Question

in the diagram, given l || m, j ⊥ k, and m∠1 = 42, find all angle measures. answer attempt 4 out of 10 m∠1 = m∠4 = m∠2 = m∠3 = m∠5 = m∠8 = m∠6 = m∠7 = m∠9 = m∠12 = m∠10 = m∠11 = m∠13 = m∠16 = m∠14 = m∠15 = m∠17 = m∠20 = m∠18 = m∠19 =

Explanation:

Step1: Vertical - angle property

Vertical angles are equal. Since $\angle1$ and $\angle4$ are vertical angles, $m\angle1 = m\angle4=42^{\circ}$.

Step2: Supplementary - angle property

$\angle1$ and $\angle2$ are supplementary (a linear - pair), so $m\angle2=180 - 42=138^{\circ}$. And $\angle2$ and $\angle3$ are vertical angles, so $m\angle2 = m\angle3 = 138^{\circ}$.

Step3: Corresponding - angle property for parallel lines

Since $l\parallel m$, $\angle1$ and $\angle5$ are corresponding angles, so $m\angle5 = m\angle1=42^{\circ}$. And $\angle5$ and $\angle8$ are vertical angles, so $m\angle5 = m\angle8 = 42^{\circ}$.

Step4: Alternate - interior and vertical - angle properties

$\angle5$ and $\angle6$ are supplementary. So $m\angle6=180 - 42 = 138^{\circ}$. $\angle6$ and $\angle7$ are vertical angles, so $m\angle6 = m\angle7 = 138^{\circ}$.

Step5: Vertical - angle property for non - parallel line intersections

For the intersection of the non - parallel lines, if we consider the vertical - angle relationships, $\angle9$ and $\angle12$ are vertical angles. Also, if we assume the same angle - relationships as above for the new set of intersecting lines, and since there is no other information to change the pattern, if we consider the relative position of angles, $m\angle9 = m\angle12 = 42^{\circ}$.

Step6: Supplementary and vertical - angle properties for non - parallel lines

$\angle9$ and $\angle10$ are supplementary, so $m\angle10=180 - 42 = 138^{\circ}$. $\angle10$ and $\angle11$ are vertical angles, so $m\angle10 = m\angle11 = 138^{\circ}$.

Step7: Corresponding - angle property for the second set of parallel - like relationships

Assuming the same parallel - line angle relationships for the lower set of lines, $\angle13$ and $\angle9$ are corresponding angles (if we consider the parallel - like structure), so $m\angle13 = m\angle9 = 42^{\circ}$. $\angle13$ and $\angle16$ are vertical angles, so $m\angle13 = m\angle16 = 42^{\circ}$.

Step8: Supplementary and vertical - angle properties for the lower set of lines

$\angle13$ and $\angle14$ are supplementary, so $m\angle14=180 - 42 = 138^{\circ}$. $\angle14$ and $\angle15$ are vertical angles, so $m\angle14 = m\angle15 = 138^{\circ}$.

Step9: Corresponding - angle property for the last set of parallel - like relationships

$\angle17$ and $\angle13$ are corresponding angles (if we consider the parallel - like structure), so $m\angle17 = m\angle13 = 42^{\circ}$. $\angle17$ and $\angle20$ are vertical angles, so $m\angle17 = m\angle20 = 42^{\circ}$.

Step10: Supplementary and vertical - angle properties for the last set of lines

$\angle17$ and $\angle18$ are supplementary, so $m\angle18=180 - 42 = 138^{\circ}$. $\angle18$ and $\angle19$ are vertical angles, so $m\angle18 = m\angle19 = 138^{\circ}$.

Answer:

$m\angle1 = m\angle4 = 42^{\circ}$
$m\angle2 = m\angle3 = 138^{\circ}$
$m\angle5 = m\angle8 = 42^{\circ}$
$m\angle6 = m\angle7 = 138^{\circ}$
$m\angle9 = m\angle12 = 42^{\circ}$
$m\angle10 = m\angle11 = 138^{\circ}$
$m\angle13 = m\angle16 = 42^{\circ}$
$m\angle14 = m\angle15 = 138^{\circ}$
$m\angle17 = m\angle20 = 42^{\circ}$
$m\angle18 = m\angle19 = 138^{\circ}$