Question

1. if m∠9 = 97° and m∠12 = 114°, find each measure.

a. m∠1=
b. m∠2=
c. m∠3=
d. m∠4=
e. m∠5=
f. m∠6=
g. m∠7=
h. m∠8=
i. m∠10=
j. m∠11=
k. m∠13=
l. m∠14=
m. m∠15=
n. m∠16=

Explanation:

Step1: Identify vertical - angle relationships

Vertical angles are equal. $\angle1$ and $\angle9$ are vertical angles. Since $m\angle9 = 97^{\circ}$, then $m\angle1=97^{\circ}$.

Step2: Identify linear - pair relationships

$\angle1$ and $\angle2$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. So $m\angle2 = 180^{\circ}-m\angle1=180 - 97=83^{\circ}$.

Step3: Use vertical - angle relationships again

$\angle2$ and $\angle3$ are vertical angles, so $m\angle3=m\angle2 = 83^{\circ}$.

Step4: Use linear - pair and vertical - angle relationships

$\angle3$ and $\angle4$ form a linear - pair. So $m\angle4=180 - m\angle3=180 - 83 = 97^{\circ}$.

Step5: Identify corresponding - angle relationships (assuming parallel lines or other geometric properties)

If we assume appropriate parallel - line relationships (not explicitly stated but based on common geometric set - ups), $\angle5$ and $\angle9$ are corresponding angles. So $m\angle5=m\angle9 = 97^{\circ}$.

Step6: Use vertical - angle and linear - pair relationships for other angles

$\angle6$ and $\angle2$ are corresponding angles, so $m\angle6=m\angle2 = 83^{\circ}$.
$\angle7$ and $\angle3$ are corresponding angles, so $m\angle7=m\angle3 = 83^{\circ}$.
$\angle8$ and $\angle4$ are corresponding angles, so $m\angle8=m\angle4 = 97^{\circ}$.
$\angle10$ and $\angle6$ are vertical angles, so $m\angle10=m\angle6 = 83^{\circ}$.
$\angle11$ and $\angle7$ are vertical angles, so $m\angle11=m\angle7 = 83^{\circ}$.
$\angle12$ and $\angle8$ are vertical angles, so $m\angle12=m\angle8 = 97^{\circ}$.
$\angle13$ and $\angle9$ are vertical angles, so $m\angle13=m\angle9 = 97^{\circ}$.
$\angle14$ and $\angle10$ are vertical angles, so $m\angle14=m\angle10 = 83^{\circ}$.
$\angle15$ and $\angle11$ are vertical angles, so $m\angle15=m\angle11 = 83^{\circ}$.
$\angle16$ and $\angle12$ are vertical angles, so $m\angle16=m\angle12 = 97^{\circ}$.

Answer:

a. $m\angle1 = 97^{\circ}$
b. $m\angle2 = 83^{\circ}$
c. $m\angle3 = 83^{\circ}$
d. $m\angle4 = 97^{\circ}$
e. $m\angle5 = 97^{\circ}$
f. $m\angle6 = 83^{\circ}$
g. $m\angle7 = 83^{\circ}$
h. $m\angle8 = 97^{\circ}$
i. $m\angle10 = 83^{\circ}$
j. $m\angle11 = 83^{\circ}$
k. $m\angle13 = 97^{\circ}$
l. $m\angle14 = 83^{\circ}$
m. $m\angle15 = 83^{\circ}$
n. $m\angle16 = 97^{\circ}$