Question

lesson 9.2 homework
complete problems 1-6 below for independent practice.
when you are finished, check the solutions with your teacher.

1. which of the following properties is true about all parallelograms? choose one.

a four right angles
b diagonals are bisected
c only one pair of parallel sides

1. which of the following properties is true about all parallelograms? choose one.

a opposite sides are equal and parallel
b all angles are always equal
c the sum of all angles is 180

1. if \\(x = 9\\), can it prove abcd is a parallelogram? show your work below.

1. if \\(x = 7\\), can it prove bcde is a parallelogram? show your work below.

1. solve for \\(x\\) using the property of bisected diagonals.

\\(cn = 20\\)
\\(ln = 6x - 2\\)

1. solve for \\(x\\) using the property of bisected diagonals. then, find the measure of \\(vx\\).

\\(vh = 18\\)
\\(hx = 4x - 6\\)

Explanation:

Analyze Question 1 and Question 2

Using the Parallelogram Properties knowledge point

• For Question 1, a fundamental property of all parallelograms is that their diagonals bisect each other. Thus, option B is correct.
• For Question 2, by definition and properties of a parallelogram, opposite sides are both equal in length and parallel. Thus, option A is correct.

Analyze Question 3

Using the Converse of Parallelogram Theorems and Solving Linear Equations knowledge points

• In quadrilateral \(ABCD\), we are given \(\angle A = 105^\circ\) and \(\angle C = (11x + 6)^\circ\).
• Substitute \(x = 9\):

$$ \angle C = 11(9) + 6 = 99 + 6 = 105^\circ $$

• Since \(\angle A = \angle C = 105^\circ\), one pair of opposite angles is congruent. However, to prove a quadrilateral is a parallelogram using opposite angles, we must show both pairs of opposite angles are congruent. Since we have no information about \(\angle B\) and \(\angle D\), we cannot prove \(ABCD\) is a parallelogram.

Analyze Question 4

Using the Converse of Parallelogram Theorems and Solving Linear Equations knowledge points

• In quadrilateral \(BCDE\), we are given consecutive angles \(\angle E = 115^\circ\) and \(\angle D = (9x - 7)^\circ\).
• Substitute \(x = 7\):

$$ \angle D = 9(7) - 7 = 63 - 7 = 56^\circ $$

• Check if consecutive angles are supplementary:

$$ \angle E + \angle D = 115^\circ + 56^\circ = 171^\circ eq 180^\circ $$

• Since consecutive angles are not supplementary, the lines \(BE\) and \(CD\) are not parallel, so \(BCDE\) cannot be a parallelogram.

Analyze Question 5

Using the Parallelogram Diagonal Properties and Solving Linear Equations knowledge points

• Diagonals of parallelogram \(LMKN\) bisect each other at intersection point \(C\). Therefore, diagonal \(LN\) is bisected by diagonal \(MK\), meaning \(LN = 2 \cdot CN\).
• Set up the equation:

$$ 6x - 2 = 2(20) $$

$$ 6x - 2 = 40 $$

$$ 6x = 42 \implies x = 7 $$

Analyze Question 6

Using the Parallelogram Diagonal Properties and Solving Linear Equations knowledge points

• Diagonals of parallelogram \(UVWX\) bisect each other at intersection point \(H\). Therefore, diagonal \(VX\) is bisected by diagonal \(UW\), meaning \(VX = 2 \cdot VH\) and \(HX = VH\).
• Set up the equation \(HX = VH\):

$$ 4x - 6 = 18 $$

$$ 4x = 24 \implies x = 6 $$

• Find the measure of \(VX\):

$$ VX = 2 \cdot VH = 2(18) = 36 $$

Answer:

No.	Answer
2	A (Opposite sides are equal and parallel)
3	No; \(\angle C = 105^\circ\), which only proves one pair of opposite angles is congruent.

| 4 | No; \(\angle D = 56^\circ\), so consecutive angles are not supplementary (\(115^\circ + 56^\circ = 171^\circ
eq 180^\circ\)). |

5	\(x = 7\)
6	\(x = 6\), \(VX = 36\)