which angles are supplementary to each other? select all that apply. ∠4 and ∠3 ∠5 and ∠4 ∠1 and ∠3 ∠1 and ∠2

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Accepted Answer

To determine which angles are supplementary, we use the definition of supplementary angles: two angles are supplementary if their sum is $180^\circ$ (a straight angle). Let's analyze each pair:

### 1. Analyze $\angle 4$ and $\angle 3$
- From the diagram, $\angle 3$ and $\angle 4$ appear to form a right angle (since $\angle 1$ is a right angle, and vertical/adjacent angle relationships suggest $\angle 3 + \angle 4 = 90^\circ$, not $180^\circ$). Thus, they are complementary, not supplementary.

### 2. Analyze $\angle 5$ and $\angle 4$
- $\angle 5$ and $\angle 4$ are adjacent angles forming a straight line. By the linear pair postulate, their sum is $180^\circ$. So, $\angle 5 + \angle 4 = 180^\circ$, meaning they are supplementary.

### 3. Analyze $\angle 1$ and $\angle 3$
- $\angle 1$ is a right angle ($90^\circ$), and $\angle 3$ is part of the right angle formed with $\angle 2$ and $\angle 1$. However, $\angle 1 + \angle 3$ would be $90^\circ + \angle 3$, which is not $180^\circ$ (since $\angle 3$ is acute). Wait, correction: If $\angle 1$ is a right angle, and the lines are intersecting, maybe $\angle 1$ and $\angle 3$ are not supplementary. Wait, no—let's re-examine. Wait, $\angle 1$ is a right angle, so $\angle 1 = 90^\circ$. If $\angle 3$ is, say, $x$, then $\angle 1 + \angle 3 = 90^\circ + x$, which is not $180^\circ$ unless $x = 90^\circ$, which it isn't. So this pair is not supplementary.

### 4. Analyze $\angle 1$ and $\angle 2$
- $\angle 1$ is a right angle ($90^\circ$), and $\angle 2$ is adjacent to it, forming a right angle (since $\angle 1$ is $90^\circ$, $\angle 1 + \angle 2 = 90^\circ$, making them complementary). Wait, no—wait, the diagram shows $\angle 1$ as a right angle, and $\angle 2$, $\angle 3$ as angles in the right angle. But for $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is acute, so $90^\circ + \angle 2 < 180^\circ$. Wait, maybe I misread. Wait, actually, if $\angle 1$ is a right angle, and the lines are intersecting, let's consider the straight line. Wait, no—$\angle 1$ is a right angle, so the lines are perpendicular? Wait, maybe the diagram has $\angle 1$ as a right angle, so the horizontal and vertical lines are perpendicular. Then:

- $\angle 5$ and $\angle 4$: Linear pair, supplementary.
- $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is part of the right angle, so $\angle 1 + \angle 2 = 90^\circ$ (complementary). No. Wait, maybe $\angle 1$ is not a right angle? Wait, the diagram shows a right angle symbol at $\angle 1$, so $\angle 1 = 90^\circ$.

Wait, let's re-express:

- **$\angle 5$ and $\angle 4$**: Linear pair, sum to $180^\circ$ (supplementary).
- **$\angle 1$ and $\angle 2$**: Wait, $\angle 1$ is $90^\circ$, $\angle 2$ is adjacent, but if $\angle 1$ is part of a straight line? No, the horizontal and vertical lines are perpendicular, so $\angle 1$ is $90^\circ$, and $\angle 2 + \angle 3 = 90^\circ$. Then:

Wait, maybe the initial analysis was wrong. Let's list all pairs:

- $\angle 4$ and $\angle 3$: Sum to $90^\circ$ (complementary).
- $\angle 5$ and $\angle 4$: Linear pair, sum to $180^\circ$ (supplementary).
- $\angle 1$ and $\angle 3$: $\angle 1 = 90^\circ$, $\angle 3$ is acute, so sum is not $180^\circ$.
- $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is acute, sum is $90^\circ + \angle 2 < 180^\circ$ (complementary).

Wait, but maybe I missed another pair. Wait, $\angle 1$ and $\angle 5$? No, $\angle 1$ and $\angle 5$ are vertical angles? No, $\angle 1$ is a right angle, $\angle 5$ is adjacent to $\angle 4$. Wait, maybe the correct supplementary pairs are $\angle 5$ and $\angle 4$, and also $\angle 1$ and $\angle 2$ is wrong. Wait, no—let's check the diagram again.

Wait, the diagram has a vertical line, a horizontal line, and a slant line. $\angle 1$ is at the intersection of vertical and horizontal (right angle). $\angle 2$, $\angle 3$ are between the slant and vertical/horizontal. $\angle 4$ and $\angle 5$ are on the horizontal line (linear pair). So:

- $\angle 5$ and $\angle 4$: Supplementary (linear pair, $180^\circ$).
- $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is part of the right angle, so $90^\circ + \angle 2 = 90^\circ$? No, $\angle 1$ is $90^\circ$, $\angle 2 + \angle 3 = 90^\circ$ (since $\angle 1$ is $90^\circ$). So $\angle 1 + \angle 2 = 90^\circ + \angle 2$, which is not $180^\circ$.

Wait, maybe the correct answer is $\angle 5$ and $\angle 4$, and also $\angle 1$ and $\angle 2$ is incorrect. Wait, no—maybe I made a mistake. Let's recall: supplementary angles sum to $180^\circ$. So:

- $\angle 5$ and $\angle 4$: Linear pair, so supplementary.
- $\angle 1$ and $\angle 2$: If $\angle 1$ is $90^\circ$, and $\angle 2$ is $90^\circ$, but no—wait, the slant line creates angles. Wait, maybe $\angle 1$ is not a right angle? The diagram shows a right angle symbol, so $\angle 1 = 90^\circ$.

Wait, perhaps the other supplementary pair is $\angle 1$ and $\angle 2$ is wrong, and the correct pairs are $\angle 5$ and $\angle 4$, and maybe $\angle 1$ and $\angle 2$ is not. Wait, no—let's check the options:

Options:
- $\angle 4$ and $\angle 3$: Complementary (sum $90^\circ$) → No.
- $\angle 5$ and $\angle 4$: Supplementary (sum $180^\circ$) → Yes.
- $\angle 1$ and $\angle 3$: $\angle 1 = 90^\circ$, $\angle 3$ is acute → Sum not $180^\circ$ → No.
- $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is acute → Sum not $180^\circ$ → No. Wait, but maybe $\angle 1$ is not a right angle? If the right angle symbol is misinterpreted, and $\angle 1$ is part of a straight line, then $\angle 1 + \angle 2 = 180^\circ$. Wait, the diagram shows a right angle symbol at $\angle 1$, so $\angle 1 = 90^\circ$.

This is confusing. Let's proceed with the linear pair: $\angle 5$ and $\angle 4$ are adjacent and form a straight line, so they are supplementary. Additionally, if $\angle 1$ is a right angle, and the slant line creates $\angle 1 + \angle 2 + \angle 3 = 180^\circ$ (since it's a straight line), but $\angle 1 = 90^\circ$, so $\angle 2 + \angle 3 = 90^\circ$. So $\angle 1 + \angle 2 + \angle 3 = 180^\circ$, but $\angle 1 + \angle 2$ would be $90^\circ + \angle 2$, not $180^\circ$.

Thus, the only pair that is supplementary is $\angle 5$ and $\angle 4$, and possibly $\angle 1$ and $\angle 2$ if $\angle 1$ is not a right angle. But the diagram shows a right angle symbol, so $\angle 1 = 90^\circ$. Therefore, the correct answer is $\angle 5$ and $\angle 4$, and maybe $\angle 1$ and $\angle 2$ is incorrect. Wait, no—let's check again.

Wait, maybe the right angle symbol is at $\angle 1$, meaning $\angle 1 = 90^\circ$, and $\angle 2 + \angle 3 = 90^\circ$ (complementary). Then:

- $\angle 5$ and $\angle 4$: Linear pair, supplementary (sum $180^\circ$).
- $\angle 1$ and $\angle 2$: $\angle 1 = 90^\circ$, $\angle 2$ is acute, so sum is $90^\circ + \angle 2 < 180^\circ$ → Not supplementary.
- $\angle 4$ and $\angle 3$: Sum $90^\circ$ → Complementary.
- $\angle 1$ and $\angle 3$: Sum $90^\circ + \angle 3 < 180^\circ$ → Not supplementary.

Thus, the only supplementary pair is $\angle 5$ and $\angle 4$. Wait, but maybe the question has multiple correct answers. Let's re-express:

From the diagram, $\angle 5$ and $\angle 4$ are adjacent and form a straight line (linear pair), so they are supplementary. Additionally, if $\angle 1$ is a right angle, and the slant line creates $\angle 1 + \angle 2 + \angle 3 = 180^\circ$, but $\angle 1 = 90^\circ$, so $\angle 2 + \angle 3 = 90^\circ$. So $\angle 1 + \angle 2 + \angle 3 = 180^\circ$, but that's three angles.

Wait, maybe the other supplementary pair is $\angle 1$ and $\angle 2$ is wrong, and the correct pairs are $\angle 5$ and $\angle 4$, and $\angle 1$ and $\angle 2$ is incorrect. I think the intended answer is $\angle 5$ and $\angle 4$, and possibly $\angle 1$ and $\angle 2$ if the right angle symbol is a mistake. But based on the diagram, $\angle 5$ and $\angle 4$ are supplementary.

So the correct answer is:

$\boldsymbol{\angle 5}$ and $\boldsymbol{\angle 4}$ (and possibly others, but based on analysis, this is one). Wait, but let's check the options again:

Options:
- $\angle 4$ and $\angle 3$: No.
- $\angle 5$ and $\angle 4$: Yes.
- $\angle 1$ and $\angle 3$: No.
- $\angle 1$ and $\angle 2$: If $\angle 1$ is $90^\circ$ and $\angle 2$ is $90^\circ$, but no—wait, maybe the right angle symbol is at $\angle 1$, so $\angle 1 = 90^\circ$, and $\angle 2 + \angle 3 = 90^\circ$, so $\angle 1 + \angle 2 + \angle 3 = 180^\circ$, but that's three angles.

I think the intended answer is $\angle 5$ and $\angle 4$, and possibly $\angle 1$ and $\angle 2$ is incorrect. So the correct pair is $\angle 5$ and $\angle 4$.

# Answer:
$\angle 5$ and $\angle 4$ (and potentially others if misinterpreted, but based on linear pair, this is correct).

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QUESTION IMAGE

Question

Explanation:

1. Analyze \(\angle 4\) and \(\angle 3\)

2. Analyze \(\angle 5\) and \(\angle 4\)

3. Analyze \(\angle 1\) and \(\angle 3\)

4. Analyze \(\angle 1\) and \(\angle 2\)

Answer: