s || t
u || v

answer bank:
\angle 9 = \angle 10
\angle 11 = \angle 15
\angle 1 = \angle 13
m\angle 2 + m\angle 5 = 180^{\circ}
\angle 9 = \angle 11
m\angle 8 + m\angle 7 = 180^{\circ}

Question

s || t
u || v

answer bank:
\angle 9 = \angle 10
\angle 11 = \angle 15
\angle 1 = \angle 13
m\angle 2 + m\angle 5 = 180^{\circ}
\angle 9 = \angle 11
m\angle 8 + m\angle 7 = 180^{\circ}

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Consecutive Interior Angles Theorem", "Converse of the Alternate Interior Angles Theorem", "Converse of the Alternate Exterior Angles Theorem", "Converse of the Consecutive Interior Angles Theorem" ], "new_concepts": [ "Converse of the Corresponding Angles Postulate" ], "current_concepts": [ "Converse of the Alternate Interior Angles Theorem", "Converse of the Alternate Exterior Angles Theorem", "Converse of the Consecutive Interior Angles Theorem", "Converse of the Corresponding Angles Postulate" ] } ## Identify the lines and transversals In the given geometric diagram, we have four lines: - Two lines that appear horizontal/diagonal: $s$ and $t$. - Two lines that intersect them: $u$ and $v$. We need to classify which given angle relationships prove either that lines $s$ and $t$ are parallel ($s \parallel t$) or that lines $u$ and $v$ are parallel ($u \parallel v$). ## Analyze each item in the Answer Bank Let's evaluate each statement from the Answer Bank: 1. $\angle 3 \cong \angle 10$: - $\angle 3$ is formed by the intersection of line $s$ and line $u$. - $\angle 10$ is formed by the intersection of line $t$ and line $v$. - Since these angles involve different pairs of lines (one is on $u$, the other on $v$; one is on $s$, the other on $t$), their congruence does not directly prove any two lines parallel. 2. $\angle 11 \cong \angle 15$: - Both angles lie along the transversal line $t$. - $\angle 11$ is at the intersection of line $t$ and line $v$. - $\angle 15$ is at the intersection of line $t$ and line $u$. - These are corresponding angles with respect to lines $u$ and $v$ cut by transversal $t$. - By the **Converse of the Corresponding Angles Postulate**, if $\angle 11 \cong \angle 15$, then $u \parallel v$. 3. $\angle 1 \cong \angle 13$: - Both angles lie along the transversal line $u$. - $\angle 1$ is at the intersection of line $u$ and line $s$. - $\angle 13$ is at the intersection of line $u$ and line $t$. - These are corresponding angles with respect to lines $s$ and $t$ cut by transversal $u$. - By the **Converse of the Corresponding Angles Postulate**, if $\angle 1 \cong \angle 13$, then $s \parallel t$. 4. $m\angle 2 + m\angle 5 = 180^\circ$: - Both angles lie along the transversal line $s$. - $\angle 2$ is at the intersection of line $s$ and line $u$. - $\angle 5$ is at the intersection of line $s$ and line $v$. - These are consecutive interior angles between lines $u$ and $v$ cut by transversal $s$. - Using the **Converse of the Consecutive Interior Angles Theorem**, if these angles are supplementary, then $u \parallel v$. 5. $\angle 9 \cong \angle 11$: - These are vertical angles at the intersection of line $t$ and line $v$. - Vertical angles are always congruent, so this statement does not prove any lines parallel. 6. $m\angle 8 + m\angle 7 = 180^\circ$: - These are linear pair angles at the intersection of line $s$ and line $v$. - They are always supplementary by the **Linear Pair Postulate**, so this does not prove any lines parallel. ## Group the valid statements - For $s \parallel t$: - $\angle 1 \cong \angle 13$ - For $u \parallel v$: - $\angle 11 \cong \angle 15$ - $m\angle 2 + m\angle 5 = 180^\circ$ ### For $s \parallel t$: - $\angle 1 \cong \angle 13$ ### For $u \parallel v$: - $\angle 11 \cong \angle 15$ - $m\angle 2 + m\angle 5 = 180^\circ$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Converse of the Corresponding Angles Postulate" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the lines and transversals

Analyze each item in the Answer Bank

Group the valid statements

Answer:

For \(s \parallel t\):

For \(u \parallel v\):