Question

which angles are congruent to each other? diagram with angles labeled 6,7,2,3,5,8,1,4,14,15,10,11,13,16,9,12 options: ∠2 and ∠4, ∠13 and ∠7, ∠16 and ∠13, ∠9 and ∠1

Explanation:

Step1: Recall congruent angle rules

Congruent angles have equal measure. Vertical angles, corresponding angles (from parallel lines cut by transversals), alternate interior/exterior angles are congruent.

Step2: Analyze each option

- $\angle2$ and $\angle4$: $\angle2$ and $\angle4$ are adjacent, not vertical/corresponding. $\angle2 + \angle4$? No, $\angle2$ and $\angle1$ are adjacent supplementary, $\angle4$ and $\angle1$ too, but $\angle2$ and $\angle4$: $\angle2$ and $\angle3$ are vertical, $\angle4$ and $\angle1$? Wait, $\angle2$ and $\angle4$: actually, $\angle2$ and $\angle1$ are adjacent, $\angle4$ and $\angle1$ are adjacent. Wait, no—$\angle2$ and $\angle4$: let's see the lines. The two vertical lines (the two parallel? Wait, the diagram: two transversals and two vertical lines? Wait, no, the two upward arrows are parallel? Wait, the first transversal (the slanted one) and the second transversal (the horizontal? No, the horizontal line? Wait, the angles: $\angle2$ and $\angle4$—$\angle2$ is adjacent to $\angle3$ (vertical), $\angle4$ adjacent to $\angle1$. $\angle2$ and $\angle4$: are they vertical? No, vertical angles are opposite. $\angle2$ and $\angle1$ are adjacent, $\angle4$ and $\angle1$ are adjacent. So $\angle2$ and $\angle4$: no, they are not congruent (unless right angles, but no info).
- $\angle13$ and $\angle7$: $\angle13$ and $\angle7$—let's track the lines. The two slanted lines (the two transversals? Wait, the first slanted line (with angles 5,6,7,8) and the second slanted line (with 13,14,15,16) are parallel? And the horizontal line and the other transversal? Wait, $\angle7$ and $\angle15$: corresponding? Wait, $\angle13$: let's see, $\angle13$ and $\angle14$ are adjacent, $\angle16$ and $\angle15$ are adjacent. $\angle7$: on the upper slanted line, $\angle13$: on the lower slanted line. If the two slanted lines are parallel, and the horizontal transversal (the one with angles 2,3,1,4) and the other transversal (with 10,11,9,12) are parallel? Wait, no—$\angle7$ and $\angle13$: are they corresponding angles? Let's see the transversals. The vertical line (with angles 1,2,3,4,9,10,11,12) and the other vertical line? Wait, no, the two vertical lines (the two upward arrows) are parallel. The slanted lines (the two with angles 5,6,7,8 and 13,14,15,16) are also parallel? Then, the transversal (the horizontal line with 2,3,1,4) cuts the two vertical lines, and the other transversal (with 10,11,9,12) cuts them. Also, the slanted transversals: the horizontal transversal (2,3,1,4) cuts the two slanted lines? Wait, $\angle7$ is on the upper slanted line, $\angle13$ is on the lower slanted line, and the transversal (the vertical line? No, the horizontal line? Wait, maybe $\angle7$ and $\angle13$ are corresponding angles because the two slanted lines are parallel and cut by a transversal (the vertical line? No, the horizontal line? Wait, maybe the key is that $\angle13$ and $\angle7$: let's check vertical angles or corresponding. Alternatively, $\angle13$ and $\angle7$: if we look at the angles, $\angle7$ and $\angle5$ are vertical, $\angle13$ and $\angle15$? No. Wait, maybe the correct one is $\angle13$ and $\angle7$ because they are corresponding angles (parallel lines cut by transversal).
- $\angle16$ and $\angle13$: $\angle16$ and $\angle13$ are adjacent, forming a linear pair? No, $\angle13$ and $\angle14$ are adjacent, $\angle16$ and $\angle15$ are adjacent. $\angle16$ and $\angle13$: vertical? No, vertical angles are opposite. $\angle16$ and $\angle13$: no, they are adjacent, supplementary? Not congruent.
- $\angle9$ and $\angle1$: $\angle9$ and $\angle1$—$\angle1$ is on the upper vertical line, $\angle9$ on the lower vertical line. Are they corresponding? The two vertical lines are parallel, and the transversal (the slanted line with 10,11,9,12) and the other transversal (with 2,3,1,4)? Wait, $\angle1$ and $\angle9$: no, $\angle1$ and $\angle10$? Maybe not. So the only one that makes sense is $\angle13$ and $\angle7$ as corresponding angles (parallel lines cut by transversal, so congruent).

Answer:

$\angle13$ and $\angle7$