Question

lesson 9.3 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. find the measure of each of the following angles in the rhombus.

\\(m\angle 1 = \\_\\_^{\circ}\\) \\(m\angle 2 = \\_\\_^{\circ}\\) \\(m\angle 3 = \\_\\_^{\circ}\\)
\\(m\angle 4 = \\_\\_^{\circ}\\) \\(m\angle 5 = \\_\\_^{\circ}\\)

1. the following figure is a rhombus. find the value of \\(m\\). then, use that value to find the measure of \\(vw\\).

Explanation:

Analyze properties of the first rhombus

Using the Rhombus Properties and Rhombus Theorems knowledge points

• A rhombus is a parallelogram with four congruent sides.
• The diagonals of a rhombus are perpendicular, meaning they intersect at right angles:

$$ m\angle 1 = 90^\circ $$

• The diagonals of a rhombus bisect its angles.
• Since the bottom-left angle is given as \(44^\circ\), the diagonal bisects it:

$$ m\angle 2 = \frac{44^\circ}{2} = 22^\circ $$

• In the right triangle formed by the diagonals at the bottom-left:

$$ m\angle 3 = 90^\circ - m\angle 2 = 90^\circ - 22^\circ = 68^\circ $$

• By alternate interior angles or angle bisector symmetry:

$$ m\angle 4 = m\angle 2 = 22^\circ $$

$$ m\angle 5 = m\angle 3 = 68^\circ $$

Solve for m in the second rhombus

Using the Rhombus Properties and Solving Linear Equations knowledge points

• All four sides of a rhombus are equal in length:

$$ VW = WX $$

• Substitute the given side expressions:

$$ 6m - 12 = 4m + 4 $$

• Solve the linear equation:

$$ 2m = 16 \implies m = 8 $$

Calculate the length of VW

Using the Rhombus Properties knowledge point

• Substitute \(m = 8\) into the expression for \(VW\):

$$ VW = 6(8) - 12 = 48 - 12 = 36 $$

Answer:

Question 1

• \(m\angle 1 = 90^\circ\)
• \(m\angle 2 = 22^\circ\)
• \(m\angle 3 = 68^\circ\)
• \(m\angle 4 = 22^\circ\)
• \(m\angle 5 = 68^\circ\)

Question 2

• \(m = 8\)
• \(VW = 36\)