Question

a diagram is shown, where ( m perp k ) and ( s parallel q ). what is the value of ( y )? the diagram includes angles ( (4x + 10)^circ ) and ( (7x + 3)^circ ).

Explanation:

Step1: Identify angle relationships

Since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \). Also, \( s \parallel q \), so we can use corresponding angles or other parallel line angle relationships. The angles \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) along with the right angle should add up to \( 180^\circ \) (since they are on a straight line related to the perpendicular and parallel lines). Wait, actually, let's correct: since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \), and the two angles \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) and the right angle? Wait, no, let's look at the diagram. The angle \( (7x + 3)^\circ \) and the angle adjacent to \( (4x + 10)^\circ \) (since \( s \parallel q \)) and the right angle: actually, the sum of \( (7x + 3)^\circ \), \( (4x + 10)^\circ \), and \( 90^\circ \)? No, wait, maybe the two angles \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) are complementary to the right angle? Wait, no, let's think again. Since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \). The lines \( s \) and \( q \) are parallel, so the angle \( (4x + 10)^\circ \) and the angle that is vertical or corresponding to the angle related to \( (7x + 3)^\circ \). Wait, actually, the sum of \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) should be \( 90^\circ \)? No, that doesn't make sense. Wait, no, let's see: the angle \( (7x + 3)^\circ \) and the angle \( (4x + 10)^\circ \) are such that when added to the right angle, they form a straight line? Wait, no, the correct approach is: since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \). The lines \( s \parallel q \), so the alternate interior angles or corresponding angles. Wait, actually, the angle \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) are complementary to the right angle? No, let's set up the equation. Wait, the sum of \( (7x + 3) \) and \( (4x + 10) \) should be \( 90 \) because \( m \perp k \), so the two angles and the right angle? Wait, no, maybe the two angles \( (7x + 3) \) and \( (4x + 10) \) are such that \( (7x + 3) + (4x + 10) = 90 \)? Wait, no, that would be if they are complementary, but let's check:

Wait, the correct equation: since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \). The lines \( s \) and \( q \) are parallel, so the angle \( (4x + 10)^\circ \) and the angle adjacent to \( (7x + 3)^\circ \) (since \( s \parallel q \)) and the right angle: actually, the sum of \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) should be \( 90^\circ \) because they are part of the right angle? Wait, no, let's do the math. Let's set \( 7x + 3 + 4x + 10 = 90 \). Wait, \( 11x + 13 = 90 \), \( 11x = 77 \), \( x = 7 \). Wait, is that correct? Let's check: \( 7x + 3 = 7*7 + 3 = 52 \), \( 4x + 10 = 4*7 + 10 = 38 \), \( 52 + 38 = 90 \), which is correct because \( m \perp k \), so those two angles add up to \( 90^\circ \). Now, to find \( y \), we need to see the relationship between \( y \) and the other angles. Since \( s \parallel q \), the angle \( y \) is equal to the angle that is supplementary to \( (4x + 10)^\circ \)? Wait, no, let's see: the angle \( y \) and the angle \( (7x + 3)^\circ \) or \( (4x + 10)^\circ \). Wait, actually, since \( s \parallel q \), the corresponding angle to \( (7x + 3)^\circ \) would be equal to \( y \)? Wait, no, let's look at the diagram again. The angle \( y \) is adjacent to the right angle and the parallel lines. Wait, actually, \( y \) is equal to \( 90^\circ + (4x + 10)^\circ \)? No, wait, let's use the value of \( x = 7 \). So \( 4x + 10 = 38^\circ \), and \( y \) is supplementary to the angle that is \( 90^\circ - 38^\circ \)? No, wait, maybe \( y \) is equal to \( 180^\circ - (90^\circ - (4x + 10)^\circ) \)? No, this is getting confusing. Wait, let's start over.

First, find \( x \). Since \( m \perp k \), the angle between \( m \) and \( k \) is \( 90^\circ \). The two angles \( (7x + 3)^\circ \) and \( (4x + 10)^\circ \) are complementary (add up to \( 90^\circ \)) because they are formed by the perpendicular line \( m \) and the parallel lines \( s \) and \( q \). So:

\( (7x + 3) + (4x + 10) = 90 \)

Step2: Solve for \( x \)

Combine like terms:

\( 11x + 13 = 90 \)

Subtract 13 from both sides:

\( 11x = 90 - 13 = 77 \)

Divide both sides by 11:

\( x = \frac{77}{11} = 7 \)

Step3: Find the angle related to \( y \)

Now, we need to find \( y \). Let's look at the diagram. The angle \( y \) is adjacent to the right angle and the angle \( (4x + 10)^\circ \). Wait, actually, since \( s \parallel q \), the angle \( y \) is equal to \( 90^\circ + (4x + 10)^\circ \)? Wait, no, let's check the angles. The angle \( (4x + 10)^\circ \) is \( 38^\circ \) (since \( x = 7 \)), and the angle \( y \) is supplementary to the angle that is \( 90^\circ - 38^\circ \)? No, wait, maybe \( y \) is equal to \( 180^\circ - (90^\circ - (4x + 10)^\circ) \)? No, that's not right. Wait, actually, the angle \( y \) and the angle \( (7x + 3)^\circ \) are supplementary? Wait, \( 7x + 3 = 52^\circ \), so \( y + 52^\circ = 180^\circ \)? No, that would make \( y = 128^\circ \), but let's check. Wait, no, let's see the diagram again. The line \( k \) is vertical, \( m \) is horizontal (since \( m \perp k \)). The lines \( s \) and \( q \) are parallel, so the angle \( y \) is equal to the angle formed by \( k \) and \( q \), and the angle \( (7x + 3)^\circ \) is formed by \( k \) and \( s \). Since \( s \parallel q \), the consecutive interior angles would be supplementary? Wait, no, \( y \) and \( (7x + 3)^\circ \): wait, \( 7x + 3 = 52^\circ \), and \( y \) is adjacent to the right angle. Wait, maybe \( y = 90^\circ + (4x + 10)^\circ \). Let's calculate \( 4x + 10 = 38^\circ \), so \( 90 + 38 = 128^\circ \). Alternatively, \( y = 180^\circ - (7x + 3)^\circ \). Since \( 7x + 3 = 52^\circ \), \( 180 - 52 = 128^\circ \). Yes, that makes sense. So \( y = 180 - (7x + 3) \). Substituting \( x = 7 \):

\( y = 180 - (7*7 + 3) = 180 - 52 = 128 \)

Answer:

\( 128 \)