Question
Consider Carmen’s plans.
Parallel lines k and l are crossed by transversal t. Line m crosses lines t, k, and l. The angles formed by lines k, t, and m, clockwise from top left are 2, 1, blank, blank, blank (6 x + 5) degrees; formed by lines l and t are blank, (7x minus 4) degrees, blank, blank; formed by lines l and m are 48 degrees, blank, blank, blank.
Carmen used her knowledge of angle relationships to find the value of x in the diagram.
What is the value of x?
x = 5
x = 9
x = 13
x = 16
Parallel lines k and l are crossed by transversal t. Line m crosses lines t, k, and l. The angles formed by lines k, t, and m, clockwise from top left are 2, 1, blank, blank, blank (6 x + 5) degrees; formed by lines l and t are blank, (7x minus 4) degrees, blank, blank; formed by lines l and m are 48 degrees, blank, blank, blank.
Carmen used her knowledge of angle relationships to find the value of x in the diagram.
What is the value of x?
x = 5
x = 9
x = 13
x = 16
Answers
GPT-4o mini
To solve for the value of \(x\) using the relationships between the angles formed by the parallel lines \(k\) and \(l\) and the transversal \(t\), we will analyze the information provided:
1. **Angles formed by lines k, t, and m**: Clockwise from top-left, we have:
- Angle 1: \( (6x + 5)^\circ \)
- Angle 2: \( 2^\circ \)
- Angle 3: (unknown)
- Angle 4: (unknown)
- Angle 5: (unknown)
2. **Angles formed by lines l and t**: Include:
- Angle 1: (unknown)
- Angle 2: \( (7x - 4)^\circ \)
- Angle 3: (unknown)
- Angle 4: (unknown)
3. **Angles formed by lines l and m**: Include:
- Angle 1: \( 48^\circ \)
- Angle 2: (unknown)
- Angle 3: (unknown)
**However, we can derive relationships**:
Since lines \(k\) and \(l\) are parallel, several angle relationships apply, including:
- Alternate interior angles are equal
- Corresponding angles are equal
1. **Transversal angle relationships**:
- The angle \( (6x + 5)^\circ \) at line \(k\) with transversal \(t\) has a corresponding angle at line \(l\):
- This corresponding angle is \( (7x - 4)^\circ \).
- Therefore, we set them equal:
\[
6x + 5 = 7x - 4
\]
Let's solve this equation:
\[
6x + 5 = 7x - 4
\]
Subtracting \(6x\) from both sides:
\[
5 = x - 4
\]
Adding \(4\) to both sides:
\[
x = 9
\]
2. **Verification**:
Now we should check to ensure \(x = 9\) works with the angle relationships. We can calculate \( (6x + 5)^\circ \) and \( (7x - 4)^\circ \):
- \( 6(9) + 5 = 54 + 5 = 59^\circ \)
- \( 7(9) - 4 = 63 - 4 = 59^\circ \)
They are equal, confirming our derived angle relationship is satisfied.
Thus, the value of \(x\) is:
\[
\boxed{9}
\]
1. **Angles formed by lines k, t, and m**: Clockwise from top-left, we have:
- Angle 1: \( (6x + 5)^\circ \)
- Angle 2: \( 2^\circ \)
- Angle 3: (unknown)
- Angle 4: (unknown)
- Angle 5: (unknown)
2. **Angles formed by lines l and t**: Include:
- Angle 1: (unknown)
- Angle 2: \( (7x - 4)^\circ \)
- Angle 3: (unknown)
- Angle 4: (unknown)
3. **Angles formed by lines l and m**: Include:
- Angle 1: \( 48^\circ \)
- Angle 2: (unknown)
- Angle 3: (unknown)
**However, we can derive relationships**:
Since lines \(k\) and \(l\) are parallel, several angle relationships apply, including:
- Alternate interior angles are equal
- Corresponding angles are equal
1. **Transversal angle relationships**:
- The angle \( (6x + 5)^\circ \) at line \(k\) with transversal \(t\) has a corresponding angle at line \(l\):
- This corresponding angle is \( (7x - 4)^\circ \).
- Therefore, we set them equal:
\[
6x + 5 = 7x - 4
\]
Let's solve this equation:
\[
6x + 5 = 7x - 4
\]
Subtracting \(6x\) from both sides:
\[
5 = x - 4
\]
Adding \(4\) to both sides:
\[
x = 9
\]
2. **Verification**:
Now we should check to ensure \(x = 9\) works with the angle relationships. We can calculate \( (6x + 5)^\circ \) and \( (7x - 4)^\circ \):
- \( 6(9) + 5 = 54 + 5 = 59^\circ \)
- \( 7(9) - 4 = 63 - 4 = 59^\circ \)
They are equal, confirming our derived angle relationship is satisfied.
Thus, the value of \(x\) is:
\[
\boxed{9}
\]