Question

lines $l_1$ and $l_2$ are parallel lines. determine the measures of $\angle1$ through $\angle12$
$m\angle7=\boxed{55}^\circ$
$m\angle1=\square^\circ$
$m\angle2=\boxed{63}^\circ$
$m\angle3=\square^\circ$
$m\angle4=\square^\circ$
$m\angle5=\square^\circ$
$m\angle6=\square^\circ$
$m\angle8=\square^\circ$
$m\angle9=\square^\circ$

Explanation:

Step1: Find ∠7 (given check)

$\angle 7$ and $62^\circ$ are supplementary:
$m\angle7 = 180^\circ - 62^\circ = 118^\circ$

Step2: Find ∠1 (corr. to ∠7)

$\angle1 \cong \angle7$ (corresponding angles):
$m\angle1 = 118^\circ$

Step3: Find ∠3 (supp. to ∠1)

$\angle1$ and $\angle3$ are supplementary:
$m\angle3 = 180^\circ - 118^\circ = 62^\circ$

Step4: Find ∠4 (vert. to ∠3)

$\angle4 \cong \angle3$ (vertical angles):
$m\angle4 = 62^\circ$

Step5: Find ∠5 (vert. to ∠1)

$\angle5 \cong \angle1$ (vertical angles):
$m\angle5 = 118^\circ$

Step6: Find ∠6 (given vertical pair)

$\angle6$ and $62^\circ$ are vertical angles:
$m\angle6 = 62^\circ$

Step7: Find ∠8 (supp. to ∠6)

$\angle6$ and $\angle8$ are supplementary:
$m\angle8 = 180^\circ - 62^\circ = 118^\circ$

Step8: Find ∠9 (vert. to ∠7)

$\angle9 \cong \angle7$ (vertical angles):
$m\angle9 = 118^\circ$

Step9: Confirm ∠2 (given vertical pair)

$\angle2$ and $63^\circ$ are vertical angles:
$m\angle2 = 63^\circ$

Answer:

$m\angle7 = 118^\circ$
$m\angle1 = 118^\circ$
$m\angle2 = 63^\circ$
$m\angle3 = 62^\circ$
$m\angle4 = 62^\circ$
$m\angle5 = 118^\circ$
$m\angle6 = 62^\circ$
$m\angle8 = 118^\circ$
$m\angle9 = 118^\circ$