Question

if m∠12 = 121° and m∠6 = 75°, find each measure. a. m∠1 = 75° b. m∠2 = c. m∠3 = d. m∠4 = e. m∠5 = f. m∠7 = g. m∠8 = h. m∠9 = i. m∠10 = j. m∠11 = k. m∠13 = l. m∠14 =

Explanation:

Step1: Vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle6$ are vertical angles. Given $m\angle6 = 75^{\circ}$, so $m\angle1=75^{\circ}$.

Step2: Linear - pair property

$\angle1$ and $\angle2$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. So $m\angle2 = 180^{\circ}-m\angle1=180 - 75=105^{\circ}$.

Step3: Vertical - angle property

$\angle2$ and $\angle3$ are vertical angles, so $m\angle3=m\angle2 = 105^{\circ}$.

Step4: Vertical - angle property

$\angle3$ and $\angle4$ are vertical angles, so $m\angle4=m\angle3 = 105^{\circ}$.

Step5: Vertical - angle property

$\angle5$ and $\angle6$ are vertical angles, so $m\angle5=m\angle6 = 75^{\circ}$.

Step6: Linear - pair with $\angle5$

$\angle5$ and $\angle7$ form a linear - pair. So $m\angle7=180^{\circ}-m\angle5 = 180 - 75=105^{\circ}$.

Step7: Vertical - angle property

$\angle7$ and $\angle8$ are vertical angles, so $m\angle8=m\angle7 = 105^{\circ}$.

Step8: Vertical - angle property

$\angle8$ and $\angle9$ are vertical angles, so $m\angle9=m\angle8 = 105^{\circ}$.

Step9: Given $\angle12 = 121^{\circ}$

$\angle11$ and $\angle12$ form a linear - pair. So $m\angle11=180^{\circ}-m\angle12=180 - 121 = 59^{\circ}$.

Step10: Vertical - angle property

$\angle11$ and $\angle10$ are vertical angles, so $m\angle10=m\angle11 = 59^{\circ}$.

Step11: Vertical - angle property

$\angle10$ and $\angle13$ are vertical angles, so $m\angle13=m\angle10 = 59^{\circ}$.

Step12: Vertical - angle property

$\angle13$ and $\angle14$ are vertical angles, so $m\angle14=m\angle13 = 59^{\circ}$.

Answer:

a. $m\angle1 = 75^{\circ}$
b. $m\angle2 = 105^{\circ}$
c. $m\angle3 = 105^{\circ}$
d. $m\angle4 = 105^{\circ}$
e. $m\angle5 = 75^{\circ}$
f. $m\angle7 = 105^{\circ}$
g. $m\angle8 = 105^{\circ}$
h. $m\angle9 = 105^{\circ}$
i. $m\angle10 = 59^{\circ}$
j. $m\angle11 = 59^{\circ}$
k. $m\angle13 = 59^{\circ}$
l. $m\angle14 = 59^{\circ}$