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

Question

Accepted Answer

# Explanation:
## Step1: Recall supplementary angles definition
Supplementary angles sum to $180^\circ$ (a straight line).

## Step2: Analyze $\angle5$ and $\angle4$
$\angle5$ and $\angle4$ form a linear pair (adjacent, on a straight line), so $\angle5 + \angle4 = 180^\circ$ (supplementary).

## Step3: Analyze $\angle5$ and $\angle1$
$\angle5$ and $\angle1$ (right angle? Wait, $\angle1$ and $\angle5$ with the right angle? Wait, from the diagram, $\angle1$ and $\angle5$ and the right angle? Wait, no—wait, $\angle5$ and $\angle1$: if $\angle1$ is a right angle? Wait, the red square is a right angle, so $\angle1 + \angle5 = 90^\circ + 90^\circ = 180^\circ$? Wait, no, maybe $\angle1$ is a right angle? Wait, the diagram has a right angle (red square) between $\angle1$ and $\angle5$? Wait, no, the red square is at $\angle1$ and $\angle5$? Wait, no, the red square is between $\angle1$ and the angle adjacent? Wait, maybe $\angle1$ and $\angle5$ are adjacent and form a linear pair? Wait, no, the red square is a right angle, so $\angle1$ is $90^\circ$, $\angle5$ is $90^\circ$, so $\angle5 + \angle1 = 180^\circ$ (supplementary).

## Step4: Analyze $\angle4$ and $\angle1$
$\angle4$ and $\angle1$: since $\angle4 + \angle5 = 180^\circ$ and $\angle5 + \angle1 = 180^\circ$, so $\angle4 = \angle1$? No, wait, $\angle4$ and $\angle1$: if $\angle1$ is $90^\circ$, and $\angle4$ is adjacent to $\angle5$ (which is $90^\circ$), so $\angle4 + \angle1 = 180^\circ$ (supplementary).

## Step5: Analyze $\angle1$ and $\angle2$
$\angle1$ is $90^\circ$ (right angle), $\angle2$ is part of the right angle? Wait, no—wait, $\angle1$ and $\angle2$: if $\angle1$ is $90^\circ$, and $\angle2$ is adjacent, but $\angle1 + \angle2$ would be less than $180^\circ$ (since $\angle1$ is a right angle and $\angle2$ is smaller). Wait, maybe the initial checks were wrong. Wait, no—wait, the red square is a right angle, so $\angle1$ is $90^\circ$, $\angle5$ is $90^\circ$. Then:

- $\angle5$ and $\angle4$: linear pair, sum to $180^\circ$ (supplementary) – correct.
- $\angle5$ and $\angle1$: $90^\circ + 90^\circ = 180^\circ$ – supplementary – correct.
- $\angle4$ and $\angle1$: $\angle4 + \angle1 = \angle4 + 90^\circ$, and since $\angle4 + \angle5 = 180^\circ$ ( $\angle5 = 90^\circ$ ), so $\angle4 = 90^\circ$, so $\angle4 + \angle1 = 180^\circ$ – supplementary – correct.
- $\angle1$ and $\angle2$: $\angle1 = 90^\circ$, $\angle2$ is part of the right angle? Wait, no, maybe the diagram has $\angle1$ as a right angle, and $\angle2$ is adjacent to $\angle1$ but $\angle1 + \angle2$ is not $180^\circ$. Wait, maybe the initial checks were incorrect. Wait, no—wait, the problem's options: let's re-express.

Wait, supplementary angles sum to $180^\circ$. Let's re-express:

- $\angle5$ and $\angle4$: linear pair, sum to $180^\circ$ – supplementary (correct).
- $\angle5$ and $\angle1$: if $\angle1$ is $90^\circ$ and $\angle5$ is $90^\circ$, sum to $180^\circ$ – supplementary (correct).
- $\angle4$ and $\angle1$: $\angle4 = 90^\circ$ (since $\angle5 = 90^\circ$ and $\angle4 + \angle5 = 180^\circ$), so $\angle4 + \angle1 = 180^\circ$ – supplementary (correct).
- $\angle1$ and $\angle2$: $\angle1 = 90^\circ$, $\angle2$ is smaller, so sum is less than $180^\circ$ – not supplementary (so initial check was wrong).

Wait, maybe the red square is a right angle, so $\angle1$ is $90^\circ$, $\angle5$ is $90^\circ$, $\angle4$ is $90^\circ$ (since $\angle4 + \angle5 = 180^\circ$), so $\angle4 = 90^\circ$. Then:

- $\angle5$ and $\angle4$: $90 + 90 = 180$ – supplementary (correct).
- $\angle5$ and $\angle1$: $90 + 90 = 180$ – supplementary (correct).
- $\angle4$ and $\angle1$: $90 + 90 = 180$ – supplementary (correct).
- $\angle1$ and $\angle2$: $90 + \angle2$ (where $\angle2$ is acute) – sum < 180 – not supplementary. So the incorrect one is $\angle1$ and $\angle2$. But the problem's initial checks had $\angle1$ and $\angle2$ checked, which is wrong. Wait, maybe the diagram is different. Wait, the diagram has a right angle (red square) at $\angle1$, so $\angle1$ is $90^\circ$, $\angle5$ is $90^\circ$, $\angle4$ is $90^\circ$, $\angle3$ and $\angle2$ are acute. So:

Supplementary pairs:

- $\angle5$ and $\angle4$ (linear pair, $180^\circ$) – correct.
- $\angle5$ and $\angle1$ ( $90 + 90 = 180$ ) – correct.
- $\angle4$ and $\angle1$ ( $90 + 90 = 180$ ) – correct.
- $\angle1$ and $\angle2$ ( $90 + \angle2 < 180$ ) – incorrect.

So the correct supplementary pairs are $\angle5$ and $\angle4$, $\angle5$ and $\angle1$, $\angle4$ and $\angle1$.

# Answer:
The supplementary angle pairs are $\boldsymbol{\angle5}$ and $\boldsymbol{\angle4}$, $\boldsymbol{\angle5}$ and $\boldsymbol{\angle1}$, $\boldsymbol{\angle4}$ and $\boldsymbol{\angle1}$ (the pair $\angle1$ and $\angle2$ is not supplementary as their sum is less than $180^\circ$).

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QUESTION IMAGE

Question

Explanation:

Step1: Recall supplementary angles definition

Step2: Analyze \(\angle5\) and \(\angle4\)

Step3: Analyze \(\angle5\) and \(\angle1\)

Step4: Analyze \(\angle4\) and \(\angle1\)

Step5: Analyze \(\angle1\) and \(\angle2\)

Answer: