Find the areas of the parallelograms whose vertices are given. A(1, 0, -1), B(1, 7, 2), C(2, 4, -1), D(0, 3, 2)

a. Using symmetry, give a justification that the diagonals of a parallelogram bisect each other.

b. Is the result stated in part (a) also true for special parallelograms? For kites? Explain.

Prove that

$$\| \mathbf { u } + \mathbf { v } \| ^ { 2 } + \| \mathbf { u } - \mathbf { v } \| ^ { 2 } = 2 \| \mathbf { u } \| ^ { 2 } + 2 \| \mathbf { v } \| ^ { 2 }$$

and interpret the result geometrically by translating it into a theorem about parallelograms.

Quadrilaterals ABCD and PQRS are parallelograms with $\overline{A B} \cong \overline{P Q}$ and $\overline{B C} \cong \overline{Q R}$. Prove that ABCD $\cong$ PQRS or draw a counterexample to show that they may not be congruent.

Which statement is not true about all parallelograms? A. Opposite angles are congruent. B. Consecutive angles are supplementary. C. Diagonals perpendicularly bisect each other. D. Opposite sides are congruent.

(a) Show that

$$\| \mathbf { u } + \mathbf { v } \| ^ { 2 } + \| \mathbf { u } - \mathbf { v } \| ^ { 2 } = 2 \left( \| \mathbf { u } \| ^ { 2 } + \| \mathbf { v } \| ^ { 2 } \right)$$

for any vectors u and v. (b) What does this say about parallelograms?

Given: All rhombuses are parallelograms. What can you conclude from the given additional statement? If no conclusion is possible, write no conclusion.

$A B C D$ is not a parallelogram.

Reasoning, find the ratio of the areas of the parallelograms (smaller to larger). Justify each answer. The bases are the same length. The height of one parallelogram is twice the height of the other.

Find the areas of the parallelograms whose vertices are given. A(0, 0, 0), B(3, 2, 4), C(5, 1, 4), D(2, -1, 0)

A polygon has an area of 48 square meters and a height of 8 meters. Draw three different triangles that fit this description and three different parallelograms. Explain your thinking.

Decide if the following statements are true or false. If a statement is false, provide a diagram of a counterexample. a. All squares are rectangles. b. All quadrilaterals are parallelograms. c. All squares are rhombi.

Which statement about parallelograms is always true?

A. The diagonals are congruent.

B. The diagonals bisect each other.

C. The diagonals are perpendicular.

D. The diagonals bisect their respective angles.

A polygon has an area of 80 square meters and a height of 10 meters. Make scale drawings of three different triangles and three different parallelograms that match this description. Label the base and the height.

Sketch two noncongruent parallelograms ABCD and EFGH such that

$$\overline { A C } \cong \overline { B D } \cong \overline { E G } \cong \overline { F H }.$$