Question

using the figure and the information given, determine which lines (if any) must be parallel. state the theorem/postulate to justify your answer. each problem is independent of the others. answer attempt 7 out of 10 if ∠15≅∠8 what lines are parallel? theorem: if ∠13≅∠9 what lines are parallel? theorem: if ∠6≅∠7 what lines are parallel? theorem: if ∠6 & ∠14 are supplementary what lines are parallel? theorem:

Explanation:

Step1: Identify corresponding - angles

Corresponding angles are equal when lines are parallel. $\angle15$ and $\angle8$ are corresponding angles formed by lines $c$ and $d$ and a transversal. If $\angle15\cong\angle8$, then by the Corresponding - Angles Postulate, $c\parallel d$.

Step2: Analyze $\angle13$ and $\angle9$

$\angle13$ and $\angle9$ are not in a position (such as corresponding, alternate - interior, or alternate - exterior) that would imply parallel lines. So, no lines are parallel when $\angle13\cong\angle9$.

Step3: Identify alternate - interior angles

$\angle6$ and $\angle7$ are alternate - interior angles formed by lines $a$ and $b$ and a transversal. If $\angle6\cong\angle7$, then by the Alternate - Interior Angles Theorem, $a\parallel b$.

Step4: Analyze supplementary angles

$\angle6$ and $\angle14$ are same - side interior angles formed by lines $c$ and $d$ and a transversal. If $\angle6$ and $\angle14$ are supplementary (i.e., $\angle6+\angle14 = 180^{\circ}$), then by the Same - Side Interior Angles Postulate, $c\parallel d$.

Answer:

If $\angle15\cong\angle8$, lines $c$ and $d$ are parallel. Theorem: Corresponding - Angles Postulate.
If $\angle13\cong\angle9$, no lines are parallel. Theorem: None (as angles are not in a relevant parallel - implying position).
If $\angle6\cong\angle7$, lines $a$ and $b$ are parallel. Theorem: Alternate - Interior Angles Theorem.
If $\angle6$ and $\angle14$ are supplementary, lines $c$ and $d$ are parallel. Theorem: Same - Side Interior Angles Postulate.