Unit Test Review

Jacob is cutting a tile in the shape of a parallelogram. Two opposite angles have measures of [tex](6n - 70)^{\circ}[/tex] and [tex](2n + 10)^{\circ}[/tex].

What are the two different angle measures of the parallelogram-shaped tile?

A. [tex]20^{\circ}[/tex] and [tex]160^{\circ}[/tex]
B. [tex]50^{\circ}[/tex] and [tex]130^{\circ}[/tex]
C. [tex]30^{\circ}[/tex] and [tex]150^{\circ}[/tex]
D. [tex]70^{\circ}[/tex] and [tex]110^{\circ}[/tex]



Answer :

To find the two different angle measures of the parallelogram-shaped tile, we first need to address the relationship between the given angles and use the properties of a parallelogram.

Given two opposite angles with expressions:
[tex]\[ \text{Angle 1} = (6n - 70)^\circ \][/tex]
[tex]\[ \text{Angle 2} = (2n + 10)^\circ \][/tex]

We know that opposite angles in a parallelogram are equal. Therefore,
[tex]\[ 6n - 70 = 2n + 10 \][/tex]

To solve for \( n \), follow these steps:
1. Set up the equation:
[tex]\[ 6n - 70 = 2n + 10 \][/tex]

2. Isolate \( n \):
Subtract \( 2n \) from both sides:
[tex]\[ 6n - 2n - 70 = 2n - 2n + 10 \][/tex]
[tex]\[ 4n - 70 = 10 \][/tex]

Now, add 70 to both sides:
[tex]\[ 4n - 70 + 70 = 10 + 70 \][/tex]
[tex]\[ 4n = 80 \][/tex]

3. Solve for \( n \):
[tex]\[ n = \frac{80}{4} \][/tex]
[tex]\[ n = 20 \][/tex]

Next, substitute \( n = 20 \) back into the expressions for the angles:

4. Calculate the measures:
[tex]\[ \text{Angle 1} = 6n - 70 = 6(20) - 70 = 120 - 70 = 50^\circ \][/tex]
[tex]\[ \text{Angle 2} = 2n + 10 = 2(20) + 10 = 40 + 10 = 50^\circ \][/tex]

Both calculated angles (opposite angles in the parallelogram) are equal to \( 50^\circ \).

In a parallelogram, the adjacent angles are supplementary (they add up to \( 180^\circ \)). Therefore, to find the measure of the adjacent angle, we subtract the given angle from \( 180^\circ \):

5. Calculate the adjacent angles:
[tex]\[ \text{Adjacent Angle to 50} = 180^\circ - 50^\circ = 130^\circ \][/tex]

Thus, the two different angle measures in the parallelogram are \( 50^\circ \) and \( 130^\circ \).

The correct option is:
[tex]\[ 50^\circ \text{ and } 130^\circ \][/tex]