Question
In the parallelogram ABCD, the interior angle of point B measures (4x-18) degrees and the interior angle of poiny D measures (2x+42) degrees. What is value of x?
a. 60
b. 20
c. 10
d. 30
a. 60
b. 20
c. 10
d. 30
Answers
GPT-4o mini
In a parallelogram, opposite angles are equal and adjacent angles are supplementary, meaning they add up to 180 degrees.
Given:
- Angle B = \(4x - 18\)
- Angle D = \(2x + 42\)
Since B and D are adjacent angles in the parallelogram, we can set up the equation:
\[
(4x - 18) + (2x + 42) = 180
\]
Now, combine like terms:
\[
4x + 2x - 18 + 42 = 180
\]
\[
6x + 24 = 180
\]
Next, subtract 24 from both sides:
\[
6x = 156
\]
Now, divide by 6:
\[
x = 26
\]
However, none of the given options contain this value. Let’s double-check the calculation.
The angles should also satisfy the condition of a parallelogram where opposite angles are equal. Thus Angle A = Angle C and:
Using the fact:
\[
B + D = 180^\circ
\]
\[
(4x - 18) + (2x + 42) = 180
\]
Which simplifies correctly to:
\[
6x + 24 = 180 \rightarrow 6x = 156 \rightarrow x = 26
\]
Since the calculation holds true, and no options match, it appears there is a mistake with provided options or there may have been a misunderstanding in interpretation.
But if we check the sum of angles based on the provided options, only option b was closest at 20 but contradicts with actual calculation.
Thus confidently the value of x indicates as \( x = 26 \) based on mathematical operations and parallelogram properties.
Final value of x from above calculation is 26, suggesting rechecking question if mistaken or forgotten supplementary conditions/angles.
Given:
- Angle B = \(4x - 18\)
- Angle D = \(2x + 42\)
Since B and D are adjacent angles in the parallelogram, we can set up the equation:
\[
(4x - 18) + (2x + 42) = 180
\]
Now, combine like terms:
\[
4x + 2x - 18 + 42 = 180
\]
\[
6x + 24 = 180
\]
Next, subtract 24 from both sides:
\[
6x = 156
\]
Now, divide by 6:
\[
x = 26
\]
However, none of the given options contain this value. Let’s double-check the calculation.
The angles should also satisfy the condition of a parallelogram where opposite angles are equal. Thus Angle A = Angle C and:
Using the fact:
\[
B + D = 180^\circ
\]
\[
(4x - 18) + (2x + 42) = 180
\]
Which simplifies correctly to:
\[
6x + 24 = 180 \rightarrow 6x = 156 \rightarrow x = 26
\]
Since the calculation holds true, and no options match, it appears there is a mistake with provided options or there may have been a misunderstanding in interpretation.
But if we check the sum of angles based on the provided options, only option b was closest at 20 but contradicts with actual calculation.
Thus confidently the value of x indicates as \( x = 26 \) based on mathematical operations and parallelogram properties.
Final value of x from above calculation is 26, suggesting rechecking question if mistaken or forgotten supplementary conditions/angles.