Question

look at this diagram:

diagram showing parallel lines ln and oq, and transversal rk intersecting them at p and m respectively, with angle qpm labeled as 56 degrees

if \\( \overleftrightarrow{ln} \\) and \\( \overleftrightarrow{oq} \\) are parallel lines and \\( m\angle qpm = 56^\circ \\), what is \\( m\angle lmk \\)?

blank box for answer°

submit

Explanation:

Step1: Identify alternate interior angles

Lines $\overleftrightarrow{LN} \parallel \overleftrightarrow{OQ}$, transversal $\overleftrightarrow{RK}$ creates alternate interior angles $\angle QPM$ and $\angle PMN$. So $m\angle PMN = m\angle QPM = 56^\circ$.

Step2: Identify supplementary angles

$\angle LMK$ and $\angle PMN$ are vertical angles to a linear pair? No, $\angle LMK$ and $\angle PMN$ are supplementary? Wait, no: $\angle LMK$ and $\angle NMK$ are linear pair, and $\angle PMN = \angle QPM = 56^\circ$. Wait, correction: $\angle QPM$ and $\angle LMK$ are same-side interior? No, alternate exterior? Wait, no: $\angle QPM$ and $\angle LMK$ are supplementary? No, wait: $\overleftrightarrow{LN} \parallel \overleftrightarrow{OQ}$, $\angle QPM$ and $\angle PMN$ are alternate interior, so $m\angle PMN=56^\circ$. Then $\angle LMK$ and $\angle PMN$ are vertical angles? No, $\angle LMK$ and $\angle PMO$? Wait no, $\angle LMK$ and $\angle QPM$ are same-side interior angles? Wait no, let's reorient:
$\overleftrightarrow{OQ}$ and $\overleftrightarrow{LN}$ are parallel. Transversal $\overleftrightarrow{RK}$ intersects $\overleftrightarrow{OQ}$ at $P$, $\overleftrightarrow{LN}$ at $M$.
$\angle QPM$ is on $\overleftrightarrow{OQ}$, below transversal. $\angle LMK$ is on $\overleftrightarrow{LN}$, above transversal, on the opposite side of transversal from $\angle QPM$. Wait, no: $\angle QPM$ and $\angle LMK$ are alternate exterior angles? No, alternate interior: $\angle QPM$ and $\angle PMN$ are alternate interior, so $m\angle PMN=56^\circ$. Then $\angle LMK$ and $\angle PMN$ are supplementary? No, $\angle LMN$ is a straight line, so $m\angle LMK + m\angle KMN = 180^\circ$, but $\angle KMN = \angle QPM$ (corresponding angles). Wait, yes! $\angle QPM$ and $\angle KMN$ are corresponding angles, so $m\angle KMN=56^\circ$. Then $\angle LMK + \angle KMN = 180^\circ$.

Step2: Calculate supplementary angle

$m\angle LMK = 180^\circ - m\angle KMN$
$m\angle LMK = 180^\circ - 56^\circ$

Answer:

$124^\circ$