Question

linear pairs and vertical angles
which are vertical angles?
∠afe and ∠bfd
∠bfc and ∠dfe
∠afe and ∠cfd
∠bfc and ∠efa

Explanation:

Step1: Recall vertical angles definition

Vertical angles are opposite angles formed by two intersecting lines, equal in measure.

Step2: Analyze each option

- $\angle AFE$ and $\angle BFD$: Formed by intersecting lines $AB$ and $DE$, opposite each other. Wait, no—wait, lines $AB$ (with points A, F, B) and $DE$ (with E, F, D) intersect at F, and $\angle AFE$ and $\angle BFD$: Wait, no, let's check the diagram. Wait, the correct vertical angles: when two lines intersect, vertical angles are opposite. Wait, the first option: $\angle AFE$ and $\angle BFD$—wait, no, maybe I misread. Wait, the diagram: lines are $AB$ (A-F-B), $DE$ (E-F-D), and $FC$ (F-C). So intersecting lines are $AB$ and $DE$ (intersect at F), and $FC$ is another line. So vertical angles from $AB$ and $DE$: $\angle AFE$ (between A-F-E) and $\angle BFD$ (between B-F-D)? Wait, no, $\angle AFE$ and $\angle BFD$: let's see, $\angle AFE$ has sides FA and FE, $\angle BFD$ has sides FB and FD. Since FA and FB are opposite rays (A-F-B is a straight line), and FE and FD are opposite rays (E-F-D is a straight line), so $\angle AFE$ and $\angle BFD$ are vertical angles? Wait, but the original selected option was $\angle BFC$ and $\angle DFE$, which is wrong. Wait, no—wait, maybe I made a mistake. Wait, vertical angles are formed by two intersecting lines, so the two angles must be opposite each other, with the same vertex, and their sides are opposite rays. Let's re-express:

- Option 1: $\angle AFE$ (vertex F, sides FA, FE) and $\angle BFD$ (vertex F, sides FB, FD). FA and FB are opposite rays (since A-F-B is straight), FE and FD are opposite rays (E-F-D is straight). So these are vertical angles. Wait, but the original selection was wrong. Wait, maybe the diagram is different. Wait, the lines: A-F-D is a straight line? Wait, no, the diagram shows A-F-D as a horizontal line (A to left, D to right), E-F-B as a diagonal line (E up-right, B down-left), and F-C as a vertical line down. So intersecting lines: A-D (horizontal) and E-B (diagonal) intersect at F. Then vertical angles from A-D and E-B: $\angle AFE$ (between A-F-E) and $\angle DFB$ (which is $\angle BFD$). So $\angle AFE$ and $\angle BFD$ are vertical angles. Wait, but the other options: $\angle BFC$ (sides FB, FC) and $\angle DFE$ (sides FD, FE). FB and FD: FB is part of E-B, FD is part of A-D. FC is vertical down, FE is part of E-B. So $\angle BFC$ and $\angle DFE$: are their sides opposite rays? FB and FD: no, FB and FD are not opposite rays (since E-B and A-D intersect at F, FB is from F to B, FD is from F to D; E-F-B and A-F-D are two lines, so FB and FD are not opposite rays). Wait, maybe I messed up. Wait, let's list all angles:

- Intersection of A-D (horizontal) and E-B (diagonal) at F: angles are $\angle AFE$, $\angle EFD$, $\angle DFB$, $\angle BFA$ (but BFA is straight? No, A-F-D is straight, E-F-B is straight, so they form vertical angles: $\angle AFE$ and $\angle DFB$ (i.e., $\angle BFD$), and $\angle EFD$ and $\angle BFA$. Then there's line F-C (vertical down), so angles with F-C: $\angle BFC$ (between FB and FC), $\angle CFD$ (between FC and FD), $\angle DFE$ (between FD and FE), $\angle EFA$ (between FE and FA). Wait, no, $\angle DFE$ is between FD and FE, which is the same as $\angle EFD$ (from A-D and E-B intersection). Wait, maybe the correct vertical angles are $\angle AFE$ and $\angle BFD$ because they are opposite angles formed by the intersection of lines A-D and E-B. Let's check the options again:

Option 1: $\angle AFE$ and $\angle BFD$ – these are vertical angles (opposite, formed by two intersecting lines).

Option 2: $\angle BFC$ (FB and FC) and $\angle DFE$ (FD and FE) – FB and FD are not opposite rays, FC and FE are not opposite rays, so not vertical.

Option 3: $\angle AFE$ and $\angle CFD$ – $\angle CFD$ is between FC and FD, not opposite to $\angle AFE$.

Option 4: $\angle BFC$ and $\angle EFA$ – $\angle EFA$ is between FE and FA, not opposite to $\angle BFC$.

So the correct answer should be $\angle AFE$ and $\angle BFD$. Wait, but the original selected option was wrong. So the correct vertical angles are $\angle AFE$ and $\angle BFD$.

Answer:

$\angle AFE$ and $\angle BFD$