Question

12 which of the following are exterior angles? check all that apply. ∠1 ∠5 ∠6 ∠3 ∠4 ∠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, lying outside the triangle.

Step2: Analyze Each Angle

- $\angle 1$: Inside the triangle, so not exterior.
- $\angle 2$: Inside the triangle (a right angle, part of the triangle), not exterior.
- $\angle 3$: Formed by extending a side, outside the triangle, so exterior.
- $\angle 4$: Formed by extending a side, outside the triangle, so exterior.
- $\angle 5$: Vertical to $\angle 4$? Wait, no—$\angle 5$ is formed by the extension, outside the triangle? Wait, no, let's re - check. Wait, the triangle has sides, and the exterior angles are those outside. Wait, $\angle 3$: when we extend the top side, $\angle 3$ is outside. $\angle 4$: when we extend the bottom side, $\angle 4$? Wait, no, $\angle 5$: Let's look at the diagram. The triangle has a vertex at the top right (with angles 2,3), a vertex at the bottom right (with angles 6,4,5), and a vertex at the left (angle 1). So, exterior angles: $\angle 3$ (formed by extending the top side, outside the triangle), $\angle 4$ (wait, no, $\angle 5$: Wait, maybe I made a mistake. Wait, the exterior angle is equal to the sum of the two non - adjacent interior angles. Let's re - define: For a triangle, an exterior angle is adjacent to an interior angle (they form a linear pair) and is outside the triangle. So, for the top right vertex: interior angle is $\angle 2$, so the exterior angle is $\angle 3$ (since $\angle 2+\angle 3 = 180^{\circ}$, linear pair, and $\angle 3$ is outside). For the bottom right vertex: interior angle is $\angle 6$, so the exterior angles would be $\angle 4$? Wait, no, $\angle 5$: Wait, $\angle 4$ and $\angle 5$: $\angle 6+\angle 4 = 180^{\circ}$ (linear pair), so $\angle 4$ is adjacent to $\angle 6$ (interior), so $\angle 4$ is exterior? Wait, no, $\angle 5$: $\angle 4$ and $\angle 5$ are vertical angles? Wait, maybe the correct exterior angles are $\angle 3$ and $\angle 5$? Wait, no, let's start over.

Wait, the triangle has three vertices. Let's label the triangle: let the left vertex be $A$ (angle 1), top right vertex be $B$ (angles 2,3), bottom right vertex be $C$ (angles 6,4,5). At vertex $B$: interior angle is $\angle 2$, so exterior angle is $\angle 3$ (since it's adjacent to $\angle 2$ and outside the triangle). At vertex $C$: interior angle is $\angle 6$, so the exterior angles would be the angles adjacent to $\angle 6$ and outside. $\angle 4$ and $\angle 5$: $\angle 6+\angle 4 = 180^{\circ}$ (linear pair), so $\angle 4$ is adjacent to $\angle 6$, but is $\angle 4$ outside? Wait, the triangle is on the left side of the line with angles 4,5,6. So $\angle 5$ is outside the triangle (since the triangle is to the left of that line), and $\angle 4$ is also outside? Wait, no, maybe I messed up. Wait, the problem is to check which are exterior angles. Let's list the angles:

- $\angle 1$: interior (part of the triangle)
- $\angle 2$: interior (part of the triangle)
- $\angle 3$: exterior (adjacent to $\angle 2$, linear pair, outside)
- $\angle 4$: Wait, no, $\angle 4$ is adjacent to $\angle 6$, but is it outside? Wait, the triangle is between the left side (angle 1), top side (angle 2), and bottom side (angle 6). So the line at the bottom right is extended, so $\angle 5$ is outside (since it's on the opposite side of $\angle 6$ from the triangle), and $\angle 3$ is outside (on the opposite side of $\angle 2$ from the triangle). Wait, maybe the correct exterior angles are $\angle 3$ and $\angle 5$? But the options include $\angle 3$, $\angle 4$, $\angle 5$? Wait, let's check the linear pairs. At vertex $B$: $\angle 2+\angle 3 = 180^{\circ}$, so $\angle 3$ is exterior. At vertex $C$: $\angle 6+\angle 4 = 180^{\circ}$, but $\angle 4$ is on the same side as the extension? Wait, no, the triangle is to the left of the line with $\angle 6$, $\angle 4$, $\angle 5$. So $\angle 5$ is vertical to $\angle 4$? No, $\angle 4$ and $\angle 5$ are supplementary? Wait, no, $\angle 4$ and $\angle 5$ are vertical angles? No, they are adjacent? Wait, maybe the diagram is a triangle with a horizontal line at the top (with $\angle 2$ and $\angle 3$) and a slanted line at the bottom (with $\angle 6$, $\angle 4$, $\angle 5$) and a left side (with $\angle 1$). So, exterior angles: $\angle 3$ (top exterior), $\angle 5$ (bottom exterior), and maybe $\angle 4$? Wait, no, let's use the definition: an exterior angle is formed by one side of the triangle and the extension of another side. So for the top side (side $B$ to the right), extending it gives $\angle 3$ (exterior). For the bottom side (side $C$ to the right), extending it gives $\angle 5$ (exterior). Wait, but the options have $\angle 3$, $\angle 4$, $\angle 5$. Wait, maybe $\angle 4$ is also an exterior angle? No, $\angle 4$ is adjacent to $\angle 6$ (interior), but if we consider the triangle, the interior angle at $C$ is $\angle 6$, so the exterior angle should be the one outside, so $\angle 5$ (since $\angle 6$ and $\angle 5$? No, $\angle 6$ and $\angle 4$ are supplementary. Wait, I think I made a mistake. Let's look at the options again. The options are $\angle 1$, $\angle 5$, $\angle 3$, $\angle 6$, $\angle 4$, $\angle 2$.

So, $\angle 1$: interior, no. $\angle 2$: interior, no. $\angle 3$: exterior (linear pair with $\angle 2$, outside triangle), yes. $\angle 4$: Wait, if we consider the side of the triangle at $C$, and extend it, then $\angle 4$ is adjacent to $\angle 6$, but is it outside? Wait, maybe the diagram has the triangle with $\angle 2$ as an interior right angle, $\angle 1$ as an acute angle, $\angle 6$ as an acute angle. Then, the line at the top is extended to form $\angle 3$ (exterior), and the line at the bottom is extended to form $\angle 4$ and $\angle 5$. So, $\angle 3$ (exterior, linear pair with $\angle 2$), $\angle 4$ (exterior? No, $\angle 4$ is adjacent to $\angle 6$, but if the triangle is on the left, then $\angle 4$ is on the right side of the extension, so maybe $\angle 5$ is exterior. Wait, maybe the correct exterior angles are $\angle 3$ and $\angle 5$ and $\angle 4$? No, 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 each vertex, there is one exterior angle (but actually, at each vertex, there are two vertical exterior angles, but in the context of a triangle, we usually consider one per vertex). Wait, at vertex $B$ (top right), the two sides are the top side and the right side (with $\angle 2$). Extending the top side gives $\angle 3$ (exterior). At vertex $C$ (bottom right), the two sides are the bottom side and the right side (with $\angle 6$). Extending the bottom side gives $\angle 5$ (exterior) and $\angle 4$ (which is vertical to $\angle 5$? No, $\angle 4$ and $\angle 5$ are supplementary? Wait, no, $\angle 4$ and $\angle 5$ are vertical angles? No, they are adjacent angles forming a linear pair with $\angle 6$? Wait, $\angle 6+\angle 4 = 180^{\circ}$, so $\angle 4$ is a linear pair with $\angle 6$, but is it outside the triangle? If the triangle is to the left of $\angle 6$, then $\angle 4$ is to the right of the extension, so it's outside. Wait, maybe I was wrong earlier. So, $\angle 3$ (exterior, linear pair with $\angle 2$), $\angle 4$ (exterior, linear pair with $\angle 6$), $\angle 5$ (exterior, vertical to $\angle 4$? No, $\angle 5$ is also a linear pair? Wait, no, $\angle 4$ and $\angle 5$ are vertical angles, so they are equal. So, the exterior angles are $\angle 3$, $\angle 4$, $\angle 5$? But let's check the options. The options are $\angle 1$, $\angle 5$, $\angle 3$, $\angle 6$, $\angle 4$, $\angle 2$. So, eliminating the interior angles ($\angle 1$, $\angle 2$, $\angle 6$), the exterior angles are $\angle 3$, $\angle 4$, $\angle 5$. Wait, but maybe the diagram is different. Wait, the user's diagram: triangle with left angle 1, top right angle 2 (and 3 outside), bottom right angle 6 (and 4,5 outside). So, $\angle 3$: outside (linear pair with $\angle 2$), $\angle 4$: outside (linear pair with $\angle 6$), $\angle 5$: outside (vertical to $\angle 4$? No, $\angle 4$ and $\angle 5$ are supplementary? Wait, no, $\angle 4$ and $\angle 5$ are vertical angles, so they are equal. So, the exterior angles are $\angle 3$, $\angle 4$, $\angle 5$. But let's confirm with the definition. An exterior angle is formed by one side of the triangle and the extension of another side. So, for the top side (side of the triangle), extending it gives $\angle 3$ (exterior). For the bottom side (side of the triangle), extending it gives $\angle 5$ (exterior) and $\angle 4$ (which is also exterior, as it's on the other side of the extension? Wait, maybe the answer is $\angle 3$, $\angle 4$, $\angle 5$. But looking at the options, the checkboxes are for $\angle 1$, $\angle 5$, $\angle 3$, $\angle 6$, $\angle 4$, $\angle 2$. So, the exterior angles are $\angle 3$, $\angle 4$, $\angle 5$.

Answer:

$\angle 3$, $\angle 4$, $\angle 5$