Question

which angles are vertical to \\(\angle 4\\)? select all that apply. image of intersecting lines with angle numbering: 14,15,6,7 on top horizontal; 13,16,5,8 below; 11,12,3,4 (with 4 arrow) and 10,9,2,1 on bottom horizontal. options: \\(\angle 9\\), \\(\angle 3\\), \\(\angle 2\\), \\(\angle 6\\) (all checked).

Explanation:

Step1: Recall vertical angles definition

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

Step2: Identify intersecting lines for ∠4

∠4 is formed by the intersection of two lines. We look for the angle opposite to ∠4 when these lines intersect. From the diagram, the lines forming ∠4 also form ∠9 (since vertical angles are opposite each other at the intersection point). Wait, actually, let's re - examine. Wait, maybe I made a mistake earlier. Wait, the correct vertical angle to ∠4: when two lines intersect, vertical angles are non - adjacent and opposite. Let's look at the intersection where ∠4 is. The angle opposite to ∠4 should be ∠9? Wait, no, maybe I misread the diagram. Wait, actually, in the diagram, ∠4 and ∠9: Wait, no, let's check the intersection points. Wait, the lines that form ∠4: one line is the one with ∠1, ∠2, ∠3, ∠4 and another line is the one with ∠9, ∠10, ∠11, ∠12. Wait, no, maybe the correct vertical angle to ∠4 is ∠9? Wait, no, let's recall the definition again. Vertical angles are formed when two lines cross each other. So if we have two intersecting lines, say line A and line B, then the angles opposite each other are vertical angles. So for ∠4, the two lines that form it: let's see the angles around the intersection. ∠4 and ∠9: are they vertical? Wait, maybe the correct answer is ∠9. Wait, but let's check the options. Wait, the options are ∠9, ∠3, ∠2, ∠6. Wait, maybe I misanalyzed. Wait, no, vertical angles are opposite. So when two lines intersect, vertical angles are equal. So if ∠4 is at an intersection, the vertical angle should be the one opposite. Let's look at the diagram again. The angle ∠4 and ∠9: are they vertical? Wait, maybe the correct vertical angle to ∠4 is ∠9. Wait, but let's confirm. Let's consider the two lines that intersect to form ∠4. One line is the transversal (the one with ∠1, ∠2, ∠3, ∠4) and the other line is the one with ∠9, ∠10, ∠11, ∠12. When these two lines intersect, the vertical angle of ∠4 is ∠9. Wait, but maybe I made a mistake. Wait, no, the definition of vertical angles is that they are opposite angles formed by two intersecting lines. So ∠4 and ∠9 are vertical angles. Wait, but let's check the other options. ∠3 is adjacent to ∠4, ∠2 is adjacent, ∠6 is not at the same intersection. So the correct vertical angle to ∠4 is ∠9.

Answer:

∠9