Question

find m∠krl.
m∠krl = \square°

Explanation:

Step1: Identify right angle

$\angle HRP$ is a right angle, so $m\angle HRP = 90^\circ$.

Step2: Analyze angle sum

We know $m\angle HRQ = 56^\circ$ and $m\angle HRQ + m\angle QRP + m\angle KRL = 90^\circ$? Wait, no, actually, since $KM$ and $HP$ are perpendicular, $\angle KRM = 90^\circ$. Also, $\angle HRQ$ and $\angle LRM$ are vertical angles? Wait, no, better: $\angle HRQ = 56^\circ$, and $\angle KRL + \angle HRQ = 90^\circ$? Wait, $HP$ is vertical, $KM$ is horizontal, so they are perpendicular. So $\angle KRP = 90^\circ$? Wait, no, $K$ is up, $P$ is down, so $KP$ is vertical, $HM$ is horizontal, so they are perpendicular, so $\angle KRM = 90^\circ$. Now, $\angle HRQ = 56^\circ$, and $\angle LRM$ is equal to $\angle HRQ$ (vertical angles), so $\angle LRM = 56^\circ$. Then $\angle KRL = 90^\circ - 56^\circ = 34^\circ$? Wait, no, wait: $KP$ is vertical, $HM$ is horizontal, so $\angle KRM = 90^\circ$. $\angle HRQ = 56^\circ$, so $\angle QRP = 90^\circ - 56^\circ = 34^\circ$? No, wait, $\angle KRL$: let's see, $R$ is the intersection. $K$ is up, $L$ is a line from $R$ to $L$, $Q$ is a line from $R$ to $Q$. $\angle HRQ = 56^\circ$, and $\angle KRL$ and $\angle HRQ$ are complementary because $KP$ and $HM$ are perpendicular (so $\angle KRH = 90^\circ$? Wait, $HM$ is horizontal, $KP$ is vertical, so they form right angles. So $\angle HRK = 90^\circ$? Wait, $H$ is left, $K$ is up, so $\angle HRK$ is a right angle? Wait, no, $HM$ is horizontal (left to right), $KP$ is vertical (up to down), so their intersection at $R$ forms four right angles. So $\angle HRK = 90^\circ$, $\angle KRM = 90^\circ$, $\angle MRP = 90^\circ$, $\angle PRH = 90^\circ$. Now, $\angle HRQ = 56^\circ$, so $\angle KRL$: since $\angle HRQ$ and $\angle LRM$ are vertical angles (equal), and $\angle KRL + \angle LRM = 90^\circ$ (because $\angle KRM = 90^\circ$), so $\angle KRL = 90^\circ - 56^\circ = 34^\circ$. Wait, no, maybe I messed up. Let's do it again. $\angle HRQ = 56^\circ$, and $\angle KRL$: since $KP$ is vertical and $HM$ is horizontal, the angle between $KP$ and $HM$ is $90^\circ$. So $\angle KRL + \angle HRQ = 90^\circ$? Wait, $\angle HRQ$ is $56^\circ$, so $\angle KRL = 90^\circ - 56^\circ = 34^\circ$. Yes, that makes sense. So the measure of $\angle KRL$ is $34^\circ$.

Step1: Recognize perpendicular lines

Lines \( KP \) (vertical) and \( HM \) (horizontal) are perpendicular, so \( \angle KRM = 90^\circ \) (right angle).

Step2: Use vertical angles or complementary angles

\( \angle HRQ = 56^\circ \). Since \( \angle KRL \) and \( \angle HRQ \) are complementary (they form a right angle with \( \angle KRM \)), we calculate:
\( m\angle KRL = 90^\circ - 56^\circ = 34^\circ \).

Answer:

\( 34 \)