Question

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

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 relationship

Vertical angles are equal. $\angle9$ and $\angle6$ are vertical angles, so $m\angle6 = 97^{\circ}$. $\angle12$ and $\angle3$ are vertical angles, so $m\angle3=114^{\circ}$.

Step2: Use linear - pair relationship

A linear - pair of angles is supplementary (sum to $180^{\circ}$). $\angle1$ and $\angle12$ form a linear pair, so $m\angle1 = 180 - 114=66^{\circ}$. $\angle2$ and $\angle3$ form a linear pair, so $m\angle2 = 180 - 114 = 66^{\circ}$. $\angle5$ and $\angle9$ form a linear pair, so $m\angle5=180 - 97 = 83^{\circ}$. $\angle4$ and $\angle3$ form a linear pair, so $m\angle4=180 - 114 = 66^{\circ}$.

Step3: Use corresponding - angle and alternate - angle relationships

$\angle7$ and $\angle3$ are corresponding angles, so $m\angle7 = 114^{\circ}$. $\angle8$ and $\angle12$ are corresponding angles, so $m\angle8 = 114^{\circ}$. $\angle10$ and $\angle6$ are corresponding angles, so $m\angle10 = 97^{\circ}$. $\angle11$ and $\angle7$ are corresponding angles, so $m\angle11 = 114^{\circ}$. $\angle13$ and $\angle5$ are corresponding angles, so $m\angle13 = 83^{\circ}$. $\angle14$ and $\angle6$ are corresponding angles, so $m\angle14 = 97^{\circ}$. $\angle15$ and $\angle7$ are corresponding angles, so $m\angle15 = 114^{\circ}$. $\angle16$ and $\angle8$ are corresponding angles, so $m\angle16 = 114^{\circ}$.

Answer:

a. $m\angle1 = 66^{\circ}$
b. $m\angle2 = 66^{\circ}$
c. $m\angle3 = 114^{\circ}$
d. $m\angle4 = 66^{\circ}$
e. $m\angle5 = 83^{\circ}$
f. $m\angle6 = 97^{\circ}$
g. $m\angle7 = 114^{\circ}$
h. $m\angle8 = 114^{\circ}$
i. $m\angle10 = 97^{\circ}$
j. $m\angle11 = 114^{\circ}$
k. $m\angle13 = 83^{\circ}$
l. $m\angle14 = 97^{\circ}$
m. $m\angle15 = 114^{\circ}$
n. $m\angle16 = 114^{\circ}$