lines l, m, and n lie in a plane and are cut by a transversal, t. ∠1 is supplementary to ∠2, and ∠2 is supplementary to ∠3. which lines, if any, are parallel? m and n what justifies your answer? linear pair postulate corresponding angles theorem same side interior angle theorem converse of same side interior angle theorem

Question

Accepted Answer

# Explanation:
## Step1: Recall Angle Relationships
Supplementary angles sum to $180^\circ$. Given $\angle1$ supplementary to $\angle2$, $\angle1 + \angle2 = 180^\circ$. Also, $\angle2$ supplementary to $\angle3$, so $\angle2 + \angle3 = 180^\circ$.
## Step2: Deduce $\angle1$ and $\angle3$ Relationship
From $\angle1 + \angle2 = 180^\circ$ and $\angle2 + \angle3 = 180^\circ$, subtract the two equations (or use transitivity) to get $\angle1 = \angle3$? Wait, no—wait, actually, since $\angle1$ and $\angle3$ are same - side interior angles? Wait, no, let's re - examine. Wait, lines $m$ and $n$ with transversal $t$, $\angle1$ and $\angle3$ (if we consider the position) — wait, actually, since $\angle1$ and $\angle2$ are supplementary, $\angle2$ and $\angle3$ are supplementary, so $\angle1=\angle3$? No, wait, supplementary to the same angle means $\angle1$ and $\angle3$ are equal? Wait, no, if $a + b=180$ and $c + b = 180$, then $a = c$. But in terms of parallel lines, the converse of the same - side interior angle theorem states that if same - side interior angles are supplementary, then the lines are parallel. Wait, let's correct: $\angle1$ and $\angle3$ — wait, actually, $\angle1$ and $\angle3$ (when considering lines $m$ and $n$ cut by $t$): since $\angle1$ and $\angle2$ are supplementary ($\angle1+\angle2 = 180^\circ$) and $\angle2$ and $\angle3$ are supplementary ($\angle2+\angle3 = 180^\circ$), then $\angle1=\angle3$? No, that's not right. Wait, $\angle1$ and $\angle2$ are adjacent supplementary (linear pair), and $\angle2$ and $\angle3$ are same - side interior angles? Wait, no, lines $m$ and $n$, transversal $t$: $\angle1$ and $\angle3$ — actually, the key is that $\angle1$ and $\angle3$ (if we look at the positions) — wait, the converse of the same - side interior angle theorem says that if two lines are cut by a transversal and the same - side interior angles are supplementary, then the lines are parallel. Wait, in this case, since $\angle1$ is supplementary to $\angle2$ and $\angle2$ is supplementary to $\angle3$, we can show that $\angle1$ and $\angle3$ (or the appropriate same - side interior angles) are supplementary? Wait, no, let's re - express:

Let's denote: $\angle1 + \angle2=180^\circ$ (given, supplementary) and $\angle2+\angle3 = 180^\circ$ (given, supplementary). By the transitive property of equality (or subtracting $\angle2$ from both equations), we get $\angle1=\angle3$? No, that's not. Wait, $\angle1 + \angle2=180$ implies $\angle1 = 180-\angle2$, and $\angle2+\angle3 = 180$ implies $\angle3=180 - \angle2$, so $\angle1=\angle3$. But that's not same - side interior. Wait, maybe I misidentified the angles. Wait, lines $m$ and $n$, transversal $t$: $\angle1$ and $\angle3$ — no, maybe $\angle1$ and the angle adjacent to $\angle3$. Wait, the correct reasoning is: Since $\angle1$ is supplementary to $\angle2$ ($\angle1+\angle2 = 180^\circ$) and $\angle2$ is supplementary to $\angle3$ ($\angle2+\angle3 = 180^\circ$), then $\angle1=\angle3$ (supplements of the same angle are equal). But actually, the converse of the same - side interior angle theorem: Wait, no, the same - side interior angle theorem says that if two parallel lines are cut by a transversal, then same - side interior angles are supplementary. The converse is that if same - side interior angles are supplementary, then the lines are parallel. Wait, in this case, $\angle1$ and $\angle3$ — no, maybe $\angle1$ and the angle that is supplementary to $\angle3$ related to line $m$ and $n$. Wait, the key is that since $\angle1$ and $\angle2$ are supplementary, and $\angle2$ and $\angle3$ are supplementary, we can conclude that $\angle1$ and $\angle3$ are equal (supplements of the same angle), but actually, the correct justification is the converse of the same - side interior angle theorem? Wait, no, let's think again.

Wait, lines $m$ and $n$ are cut by transversal $t$. $\angle1$ and $\angle3$ (or the appropriate same - side interior angles): Since $\angle1$ is supplementary to $\angle2$ ($\angle1+\angle2 = 180^\circ$) and $\angle2$ is supplementary to $\angle3$ ($\angle2+\angle3 = 180^\circ$), then $\angle1=\angle3$? No, that's not. Wait, $\angle1$ and $\angle2$ are adjacent (linear pair), so they are supplementary. $\angle2$ and $\angle3$ are same - side interior angles for lines $m$ and $n$ cut by $t$. Wait, if $\angle2$ and $\angle3$ are supplementary, then by the converse of the same - side interior angle theorem, lines $m$ and $n$ are parallel. Wait, yes! Because $\angle2$ and $\angle3$ are same - side interior angles (for lines $m$ and $n$ with transversal $t$) and they are supplementary (since $\angle2$ is supplementary to $\angle3$ as given: $\angle2+\angle3 = 180^\circ$). So the converse of the same - side interior angle theorem states that if two lines are cut by a transversal and the same - side interior angles are supplementary, then the lines are parallel. So that's the justification.

# Answer:
converse of same side interior angle theorem

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall Angle Relationships

Step2: Deduce \(\angle1\) and \(\angle3\) Relationship

Answer: