram. However, since we are focusing on proving with sides and diagonals due to the available information, statements 1 and 3 are correct.