Question

find m∠1 and m∠2. justify each answer.
12.
13.
14.

Explanation:

Step1: Identify angle - pair relationship in 12

$\ell\parallel m$, $\angle1$ and the $75^{\circ}$ angle are corresponding angles.
Since corresponding angles are congruent when two parallel lines are cut by a transversal, $m\angle1 = 75^{\circ}$.
$\angle1$ and $\angle2$ are vertical angles.
Since vertical angles are congruent, $m\angle2=m\angle1 = 75^{\circ}$.

Step2: Identify angle - pair relationship in 13

$a\parallel b$, the $120^{\circ}$ angle and $\angle2$ are same - side interior angles.
Same - side interior angles are supplementary when two parallel lines are cut by a transversal. So $m\angle2=180 - 120=60^{\circ}$.
$\angle1$ and $\angle2$ are linear - pair angles.
Since linear - pair angles are supplementary, $m\angle1 = 180 - m\angle2=120^{\circ}$.

Step3: Identify angle - pair relationship in 14

In the quadrilateral $ABCD$, the sum of the interior angles of a quadrilateral is $360^{\circ}$.
We know two angles are $80^{\circ}$ and $70^{\circ}$. Let's assume the quadrilateral is a parallelogram (from the arrow - head markings on the sides indicating parallel sides).
$\angle1$ and the $80^{\circ}$ angle are supplementary (adjacent angles in a parallelogram are supplementary), so $m\angle1 = 180 - 80=100^{\circ}$.
$\angle2$ and the $70^{\circ}$ angle are supplementary (adjacent angles in a parallelogram are supplementary), so $m\angle2 = 180 - 70 = 110^{\circ}$.

Answer:

1. $m\angle1 = 75^{\circ}$, $m\angle2 = 75^{\circ}$
2. $m\angle1 = 120^{\circ}$, $m\angle2 = 60^{\circ}$
3. $m\angle1 = 100^{\circ}$, $m\angle2 = 110^{\circ}$