Question

1. name each angle pair as corresponding, alternate interior, alternate exterior, consecutive interior, consecutive exterior, or no relationship. identify the transversal that connects each angle pair.

a) ∠4 and ∠10 ____________; transversal: ____
b) ∠8 and ∠11 ____________; transversal: ____
c) ∠1 and ∠4 ____________; transversal: ____
d) ∠2 and ∠12 ____________; transversal: ____
e) ∠5 and ∠7 ____________; transversal: ____
f) ∠2 and ∠13 ____________; transversal: ____

Explanation:

Step1: Analyze ∠4 and ∠10

∠4 is on line \( k \), ∠10 is on line \( k \) and \( l \). Wait, no, transversal is the line cutting two lines. Lines \( j \) and \( k \) are cut by \( l \)? Wait, ∠4 is on \( k \) (between \( m \) and \( k \)? Wait, the diagram: lines \( j \) (top), \( k \) (middle), \( l \) (left vertical), \( m \) (right slant). ∠4 is on \( k \), ∠10 is on \( k \) and \( l \). Wait, ∠4 and ∠10: lines \( l \) and \( m \) are cut by \( k \)? No, ∠4 is at \( k \) and \( m \), ∠10 is at \( k \) and \( l \). Wait, transversal for ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)): transversal is \( k \)? No, transversal is the line intersecting two lines. Wait, ∠4 and ∠10: let's see positions. ∠4 is below \( m \) on \( k \), ∠10 is above \( l \) on \( k \). Wait, maybe alternate interior? Wait, lines \( l \) and \( m \) are cut by \( k \). ∠10 is between \( l \) and \( k \), ∠4 is between \( m \) and \( k \). So alternate interior? Wait, no, ∠4 and ∠10: let's check the angles. ∠4 and ∠10: transversal is \( k \)? No, transversal is the line that connects them. Wait, maybe I messed up. Let's start with part a: ∠4 and ∠10.

∠4 is at the intersection of \( k \) and \( m \), ∠10 is at the intersection of \( k \) and \( l \). So the two lines being cut are \( l \) and \( m \), and the transversal is \( k \). Now, ∠10 is on \( k \), above \( l \), ∠4 is on \( k \), above \( m \)? Wait, no, the diagram: \( j \) is top horizontal, \( k \) is middle horizontal, \( l \) is left vertical (downward), \( m \) is right slant (downward to the right). So ∠10 is at \( k \) and \( l \) (top of \( l \) on \( k \)), ∠4 is at \( k \) and \( m \) (top of \( m \) on \( k \)). So lines \( l \) and \( m \) are parallel? Wait, no, transversal \( k \) cuts \( l \) and \( m \). ∠10 and ∠4: are they alternate interior? Wait, alternate interior angles are between the two lines, on opposite sides of transversal. ∠10 is between \( l \) and \( k \)? No, \( l \) is vertical, \( k \) is horizontal. Wait, maybe I need to re-express.

Wait, let's define:

- Transversal: a line that intersects two or more other lines.

For ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)):

- The two lines intersected by transversal: \( l \) and \( m \) (since ∠10 is on \( l \) and \( k \), ∠4 is on \( m \) and \( k \); so transversal is \( k \), intersecting \( l \) and \( m \)).

- ∠10 is on \( k \), above \( l \); ∠4 is on \( k \), above \( m \). Wait, no, ∠10 is at \( k \) and \( l \) (so between \( l \) and \( k \), but \( l \) is vertical, \( k \) is horizontal. Wait, maybe the angles: ∠10 is adjacent to ∠2 (on \( k \) and \( l \), left side), ∠4 is adjacent to ∠12 (on \( k \) and \( m \), left side). Wait, maybe ∠4 and ∠10 are alternate interior angles? Wait, no, alternate interior would be between the two lines ( \( l \) and \( m \)) and on opposite sides of transversal \( k \). But \( l \) and \( m \) are not parallel, but the angle positions: ∠10 is on \( k \), between \( l \) and the space, ∠4 is on \( k \), between \( m \) and the space. Wait, maybe the answer is alternate interior, transversal \( k \)? Wait, no, maybe I made a mistake. Let's check part a again.

Wait, maybe the lines are \( j \) (top), \( k \) (middle), \( l \) (left vertical), \( m \) (right slant). So ∠4 is at \( k \) and \( m \) (so on \( k \), below \( j \), above \( m \)? No, the diagram labels: \( j \) has 6,14,8,16; \( k \) has 2,10,4,12; \( l \) has 5,13,2,10 (wait, no, \( l \) is vertical: 6 (top left), 5 (bottom left), 14 (top right), 13 (bottom right) on \( j \) and \( k \); \( m \) is slant: 8 (top left),7 (bottom left),16 (top right),15 (bottom right) on \( j \); 4 (top left),3 (bottom left),12 (top right),11 (bottom right) on \( k \). So ∠4 is at \( k \) and \( m \) (coordinates: \( k \) (middle horizontal), \( m \) (slant), so ∠4 is the angle at \( k \) above \( m \), ∠10 is at \( k \) above \( l \) (vertical). So lines \( l \) (vertical) and \( m \) (slant) are cut by transversal \( k \) (horizontal). So ∠10 (at \( k, l \)) and ∠4 (at \( k, m \)): are they alternate interior? Alternate interior angles are between the two lines ( \( l \) and \( m \)) and on opposite sides of transversal \( k \). ∠10 is between \( l \) and \( k \) (but \( l \) is vertical, \( k \) is horizontal, so between \( l \) and the area to the right of \( l \)), ∠4 is between \( m \) and the area to the right of \( m \). Wait, maybe they are alternate interior. So part a: alternate interior, transversal \( k \).

Step2: Analyze ∠8 and ∠11

∠8 is on \( j \) (top horizontal) and \( m \) (slant), ∠11 is on \( k \) (middle horizontal) and \( m \) (slant). So transversal is \( m \), cutting \( j \) and \( k \). ∠8 is above \( m \) on \( j \), ∠11 is below \( m \) on \( k \). So are they consecutive exterior? No, consecutive exterior would be outside the two lines. Wait, ∠8 is on \( j \) (top), ∠11 is on \( k \) (middle), both on \( m \). So transversal \( m \), lines \( j \) and \( k \). ∠8 is above \( j \) and \( m \), ∠11 is below \( k \) and \( m \). Wait, maybe no relationship? Wait, no, ∠8 and ∠11: ∠8 is at \( j, m \) (top left of \( m \) on \( j \)), ∠11 is at \( k, m \) (bottom right of \( m \) on \( k \)). So transversal \( m \), lines \( j \) and \( k \). The angles: ∠8 is above \( j \), ∠11 is below \( k \). Maybe no relationship? Wait, maybe I'm wrong. Let's think again.

Step3: Analyze ∠1 and ∠4

∠1 is on \( l \) (vertical) and \( k \) (horizontal), bottom left; ∠4 is on \( m \) (slant) and \( k \) (horizontal), top left. So transversal is \( k \), cutting \( l \) and \( m \). ∠1 is below \( k \), ∠4 is above \( k \). So no relationship? Wait, ∠1 is at \( l, k \) (bottom), ∠4 is at \( m, k \) (top). So transversal \( k \), lines \( l \) and \( m \). No relationship?

Step4: Analyze ∠2 and ∠12

∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠12 is on \( m \) (slant) and \( k \) (horizontal), top right. So transversal is \( k \), cutting \( l \) and \( m \). ∠2 is on \( k \) left, ∠12 is on \( k \) right. So no relationship? Wait, ∠2 is at \( l, k \) (top), ∠12 is at \( m, k \) (top). So transversal \( k \), lines \( l \) and \( m \). Maybe corresponding? Corresponding angles are in the same position relative to transversal and lines. ∠2 is top left of \( l \) on \( k \), ∠12 is top right of \( m \) on \( k \). No, not corresponding. Wait, maybe no relationship.

Step5: Analyze ∠5 and ∠7

∠5 is on \( l \) (vertical) and \( j \) (top horizontal), bottom left; ∠7 is on \( m \) (slant) and \( j \) (top horizontal), bottom left. So transversal is \( j \), cutting \( l \) and \( m \). ∠5 is below \( j \), left of \( l \); ∠7 is below \( j \), left of \( m \). So are they consecutive interior? Consecutive interior angles are between the two lines and on the same side of transversal. Lines \( l \) (vertical) and \( m \) (slant) are cut by \( j \) (horizontal). ∠5 is between \( l \) and \( j \) (below \( j \)), ∠7 is between \( m \) and \( j \) (below \( j \)). So same side of transversal \( j \), between \( l \) and \( m \). So consecutive interior, transversal \( j \).

Step6: Analyze ∠2 and ∠13

∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠13 is on \( l \) (vertical) and \( j \) (top horizontal), bottom right. So transversal is \( l \), cutting \( j \) and \( k \). ∠2 is on \( k \), above \( l \); ∠13 is on \( j \), below \( l \). So are they alternate interior? Alternate interior angles are between the two lines ( \( j \) and \( k \)) and on opposite sides of transversal \( l \). \( j \) (top) and \( k \) (middle) are horizontal, cut by \( l \) (vertical). ∠2 is between \( j \) and \( k \), right of \( l \); ∠13 is between \( j \) and \( k \), left of \( l \)? Wait, no, ∠2 is on \( k \), left of \( l \) (since \( l \) is vertical, \( k \) is horizontal: ∠2 is at \( l \) and \( k \), top left, so between \( l \) and the left, but \( j \) is above \( k \). Wait, ∠13 is at \( l \) and \( j \), bottom right (so between \( l \) and the right, below \( j \)). ∠2 is at \( l \) and \( k \), top left (between \( l \) and the left, above \( k \)). So lines \( j \) and \( k \) are cut by \( l \). ∠2 is on \( k \), below \( j \), left of \( l \); ∠13 is on \( j \), above \( k \), right of \( l \). Wait, maybe alternate interior? Because they are between \( j \) and \( k \) (the two horizontal lines) and on opposite sides of transversal \( l \) (vertical). So alternate interior, transversal \( l \).

Now, let's summarize each part:

a) ∠4 and ∠10: alternate interior, transversal \( k \)

b) ∠8 and ∠11: Let's recheck. ∠8 is on \( j \) (top) and \( m \) (slant), ∠11 is on \( k \) (middle) and \( m \) (slant). Transversal \( m \), lines \( j \) and \( k \). ∠8 is above \( j \), ∠11 is below \( k \). Wait, maybe no relationship? Or maybe alternate exterior? No, alternate exterior would be outside. Wait, ∠8 is on \( j \), above \( m \); ∠11 is on \( k \), below \( m \). So they are on opposite sides of \( m \) and outside the lines \( j \) and \( k \)? No, \( j \) and \( k \) are horizontal, \( m \) is slant. Maybe no relationship. Wait, maybe the answer is no relationship, transversal \( m \).

c) ∠1 and ∠4: ∠1 is on \( l \) (vertical) and \( k \) (horizontal), bottom left; ∠4 is on \( m \) (slant) and \( k \) (horizontal), top left. Transversal \( k \), lines \( l \) and \( m \). ∠1 is below \( k \), ∠4 is above \( k \). No relationship, transversal \( k \).

d) ∠2 and ∠12: ∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠12 is on \( m \) (slant) and \( k \) (horizontal), top right. Transversal \( k \), lines \( l \) and \( m \). ∠2 is left of \( l \), ∠12 is right of \( m \). No relationship, transversal \( k \).

e) ∠5 and ∠7: consecutive interior, transversal \( j \) (since they are between \( l \) and \( m \), same side of \( j \))

f) ∠2 and ∠13: alternate interior, transversal \( l \) (between \( j \) and \( k \), opposite sides of \( l \))

But maybe I made mistakes. Let's check standard angle pair definitions:

- Corresponding: same position relative to transversal and lines.

- Alternate interior: between two lines, opposite sides of transversal.

- Alternate exterior: outside two lines, opposite sides of transversal.

- Consecutive interior: between two lines, same side of transversal.

- Consecutive exterior: outside two lines, same side of transversal.

So for part a: ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)): lines \( l \) and \( m \) (two lines) cut by \( k \) (transversal). ∠10 is between \( l \) and \( k \) (wait, \( l \) is vertical, \( k \) is horizontal, so between \( l \) and the right), ∠4 is between \( m \) and the right. So same side? No, alternate interior would be opposite sides. Wait, maybe I had it wrong. Maybe transversal is \( l \)? No, ∠4 is not on \( l \). Wait, maybe the lines are \( j \) and \( k \) cut by \( l \)? No, ∠4 is on \( m \). I think I need to re-express the diagram:

- Line \( l \): vertical, intersects \( j \) (top horizontal) at (14,13) and \( k \) (middle horizontal) at (10,9).

- Line \( m \): slant, intersects \( j \) (top horizontal) at (16,15) and \( k \) (middle horizontal) at (12,11).

- So ∠4 is at \( k \) and \( m \): angle between \( k \) (right) and \( m \) (slant down to right), so ∠4 is above \( m \) on \( k \).

- ∠10 is at \( k \) and \( l \): angle between \( k \) (right) and \( l \) (vertical down), so ∠10 is above \( l \) on \( k \).

So lines \( l \) (vertical) and \( m \) (slant) are cut by transversal \( k \) (horizontal). So ∠10 (at \( k, l \)) and ∠4 (at \( k, m \)): are they on the same side of \( k \)? Yes, both above \( k \) (wait, \( k \) is horizontal, so above \( k \) is towards \( j \), below is

Answer:

Step1: Analyze ∠4 and ∠10

∠4 is on line \( k \), ∠10 is on line \( k \) and \( l \). Wait, no, transversal is the line cutting two lines. Lines \( j \) and \( k \) are cut by \( l \)? Wait, ∠4 is on \( k \) (between \( m \) and \( k \)? Wait, the diagram: lines \( j \) (top), \( k \) (middle), \( l \) (left vertical), \( m \) (right slant). ∠4 is on \( k \), ∠10 is on \( k \) and \( l \). Wait, ∠4 and ∠10: lines \( l \) and \( m \) are cut by \( k \)? No, ∠4 is at \( k \) and \( m \), ∠10 is at \( k \) and \( l \). Wait, transversal for ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)): transversal is \( k \)? No, transversal is the line intersecting two lines. Wait, ∠4 and ∠10: let's see positions. ∠4 is below \( m \) on \( k \), ∠10 is above \( l \) on \( k \). Wait, maybe alternate interior? Wait, lines \( l \) and \( m \) are cut by \( k \). ∠10 is between \( l \) and \( k \), ∠4 is between \( m \) and \( k \). So alternate interior? Wait, no, ∠4 and ∠10: let's check the angles. ∠4 and ∠10: transversal is \( k \)? No, transversal is the line that connects them. Wait, maybe I messed up. Let's start with part a: ∠4 and ∠10.

∠4 is at the intersection of \( k \) and \( m \), ∠10 is at the intersection of \( k \) and \( l \). So the two lines being cut are \( l \) and \( m \), and the transversal is \( k \). Now, ∠10 is on \( k \), above \( l \), ∠4 is on \( k \), above \( m \)? Wait, no, the diagram: \( j \) is top horizontal, \( k \) is middle horizontal, \( l \) is left vertical (downward), \( m \) is right slant (downward to the right). So ∠10 is at \( k \) and \( l \) (top of \( l \) on \( k \)), ∠4 is at \( k \) and \( m \) (top of \( m \) on \( k \)). So lines \( l \) and \( m \) are parallel? Wait, no, transversal \( k \) cuts \( l \) and \( m \). ∠10 and ∠4: are they alternate interior? Wait, alternate interior angles are between the two lines, on opposite sides of transversal. ∠10 is between \( l \) and \( k \)? No, \( l \) is vertical, \( k \) is horizontal. Wait, maybe I need to re-express.

Wait, let's define:

• Transversal: a line that intersects two or more other lines.

For ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)):

• The two lines intersected by transversal: \( l \) and \( m \) (since ∠10 is on \( l \) and \( k \), ∠4 is on \( m \) and \( k \); so transversal is \( k \), intersecting \( l \) and \( m \)).

• ∠10 is on \( k \), above \( l \); ∠4 is on \( k \), above \( m \). Wait, no, ∠10 is at \( k \) and \( l \) (so between \( l \) and \( k \), but \( l \) is vertical, \( k \) is horizontal. Wait, maybe the angles: ∠10 is adjacent to ∠2 (on \( k \) and \( l \), left side), ∠4 is adjacent to ∠12 (on \( k \) and \( m \), left side). Wait, maybe ∠4 and ∠10 are alternate interior angles? Wait, no, alternate interior would be between the two lines ( \( l \) and \( m \)) and on opposite sides of transversal \( k \). But \( l \) and \( m \) are not parallel, but the angle positions: ∠10 is on \( k \), between \( l \) and the space, ∠4 is on \( k \), between \( m \) and the space. Wait, maybe the answer is alternate interior, transversal \( k \)? Wait, no, maybe I made a mistake. Let's check part a again.

Wait, maybe the lines are \( j \) (top), \( k \) (middle), \( l \) (left vertical), \( m \) (right slant). So ∠4 is at \( k \) and \( m \) (so on \( k \), below \( j \), above \( m \)? No, the diagram labels: \( j \) has 6,14,8,16; \( k \) has 2,10,4,12; \( l \) has 5,13,2,10 (wait, no, \( l \) is vertical: 6 (top left), 5 (bottom left), 14 (top right), 13 (bottom right) on \( j \) and \( k \); \( m \) is slant: 8 (top left),7 (bottom left),16 (top right),15 (bottom right) on \( j \); 4 (top left),3 (bottom left),12 (top right),11 (bottom right) on \( k \). So ∠4 is at \( k \) and \( m \) (coordinates: \( k \) (middle horizontal), \( m \) (slant), so ∠4 is the angle at \( k \) above \( m \), ∠10 is at \( k \) above \( l \) (vertical). So lines \( l \) (vertical) and \( m \) (slant) are cut by transversal \( k \) (horizontal). So ∠10 (at \( k, l \)) and ∠4 (at \( k, m \)): are they alternate interior? Alternate interior angles are between the two lines ( \( l \) and \( m \)) and on opposite sides of transversal \( k \). ∠10 is between \( l \) and \( k \) (but \( l \) is vertical, \( k \) is horizontal, so between \( l \) and the area to the right of \( l \)), ∠4 is between \( m \) and the area to the right of \( m \). Wait, maybe they are alternate interior. So part a: alternate interior, transversal \( k \).

Step2: Analyze ∠8 and ∠11

∠8 is on \( j \) (top horizontal) and \( m \) (slant), ∠11 is on \( k \) (middle horizontal) and \( m \) (slant). So transversal is \( m \), cutting \( j \) and \( k \). ∠8 is above \( m \) on \( j \), ∠11 is below \( m \) on \( k \). So are they consecutive exterior? No, consecutive exterior would be outside the two lines. Wait, ∠8 is on \( j \) (top), ∠11 is on \( k \) (middle), both on \( m \). So transversal \( m \), lines \( j \) and \( k \). ∠8 is above \( j \) and \( m \), ∠11 is below \( k \) and \( m \). Wait, maybe no relationship? Wait, no, ∠8 and ∠11: ∠8 is at \( j, m \) (top left of \( m \) on \( j \)), ∠11 is at \( k, m \) (bottom right of \( m \) on \( k \)). So transversal \( m \), lines \( j \) and \( k \). The angles: ∠8 is above \( j \), ∠11 is below \( k \). Maybe no relationship? Wait, maybe I'm wrong. Let's think again.

Step3: Analyze ∠1 and ∠4

∠1 is on \( l \) (vertical) and \( k \) (horizontal), bottom left; ∠4 is on \( m \) (slant) and \( k \) (horizontal), top left. So transversal is \( k \), cutting \( l \) and \( m \). ∠1 is below \( k \), ∠4 is above \( k \). So no relationship? Wait, ∠1 is at \( l, k \) (bottom), ∠4 is at \( m, k \) (top). So transversal \( k \), lines \( l \) and \( m \). No relationship?

Step4: Analyze ∠2 and ∠12

∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠12 is on \( m \) (slant) and \( k \) (horizontal), top right. So transversal is \( k \), cutting \( l \) and \( m \). ∠2 is on \( k \) left, ∠12 is on \( k \) right. So no relationship? Wait, ∠2 is at \( l, k \) (top), ∠12 is at \( m, k \) (top). So transversal \( k \), lines \( l \) and \( m \). Maybe corresponding? Corresponding angles are in the same position relative to transversal and lines. ∠2 is top left of \( l \) on \( k \), ∠12 is top right of \( m \) on \( k \). No, not corresponding. Wait, maybe no relationship.

Step5: Analyze ∠5 and ∠7

∠5 is on \( l \) (vertical) and \( j \) (top horizontal), bottom left; ∠7 is on \( m \) (slant) and \( j \) (top horizontal), bottom left. So transversal is \( j \), cutting \( l \) and \( m \). ∠5 is below \( j \), left of \( l \); ∠7 is below \( j \), left of \( m \). So are they consecutive interior? Consecutive interior angles are between the two lines and on the same side of transversal. Lines \( l \) (vertical) and \( m \) (slant) are cut by \( j \) (horizontal). ∠5 is between \( l \) and \( j \) (below \( j \)), ∠7 is between \( m \) and \( j \) (below \( j \)). So same side of transversal \( j \), between \( l \) and \( m \). So consecutive interior, transversal \( j \).

Step6: Analyze ∠2 and ∠13

∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠13 is on \( l \) (vertical) and \( j \) (top horizontal), bottom right. So transversal is \( l \), cutting \( j \) and \( k \). ∠2 is on \( k \), above \( l \); ∠13 is on \( j \), below \( l \). So are they alternate interior? Alternate interior angles are between the two lines ( \( j \) and \( k \)) and on opposite sides of transversal \( l \). \( j \) (top) and \( k \) (middle) are horizontal, cut by \( l \) (vertical). ∠2 is between \( j \) and \( k \), right of \( l \); ∠13 is between \( j \) and \( k \), left of \( l \)? Wait, no, ∠2 is on \( k \), left of \( l \) (since \( l \) is vertical, \( k \) is horizontal: ∠2 is at \( l \) and \( k \), top left, so between \( l \) and the left, but \( j \) is above \( k \). Wait, ∠13 is at \( l \) and \( j \), bottom right (so between \( l \) and the right, below \( j \)). ∠2 is at \( l \) and \( k \), top left (between \( l \) and the left, above \( k \)). So lines \( j \) and \( k \) are cut by \( l \). ∠2 is on \( k \), below \( j \), left of \( l \); ∠13 is on \( j \), above \( k \), right of \( l \). Wait, maybe alternate interior? Because they are between \( j \) and \( k \) (the two horizontal lines) and on opposite sides of transversal \( l \) (vertical). So alternate interior, transversal \( l \).

Now, let's summarize each part:

a) ∠4 and ∠10: alternate interior, transversal \( k \)

b) ∠8 and ∠11: Let's recheck. ∠8 is on \( j \) (top) and \( m \) (slant), ∠11 is on \( k \) (middle) and \( m \) (slant). Transversal \( m \), lines \( j \) and \( k \). ∠8 is above \( j \), ∠11 is below \( k \). Wait, maybe no relationship? Or maybe alternate exterior? No, alternate exterior would be outside. Wait, ∠8 is on \( j \), above \( m \); ∠11 is on \( k \), below \( m \). So they are on opposite sides of \( m \) and outside the lines \( j \) and \( k \)? No, \( j \) and \( k \) are horizontal, \( m \) is slant. Maybe no relationship. Wait, maybe the answer is no relationship, transversal \( m \).

c) ∠1 and ∠4: ∠1 is on \( l \) (vertical) and \( k \) (horizontal), bottom left; ∠4 is on \( m \) (slant) and \( k \) (horizontal), top left. Transversal \( k \), lines \( l \) and \( m \). ∠1 is below \( k \), ∠4 is above \( k \). No relationship, transversal \( k \).

d) ∠2 and ∠12: ∠2 is on \( l \) (vertical) and \( k \) (horizontal), top left; ∠12 is on \( m \) (slant) and \( k \) (horizontal), top right. Transversal \( k \), lines \( l \) and \( m \). ∠2 is left of \( l \), ∠12 is right of \( m \). No relationship, transversal \( k \).

e) ∠5 and ∠7: consecutive interior, transversal \( j \) (since they are between \( l \) and \( m \), same side of \( j \))

f) ∠2 and ∠13: alternate interior, transversal \( l \) (between \( j \) and \( k \), opposite sides of \( l \))

But maybe I made mistakes. Let's check standard angle pair definitions:

• Corresponding: same position relative to transversal and lines.

• Alternate interior: between two lines, opposite sides of transversal.

• Alternate exterior: outside two lines, opposite sides of transversal.

• Consecutive interior: between two lines, same side of transversal.

• Consecutive exterior: outside two lines, same side of transversal.

So for part a: ∠4 (at \( k, m \)) and ∠10 (at \( k, l \)): lines \( l \) and \( m \) (two lines) cut by \( k \) (transversal). ∠10 is between \( l \) and \( k \) (wait, \( l \) is vertical, \( k \) is horizontal, so between \( l \) and the right), ∠4 is between \( m \) and the right. So same side? No, alternate interior would be opposite sides. Wait, maybe I had it wrong. Maybe transversal is \( l \)? No, ∠4 is not on \( l \). Wait, maybe the lines are \( j \) and \( k \) cut by \( l \)? No, ∠4 is on \( m \). I think I need to re-express the diagram:

• Line \( l \): vertical, intersects \( j \) (top horizontal) at (14,13) and \( k \) (middle horizontal) at (10,9).

• Line \( m \): slant, intersects \( j \) (top horizontal) at (16,15) and \( k \) (middle horizontal) at (12,11).

• So ∠4 is at \( k \) and \( m \): angle between \( k \) (right) and \( m \) (slant down to right), so ∠4 is above \( m \) on \( k \).

• ∠10 is at \( k \) and \( l \): angle between \( k \) (right) and \( l \) (vertical down), so ∠10 is above \( l \) on \( k \).

So lines \( l \) (vertical) and \( m \) (slant) are cut by transversal \( k \) (horizontal). So ∠10 (at \( k, l \)) and ∠4 (at \( k, m \)): are they on the same side of \( k \)? Yes, both above \( k \) (wait, \( k \) is horizontal, so above \( k \) is towards \( j \), below is