Question

mathematical connections
use the diagram.

1. a) find the value of x that makes ( p parallel q ).

b) find the value of y that makes ( r parallel s ).
diagram: two horizontal lines ( p ) (top) and ( q ) (bottom). transversal ( r ) intersects ( p ) at ( (2x + 2)^circ ) and ( q ) at ( (x + 56)^circ ) and ( (y + 7)^circ ). transversal ( s ) intersects ( q ) at ( (3y - 17)^circ ).

Explanation:

Part (a)

Step1: Identify the angle relationship

When two lines \( p \) and \( q \) are parallel, the consecutive interior angles are supplementary (they add up to \( 180^\circ \)). The angles \( (2x + 2)^\circ \) and \( (x + 56)^\circ \) are consecutive interior angles. So, we set up the equation:
\( (2x + 2)+(x + 56)=180 \)

Step2: Simplify and solve for \( x \)

Combine like terms:
\( 2x + 2+x + 56 = 180 \)
\( 3x+58 = 180 \)

Subtract 58 from both sides:
\( 3x=180 - 58 \)
\( 3x = 122 \)? Wait, no, 180 - 58 is 122? Wait, 180 - 58: 180 - 50 = 130, 130 - 8 = 122? Wait, no, wait, 2 + 56 is 58? Wait, 2x + x is 3x, 2 + 56 is 58. So 3x + 58 = 180. Then 3x = 180 - 58 = 122? Wait, that can't be, maybe I made a mistake. Wait, maybe the angles are alternate interior angles? Wait, looking at the diagram, if \( p \parallel q \), and the transversal is \( r \), then the angles \( (2x + 2)^\circ \) and \( (x + 56)^\circ \) might be same - side interior angles? Wait, no, maybe they are supplementary. Wait, let's check again. Wait, maybe the angles are same - side interior angles, so their sum is 180. So:

\( 2x + 2+x + 56 = 180 \)

\( 3x+58 = 180 \)

\( 3x=180 - 58=122 \)? No, 180 - 58 is 122? Wait, 58 + 122 = 180. Then \( x=\frac{122}{3}\approx40.67 \)? That seems odd. Wait, maybe the angles are alternate exterior angles or something else. Wait, maybe I misidentified the angle relationship. Wait, if \( p \parallel q \), and the transversal is \( r \), then the angle \( (2x + 2)^\circ \) and \( (x + 56)^\circ \) could be same - side interior angles, but maybe they are actually supplementary. Wait, maybe I made a mistake in the sign. Wait, let's re - examine the diagram. The angle \( (2x + 2)^\circ \) is above line \( p \), and \( (x + 56)^\circ \) is below line \( q \), with the transversal \( r \). Wait, maybe they are same - side interior angles, so their sum is 180. So:

\( 2x + 2+x + 56 = 180 \)

\( 3x+58 = 180 \)

\( 3x = 122 \)

\( x=\frac{122}{3}\approx40.67 \). But this is a fraction. Maybe the angle relationship is different. Wait, maybe the angles are alternate interior angles, so they are equal. Let's try that. If \( p \parallel q \), and the transversal is \( r \), then alternate interior angles are equal. So \( 2x + 2=x + 56 \)

Step3: Solve the new equation

Subtract \( x \) from both sides:
\( 2x - x+2=x - x + 56 \)
\( x+2 = 56 \)

Subtract 2 from both sides:
\( x=56 - 2=54 \)

Ah, that makes sense. I must have misidentified the angle relationship. Alternate interior angles are equal when lines are parallel. So the correct equation is \( 2x + 2=x + 56 \)

Part (b)

Step1: Identify the angle relationship

When two lines \( r \) and \( s \) are parallel, the alternate interior angles are equal. The angles \( (y + 7)^\circ \) and \( (3y - 17)^\circ \) are alternate interior angles. So we set up the equation:
\( y + 7=3y - 17 \)

Step2: Solve for \( y \)

Subtract \( y \) from both sides:
\( y - y+7=3y - y - 17 \)
\( 7 = 2y-17 \)

Add 17 to both sides:
\( 7 + 17=2y-17 + 17 \)
\( 24 = 2y \)

Divide both sides by 2:
\( y=\frac{24}{2}=12 \)

Answer:

s:
a) \( x = 54 \)

b) \( y = 12 \)