Question

Vertical lines m and n are intersected by lines k and j. At the intersection of lines m and k, the bottom right angle is (x minus 30) degrees. At the intersection of m and j, the uppercase right angle is y. At the intersection of lines k and n, the bottom left angle is (x + 50) degrees.
Find the values of x and y that make k || j and
m || n.

Answer 1

To solve for \( x \) and \( y \) such that lines \( k \parallel j \) and \( m \parallel n \), we can make use of the properties of angles formed by parallel lines and a transversal.

1. **Identify the angles and relationships:**
- At the intersection of lines \( m \) and \( k \), the bottom right angle is \( x - 30 \) degrees.
- At the intersection of lines \( m \) and \( j \), the uppercase right angle is \( y \).
- At the intersection of lines \( k \) and \( n \), the bottom left angle is \( x + 50 \) degrees.

2. **Setting up relationships for parallel lines:**
Since \( k \parallel j \), corresponding angles are equal.
- The angle \( x - 30 \) at \( m \) and \( k \) corresponds to angle \( y \) at \( m \) and \( j \). Thus, we can write the equation:
\[
x - 30 = y \tag{1}
\]

3. **Consider the angles around intersection of \( n \) and \( k \):**
Since \( m \parallel n \) and \( k \) is a transversal, the angle \( x - 30 \) (from \( m \) and \( k \)) and the angle \( x + 50 \) (from \( k \) and \( n \)) also have a relationship:
- They are supplementary (add up to \( 180^\circ \)):
\[
(x - 30) + (x + 50) = 180 \tag{2}
\]
Simplifying equation (2):
\[
2x + 20 = 180
\]
\[
2x = 160
\]
\[
x = 80
\]

4. **Now substitute \( x \) back into equation (1) to find \( y \):**
- From equation (1), substituting \( x = 80 \):
\[
80 - 30 = y
\]
\[
y = 50
\]

5. **Final solution:**
The values of \( x \) and \( y \) that satisfy \( k \parallel j \) and \( m \parallel n \) are:
\[
x = 80, \quad y = 50
\]