Question

which of the following are exterior angles? check all that apply. there is an image of a triangle with intersecting lines and labeled angles 1,2,3,4,5,6. below the image, there are checkboxes for ∠1, ∠5, ∠3, ∠4, ∠6, ∠2

Explanation:

Step1: Recall Exterior Angle Definition

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. It is supplementary to the adjacent interior angle and lies outside the triangle.

Step2: Analyze Each Angle

- $\angle 1$: Interior angle (inside the triangle), not exterior.
- $\angle 5$: Let's see the triangle. The angle adjacent to a side extension? Wait, no. Wait, looking at the diagram, the triangle has two lines intersecting its sides. Let's re - check. Wait, the exterior angles should be formed by extending a side. For the left - hand intersection, $\angle 3$ and $\angle 2$: Wait, no, let's look at the right - hand intersection. The triangle's side is extended, and $\angle 4$, $\angle 5$, $\angle 6$: Wait, no, maybe I messed up. Wait, the standard exterior angle: when you extend a side of the triangle, the angle outside is exterior. So for the triangle, at the left vertex, extending the side gives $\angle 3$ (exterior), $\angle 2$? Wait, no, $\angle 2$ is adjacent to $\angle 1$, but $\angle 3$ is vertical to an exterior angle? Wait, maybe better to list:
- $\angle 3$: Formed by extending a side of the triangle, outside the triangle, so exterior.
- $\angle 4$: Let's see, the triangle's side is extended, and $\angle 4$ is outside? Wait, no, the triangle has an interior angle, and $\angle 4$ is adjacent to a side extension. Wait, maybe the correct exterior angles are $\angle 3$, $\angle 4$, $\angle 6$? Wait, no, let's re - examine. The triangle has three vertices. At each vertex, when you extend a side, you get an exterior angle.
- At the left vertex: the two lines intersect, forming $\angle 1$ (interior), $\angle 2$ (adjacent, but is it exterior? Wait, no, $\angle 3$ is vertical to the angle that is exterior. Wait, maybe the exterior angles are $\angle 3$, $\angle 4$, $\angle 6$? Wait, no, let's check the options. The options are $\angle 1$, $\angle 5$, $\angle 3$, $\angle 4$, $\angle 6$, $\angle 2$.
- $\angle 1$: Interior, so no.
- $\angle 5$: Let's see, is it inside or outside? The triangle's side is extended, and $\angle 5$ is adjacent to an interior angle? Wait, no, maybe $\angle 3$: when we extend the left - most side of the triangle, $\angle 3$ is outside. $\angle 4$: when we extend the right - hand side of the triangle, $\angle 4$ is outside? Wait, no, $\angle 4$ is adjacent to an interior angle, but is it exterior? Wait, maybe the correct exterior angles are $\angle 3$, $\angle 4$, $\angle 6$? Wait, no, let's think again. The exterior angle theorem: an exterior angle is equal to the sum of two non - adjacent interior angles. So for the triangle, the exterior angles should be $\angle 3$, $\angle 4$, $\angle 6$? Wait, no, looking at the diagram, the angles at the intersections:
- At the left intersection: the angles are $\angle 1$, $\angle 2$, $\angle 3$. $\angle 1$ is interior, $\angle 2$ is adjacent (supplementary to $\angle 1$), but $\angle 3$ is vertical to the angle that is exterior? Wait, no, $\angle 3$ is an exterior angle because it is formed by extending a side of the triangle and is outside the triangle.
- At the right intersection: the angles are $\angle 4$, $\angle 5$, $\angle 6$. $\angle 4$ and $\angle 6$ are formed by extending the side of the triangle, outside the triangle, so they are exterior. $\angle 5$ is interior? Wait, no, maybe $\angle 5$ is adjacent to an interior angle, but no. Wait, maybe I made a mistake. Let's list again:
- $\angle 3$: Exterior (formed by extending a side, outside the triangle).
- $\angle 4$: Exterior (formed by extending a side, outside the triangle).
- $\angle 6$: Exterior (formed by extending a side, outside the triangle).
- $\angle 2$: Let's see, $\angle 2$ is adjacent to $\angle 1$, but is it exterior? No, because it is on the same side as the triangle? Wait, no, the triangle is between the two lines. So $\angle 2$ is adjacent to $\angle 1$, but $\angle 3$ is outside.
- $\angle 1$: Interior.
- $\angle 5$: Interior (or adjacent to an interior angle, not exterior).

So the exterior angles are $\angle 3$, $\angle 4$, $\angle 6$. Wait, but let's check the options. The options are $\angle 1$, $\angle 5$, $\angle 3$, $\angle 4$, $\angle 6$, $\angle 2$. So we need to check which are exterior.

Wait, maybe another approach: the exterior angle is equal to the sum of the two remote interior angles. So for the triangle, the exterior angles should be the ones that are not inside the triangle. So:

- $\angle 3$: Outside the triangle, formed by extending a side, so exterior.
- $\angle 4$: Outside the triangle, formed by extending a side, so exterior.
- $\angle 6$: Outside the triangle, formed by extending a side, so exterior.
- $\angle 2$: Let's see, is $\angle 2$ outside? No, it is adjacent to $\angle 1$, inside the "angle" formed by the two lines, but the triangle is below. Wait, maybe I was wrong. Let's look at the diagram again (mentally). The triangle has a vertex on the left, connected to the right vertex. The left vertex has two lines intersecting: one line is the side of the triangle, the other is an extension. So the angle outside the triangle at the left vertex is $\angle 3$. At the right vertex, the side of the triangle is extended, and the angles outside are $\angle 4$ and $\angle 6$. So the exterior angles are $\angle 3$, $\angle 4$, $\angle 6$. Also, wait, at the left vertex, $\angle 2$: is $\angle 2$ an exterior angle? No, because $\angle 2$ and $\angle 1$ are adjacent, and $\angle 1$ is interior. $\angle 3$ is vertical to the angle that is exterior (the one supplementary to $\angle 1$). Wait, maybe $\angle 2$ is not exterior, $\angle 3$ is. At the right vertex, the angles: the triangle's interior angle is adjacent to $\angle 4$ (supplementary), so $\angle 4$ is exterior. $\angle 6$ is also exterior (vertical to an exterior angle). $\angle 5$ is interior. $\angle 1$ is interior. So the exterior angles are $\angle 3$, $\angle 4$, $\angle 6$. Also, wait, what about $\angle 2$? No, $\angle 2$ is adjacent to $\angle 1$, and the triangle is below, so $\angle 2$ is not exterior. Wait, maybe I made a mistake with $\angle 2$. Let's check the definition again: An exterior angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side. So for the left - hand side: the triangle has a side, and we extend the other side (the one forming $\angle 1$) beyond the vertex. Then the angle between the extended side and the other side (not the triangle's side) is $\angle 3$ (exterior), and the angle between the extended side and the triangle's side is $\angle 2$ (adjacent, but is it exterior? Wait, no, the exterior angle is the one outside the triangle. So if the triangle is below $\angle 1$, then $\angle 2$ is above $\angle 1$, and the triangle is below, so $\angle 2$ is not outside the triangle. $\angle 3$ is outside. So the exterior angles are $\angle 3$, $\angle 4$, $\angle 6$.

Answer:

$\angle 3$, $\angle 4$, $\angle 6$ (So we check the boxes for $\angle 3$, $\angle 4$, $\angle 6$)