Question

find the measures of the unknown angles.find the measure of each angle. m∠jkf = m∠gkh = m∠hkj =

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle JKF$ and $\angle GKH$ are vertical angles, and $\angle HKJ$ and the $111^{\circ}$ angle are vertical angles. So $m\angle HKJ = 111^{\circ}$.

Step2: Use linear - pair property

$\angle JKF$ and $\angle HKJ$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. Let $m\angle JKF=y^{\circ}$ and $m\angle HKJ = 111^{\circ}$, then $y + 111=180$, so $y=m\angle JKF=180 - 111=69^{\circ}$. Since $m\angle JKF = m\angle GKH$, $m\angle GKH = 69^{\circ}$.

Answer:

$m\angle JKF = 69^{\circ}$
$m\angle GKH = 69^{\circ}$
$m\angle HKJ = 111^{\circ}$