Question

1. (7x - 23)° 49° (3x)° (11y - 1)° l m

Explanation:

To solve for \( x \) and \( y \), we analyze the angles formed by intersecting lines.

Step 1: Solve for \( x \)

The angles \( 49^\circ \), \( (3x)^\circ \), and \( (7x - 23)^\circ \) are related such that the sum of angles on a straight line is \( 180^\circ \). However, we can also use the property of vertical angles or supplementary angles. Notice that \( (7x - 23)^\circ \) and \( (3x + 49)^\circ \) are supplementary (they form a linear pair). Wait, actually, looking at the diagram, the angle \( (7x - 23)^\circ \) and the angle formed by \( 49^\circ \) and \( 3x^\circ \) are vertical angles? Wait, no, let's re-examine.

Wait, actually, the angle \( (7x - 23)^\circ \) and the angle adjacent to \( 49^\circ \) and \( 3x^\circ \) are vertical angles. Wait, maybe a better approach: the sum of angles around a point is \( 360^\circ \), but maybe we can use the fact that \( (7x - 23)^\circ \) and \( (3x + 49)^\circ \) are equal? Wait, no, let's check the linear pair.

Wait, actually, the angle \( (7x - 23)^\circ \) and the angle formed by \( 49^\circ \) and \( 3x^\circ \) are supplementary? Wait, no, let's look at the straight line. The angle \( (7x - 23)^\circ \) and the angle below it (formed by \( 49^\circ \) and \( 3x^\circ \)) are supplementary. Wait, actually, the correct approach is:

The angle \( (7x - 23)^\circ \) and the angle \( (3x + 49)^\circ \) are vertical angles? No, wait, let's see: when two lines intersect, vertical angles are equal. Wait, maybe the angle \( (7x - 23)^\circ \) is equal to \( (3x + 49)^\circ \)? Wait, no, let's check the sum.

Wait, actually, the sum of \( 49^\circ \), \( 3x^\circ \), and \( (7x - 23)^\circ \) is \( 180^\circ \) because they form a straight line. So:

\[
49 + 3x + (7x - 23) = 180
\]

Simplify the left side:

\[
49 - 23 + 3x + 7x = 180
\]
\[
26 + 10x = 180
\]

Subtract 26 from both sides:

\[
10x = 180 - 26
\]
\[
10x = 154
\]

Wait, that gives \( x = 15.4 \), which doesn't seem right. Maybe I made a mistake. Wait, maybe the angle \( (7x - 23)^\circ \) is equal to \( (3x + 49)^\circ \) because they are vertical angles. Let's try that:

\[
7x - 23 = 3x + 49
\]

Subtract \( 3x \) from both sides:

\[
4x - 23 = 49
\]

Add 23 to both sides:

\[
4x = 72
\]

Divide by 4:

\[
x = 18
\]

Ah, that makes sense. So \( x = 18 \).

Step 2: Solve for \( y \)

Now that we know \( x = 18 \), we can find \( 3x \):

\[
3x = 3 \times 18 = 54^\circ
\]

Now, the angle \( (11y - 1)^\circ \) is equal to \( 54^\circ \) because they are corresponding angles (since lines \( l \) and \( m \) are parallel? Wait, the diagram shows lines \( l \) and \( m \) with arrows, so they are parallel. So the angle \( (11y - 1)^\circ \) is equal to \( 3x^\circ \) (which is \( 54^\circ \)) because they are corresponding angles.

So:

\[
11y - 1 = 54
\]

Add 1 to both sides:

\[
11y = 55
\]

Divide by 11:

\[
y = 5
\]

Final Answers

For \( x \): \( \boxed{18} \)

For \( y \): \( \boxed{5} \)

Answer:

To solve for \( x \) and \( y \), we analyze the angles formed by intersecting lines.

Step 1: Solve for \( x \)

The angles \( 49^\circ \), \( (3x)^\circ \), and \( (7x - 23)^\circ \) are related such that the sum of angles on a straight line is \( 180^\circ \). However, we can also use the property of vertical angles or supplementary angles. Notice that \( (7x - 23)^\circ \) and \( (3x + 49)^\circ \) are supplementary (they form a linear pair). Wait, actually, looking at the diagram, the angle \( (7x - 23)^\circ \) and the angle formed by \( 49^\circ \) and \( 3x^\circ \) are vertical angles? Wait, no, let's re-examine.

Wait, actually, the angle \( (7x - 23)^\circ \) and the angle adjacent to \( 49^\circ \) and \( 3x^\circ \) are vertical angles. Wait, maybe a better approach: the sum of angles around a point is \( 360^\circ \), but maybe we can use the fact that \( (7x - 23)^\circ \) and \( (3x + 49)^\circ \) are equal? Wait, no, let's check the linear pair.

Wait, actually, the angle \( (7x - 23)^\circ \) and the angle formed by \( 49^\circ \) and \( 3x^\circ \) are supplementary? Wait, no, let's look at the straight line. The angle \( (7x - 23)^\circ \) and the angle below it (formed by \( 49^\circ \) and \( 3x^\circ \)) are supplementary. Wait, actually, the correct approach is:

The angle \( (7x - 23)^\circ \) and the angle \( (3x + 49)^\circ \) are vertical angles? No, wait, let's see: when two lines intersect, vertical angles are equal. Wait, maybe the angle \( (7x - 23)^\circ \) is equal to \( (3x + 49)^\circ \)? Wait, no, let's check the sum.

Wait, actually, the sum of \( 49^\circ \), \( 3x^\circ \), and \( (7x - 23)^\circ \) is \( 180^\circ \) because they form a straight line. So:

\[
49 + 3x + (7x - 23) = 180
\]

Simplify the left side:

\[
49 - 23 + 3x + 7x = 180
\]
\[
26 + 10x = 180
\]

Subtract 26 from both sides:

\[
10x = 180 - 26
\]
\[
10x = 154
\]

Wait, that gives \( x = 15.4 \), which doesn't seem right. Maybe I made a mistake. Wait, maybe the angle \( (7x - 23)^\circ \) is equal to \( (3x + 49)^\circ \) because they are vertical angles. Let's try that:

\[
7x - 23 = 3x + 49
\]

Subtract \( 3x \) from both sides:

\[
4x - 23 = 49
\]

Add 23 to both sides:

\[
4x = 72
\]

Divide by 4:

\[
x = 18
\]

Ah, that makes sense. So \( x = 18 \).

Step 2: Solve for \( y \)

Now that we know \( x = 18 \), we can find \( 3x \):

\[
3x = 3 \times 18 = 54^\circ
\]

Now, the angle \( (11y - 1)^\circ \) is equal to \( 54^\circ \) because they are corresponding angles (since lines \( l \) and \( m \) are parallel? Wait, the diagram shows lines \( l \) and \( m \) with arrows, so they are parallel. So the angle \( (11y - 1)^\circ \) is equal to \( 3x^\circ \) (which is \( 54^\circ \)) because they are corresponding angles.

So:

\[
11y - 1 = 54
\]

Add 1 to both sides:

\[
11y = 55
\]

Divide by 11:

\[
y = 5
\]

Final Answers

For \( x \): \( \boxed{18} \)

For \( y \): \( \boxed{5} \)