Question
In the parallelogram ABCD, the interior angle of point B measures (4x - 18) degrees and the interior angle of point D measures (2x + 42) degrees. What is the value of x?
a. 10
b. 60
c. 20
d. 30
a. 10
b. 60
c. 20
d. 30
Answers
GPT-4o mini
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. This means that the sum of the angles at points B and D must equal 180 degrees.
Given:
- Angle B = \( 4x - 18 \) degrees
- Angle D = \( 2x + 42 \) degrees
We can set up the equation:
\[
(4x - 18) + (2x + 42) = 180
\]
Now, combine like terms:
\[
4x - 18 + 2x + 42 = 180
\]
\[
6x + 24 = 180
\]
Next, subtract 24 from both sides:
\[
6x = 180 - 24
\]
\[
6x = 156
\]
Now, divide by 6:
\[
x = \frac{156}{6}
\]
\[
x = 26
\]
It seems that 26 is not one of the provided answer choices of 10, 60, 20, or 30. Let's re-evaluate the angles to check calculations.
1. \( (4x - 18) + (2x + 42) = 180 \)
2. \( 4x - 18 + 2x + 42 = 180 \)
3. Combine to get \( 6x + 24 = 180 \)
4. Solving this gives \( x = 26 \) again.
Based on this, it's clear the solution holds, but no options are matching. We need to ensure clarity in possible examination issues or options:
1. Let’s re-test values in answer choices:
- If \(x = 10\), Angle B = \(4(10) - 18 = 40 - 18 = 22\), and Angle D = \(2(10) + 42 = 20 + 42 = 62\) (Sum = 84)
- If \(x = 20\), Angle B = \(4(20) - 18 = 80 - 18 = 62\), and Angle D = \(2(20) + 42 = 40 + 42 = 82\) (Sum = 144)
- If \(x = 30\), Angle B = \(4(30) - 18 = 120 - 18 = 102\), and Angle D = \(2(30) + 42 = 60 + 42 = 102\) (Sum = 204)
- If \(x = 60\), Angle B = \(4(60) - 18 = 240 - 18 = 222\), and Angle D = \(2(60) + 42 = 120 + 42 = 162\) (Not valid.)
No combinations yield expected outcomes of 180, and original calculations yield consistent results for x without error, reinforcing mismatch on choices given.
To conclude, if we accept accuracy, \(x\) computed as \(26\) reflects foundational properties upheld despite mismatches on expected options listed.
**No option provided against consistently gravitational results, x ultimately \(= 26\).**
Given:
- Angle B = \( 4x - 18 \) degrees
- Angle D = \( 2x + 42 \) degrees
We can set up the equation:
\[
(4x - 18) + (2x + 42) = 180
\]
Now, combine like terms:
\[
4x - 18 + 2x + 42 = 180
\]
\[
6x + 24 = 180
\]
Next, subtract 24 from both sides:
\[
6x = 180 - 24
\]
\[
6x = 156
\]
Now, divide by 6:
\[
x = \frac{156}{6}
\]
\[
x = 26
\]
It seems that 26 is not one of the provided answer choices of 10, 60, 20, or 30. Let's re-evaluate the angles to check calculations.
1. \( (4x - 18) + (2x + 42) = 180 \)
2. \( 4x - 18 + 2x + 42 = 180 \)
3. Combine to get \( 6x + 24 = 180 \)
4. Solving this gives \( x = 26 \) again.
Based on this, it's clear the solution holds, but no options are matching. We need to ensure clarity in possible examination issues or options:
1. Let’s re-test values in answer choices:
- If \(x = 10\), Angle B = \(4(10) - 18 = 40 - 18 = 22\), and Angle D = \(2(10) + 42 = 20 + 42 = 62\) (Sum = 84)
- If \(x = 20\), Angle B = \(4(20) - 18 = 80 - 18 = 62\), and Angle D = \(2(20) + 42 = 40 + 42 = 82\) (Sum = 144)
- If \(x = 30\), Angle B = \(4(30) - 18 = 120 - 18 = 102\), and Angle D = \(2(30) + 42 = 60 + 42 = 102\) (Sum = 204)
- If \(x = 60\), Angle B = \(4(60) - 18 = 240 - 18 = 222\), and Angle D = \(2(60) + 42 = 120 + 42 = 162\) (Not valid.)
No combinations yield expected outcomes of 180, and original calculations yield consistent results for x without error, reinforcing mismatch on choices given.
To conclude, if we accept accuracy, \(x\) computed as \(26\) reflects foundational properties upheld despite mismatches on expected options listed.
**No option provided against consistently gravitational results, x ultimately \(= 26\).**