if $\angle 6 \cong \angle 10$ and $\angle 5 \cong \angle 7$, which describes all the lines that must be parallel?

- only lines $r$ and $s$ must be parallel.
- only lines $t$ and $u$ must be parallel.
- lines $r$ and $s$ and lines $t$ and $u$ must be parallel.
- neither lines $r$ and $s$ nor lines $t$ and $u$ must be parallel.

Question

if $\angle 6 \cong \angle 10$ and $\angle 5 \cong \angle 7$, which describes all the lines that must be parallel?

- only lines $r$ and $s$ must be parallel.
- only lines $t$ and $u$ must be parallel.
- lines $r$ and $s$ and lines $t$ and $u$ must be parallel.
- neither lines $r$ and $s$ nor lines $t$ and $u$ must be parallel.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Corresponding Angles", "Vertical Angles", "Parallel Line Proofs" ], "new_concepts": [], "current_concepts": [ "Corresponding Angles", "Vertical Angles", "Parallel Line Proofs" ] } ## Analyze the relationship between $\angle 6$ and $\angle 10$ $$ \begin{aligned} &\angle 6 \cong \angle 10 \ &\text{Transversal line: } s \ &\text{Lines intersected: } t \text{ and } u \ &\angle 6 \text{ and } \angle 10 \text{ are corresponding angles} \ &\Rightarrow t \parallel u \end{aligned} $$ ## Analyze the relationship between $\angle 5$ and $\angle 7$ $$ \begin{aligned} &\angle 5 \cong \angle 7 \ &\angle 5 \text{ and } \angle 7 \text{ are vertical angles formed by the intersection of lines } s \text{ and } t \ &\text{Vertical angles are always congruent regardless of parallel lines} \ &\Rightarrow \text{No parallel lines can be proven from } \angle 5 \cong \angle 7 \end{aligned} $$ ## Determine the final parallel lines $$ \begin{aligned} &\text{Only lines } t \text{ and } u \text{ must be parallel} \end{aligned} $$

Only lines r and s must be parallel.

Only lines t and u must be parallel.

Lines r and s and lines t and u must be parallel.

Neither lines r and s nor lines t and u must be parallel.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Line Proofs" ] }

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Question

Explanation:

Analyze the relationship between \(\angle 6\) and \(\angle 10\)

Analyze the relationship between \(\angle 5\) and \(\angle 7\)

Determine the final parallel lines

Answer: