Question

4520va v. geometry sem a geometry linear pairs and vertical angles which angle is a vertical angle with ∠efd? ∠bfc ∠afb ∠ced ∠dfb

Explanation:

Step1: Recall vertical angles definition

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

Step2: Analyze ∠EFD and options

- ∠EFD is formed by lines ED and EB (or EF and FD). The vertical angle should be opposite, sharing vertex F, with sides opposite to ∠EFD's sides.
- ∠BFC: Check if it's opposite. ∠EFD and ∠BFC: Wait, no, re - check. Wait, lines: ED (A - D) and EB (E - B) intersect at F. So ∠EFD (sides FE and FD) and ∠BFA? Wait, no, looking at the diagram, ∠EFD and ∠BFC? Wait, no, correct: When two lines intersect, vertical angles are opposite. The lines are AD (A - D) and EB (E - B) intersecting at F. So ∠EFD (between FE and FD) and ∠BFA? Wait, no, the options: ∠BFC, ∠AFB, ∠CED, ∠DFB. Wait, ∠EFD and ∠BFC: Wait, no, let's see the vertices. ∠EFD has vertex F, sides FE and FD. The vertical angle should have sides FB and FC? No, wait, lines AD and EB intersect at F. So the angles opposite to ∠EFD: ∠EFD and ∠BFC? Wait, no, maybe I made a mistake. Wait, ∠EFD: FE and FD. The other pair: FB and FC? No, wait, AD is a straight line (A - F - D), EB is a straight line (E - F - B). So when AD and EB intersect at F, the vertical angles are ∠EFD and ∠BFA? No, the options: ∠BFC, ∠AFB, ∠CED, ∠DFB. Wait, maybe the lines are AD (A - D) and FC? No, FC is a vertical line. Wait, no, the diagram: points A, F, D are colinear; E, F, B are colinear; F, C is vertical. So ∠EFD: between FE (going to E) and FD (going to D). The vertical angle would be between FB (going to B) and FC? No, ∠BFC: between FB and FC. Wait, no, vertical angles are formed by two intersecting lines. So lines EB (E - F - B) and AD (A - F - D) intersect at F. So the vertical angles are ∠EFD (E - F - D) and ∠BFA (B - F - A)? But ∠BFA is ∠AFB. Wait, no, ∠EFD and ∠BFC? Wait, maybe I misread. Wait, the options: ∠BFC, ∠AFB, ∠CED, ∠DFB. Let's check each:
- ∠BFC: formed by FB and FC. ∠EFD: formed by FE and FD. Are they vertical? No, because FC is a different line. Wait, no, maybe the lines are EB (E - F - B) and FC? No, FC is perpendicular? Wait, no, the key is vertical angles are opposite when two lines intersect. So two lines: let's say line 1: E - F - B, line 2: A - F - D. Intersect at F. Then vertical angles are ∠EFD (between line 2 (FD) and line 1 (FE)) and ∠BFA (between line 2 (FA) and line 1 (FB)). But ∠BFA is ∠AFB. Wait, but ∠AFB is an option. Wait, no, maybe the other pair: line 1: E - F - B, line 2: F - C? No, F - C is a ray. Wait, maybe I made a mistake. Wait, the correct vertical angle for ∠EFD: when two lines intersect, vertical angles are equal and opposite. So ∠EFD and ∠BFC? Wait, no, let's look at the diagram again. The angle ∠EFD: F is the vertex, E and D are on the sides. The vertical angle should have F as vertex, and the sides opposite to E and D. So E is on one line, D on another. The opposite sides would be B (opposite E) and A (opposite D)? No, B is on the same line as E (E - F - B), D is on A - F - D. So the vertical angle of ∠EFD (E - F - D) is ∠B - F - A, which is ∠AFB. Wait, but let's check the options. Wait, the options are ∠BFC, ∠AFB, ∠CED, ∠DFB. Wait, maybe I messed up. Wait, ∠EFD: FE and FD. The vertical angle should be FB and FA? No, ∠AFB is between FA and FB. So ∠EFD (FE, FD) and ∠AFB (FA, FB): are these vertical? Yes, because FE and FB are opposite rays (E - F - B), FD and FA are opposite rays (A - F - D). So ∠EFD and ∠AFB are vertical angles. Wait, but let's check again. Wait, maybe the answer is ∠AFB. Wait, no, maybe I made a mistake. Wait, the correct answer: vertical angles are formed by two intersecting lines, so ∠EFD and ∠BFC? No, ∠BFC is with FC. Wait, maybe the lines are EB (E - F - B) and FC (F - C)? No, that's not two lines intersecting. Wait, I think I made a mistake. Let's recall: vertical angles are opposite angles when two lines cross. So if we have two lines, say L1: E - F - B and L2: A - F - D, intersecting at F. Then the four angles are ∠EFA, ∠AFB, ∠BFD, ∠DFE? No, wait, ∠EFD is ∠DFE. So ∠DFE (E - F - D) and ∠BFA (B - F - A) are vertical angles. So ∠AFB is ∠BFA. So the answer is ∠AFB.

Answer:

$\angle AFB$