Question

question 5. in the figure shown, line l intersects parallel lines j and k. if q = 10c - 11 and r = 15c - 41, what is the value of p? note: figure not drawn to scale. a. 6 b. 49 c. 115 d. 131

Explanation:

Step1: Identify q and r relationship

Since lines \( j \) and \( k \) are parallel, and line \( l \) is a transversal, \( q \) and \( r \) are alternate interior angles (or corresponding, but here since they are equal as alternate interior). So \( q = r \).
Set \( 10c - 11 = 15c - 41 \).

Step2: Solve for c

Subtract \( 10c \) from both sides: \( -11 = 5c - 41 \).
Add 41 to both sides: \( 30 = 5c \).
Divide by 5: \( c = 6 \).

Step3: Find q (or r)

Substitute \( c = 6 \) into \( q = 10c - 11 \): \( q = 10(6) - 11 = 60 - 11 = 49 \).

Step4: Find p

\( p \) and \( q \) are supplementary (linear pair), so \( p + q = 180 \).
Thus, \( p = 180 - 49 = 131 \)? Wait, no, wait. Wait, in the diagram, \( p \) and \( q \) are vertical angles? Wait, no, looking at the diagram: line \( l \) intersects line \( j \), so \( p \) and \( q \) are adjacent angles forming a linear pair? Wait, no, maybe \( p \) and \( q \) are vertical angles? Wait, no, the diagram: line \( l \) crosses line \( j \), so \( p \) and \( q \) are supplementary? Wait, no, let's re-examine. Wait, when two lines intersect, vertical angles are equal, linear pairs are supplementary. Wait, in the diagram, line \( l \) intersects line \( j \), so angle \( p \) and angle \( q \): if \( q \) is above the line \( l \), and \( p \) is below, then they are vertical angles? No, wait, maybe \( q \) and \( r \) are equal (alternate interior angles), then \( p \) and \( q \) are supplementary. Wait, let's redo:

Wait, lines \( j \) and \( k \) are parallel, line \( l \) is transversal. So \( q \) and \( r \) are alternate interior angles, so \( q = r \). So \( 10c - 11 = 15c - 41 \), solve for \( c \):

\( 15c - 10c = 41 - 11 \)

\( 5c = 30 \)

\( c = 6 \). Then \( q = 10*6 -11 = 49 \). Then, angle \( p \) and angle \( q \): are they supplementary? Wait, in the diagram, line \( l \) intersects line \( j \), so angle \( p \) and angle \( q \) are adjacent, forming a linear pair, so \( p + q = 180 \). So \( p = 180 - 49 = 131 \)? But the options have D as 131? Wait, but let's check again. Wait, maybe \( p \) and \( q \) are vertical angles? No, that can't be. Wait, maybe I misread the diagram. Wait, the diagram: line \( l \) is horizontal, line \( j \) is slanting, intersecting line \( l \). So angle \( p \) is below line \( l \), angle \( q \) is above line \( l \), so they are supplementary (linear pair), so \( p + q = 180 \). So \( p = 180 - 49 = 131 \). But wait, the options: D is 131. Wait, but earlier I thought maybe mistake, but let's check again.

Wait, let's confirm:

1. \( j \parallel k \), transversal \( l \), so \( q = r \) (alternate interior angles). So \( 10c -11 = 15c -41 \). Solve: \( 5c = 30 \), \( c=6 \). Then \( q = 10*6 -11 = 49 \). Then, angle \( p \) and angle \( q \): since they are adjacent angles on a straight line (linear pair), they are supplementary. So \( p = 180 - 49 = 131 \). So the answer is D. 131.

Wait, but the options are A.6, B.49, C.115, D.131. So yes, D is 131.

Answer:

D. 131