Question

1. if m∠3 = 54°, find each measure.

a. m∠1=
b. m∠2=
c. m∠4=
d. m∠5=
e. m∠6=
f. m∠7=
g. m∠8=
h. m∠9=
i. m∠10=
j. m∠11=
k. m∠12=
l. m∠13=
m. m∠14=

Explanation:

Step1: Vertical - angle property

Vertical angles are equal. Since $\angle3$ and $\angle6$ are vertical angles, if $m\angle3 = 54^{\circ}$, then $m\angle6=54^{\circ}$.

Step2: Complementary - angle property

$\angle3$ and $\angle4$ are complementary (assuming the right - angle relationship shown). Since the sum of complementary angles is $90^{\circ}$, if $m\angle3 = 54^{\circ}$, then $m\angle4=90^{\circ}-54^{\circ}=36^{\circ}$.

Step3: Supplementary - angle property

$\angle3$ and $\angle7$ are supplementary (a straight - line forms an angle of $180^{\circ}$). So $m\angle7 = 180^{\circ}-54^{\circ}=126^{\circ}$.

Step4: Vertical - angle property for other pairs

$\angle8$ and $\angle3$ are vertical angles, so $m\angle8 = 54^{\circ}$. $\angle9$ and $\angle7$ are vertical angles, so $m\angle9 = 126^{\circ}$. $\angle10$ and $\angle3$ are vertical angles, so $m\angle10 = 54^{\circ}$. $\angle11$ and $\angle4$ are vertical angles, so $m\angle11 = 36^{\circ}$.

Step5: Linear - pair and vertical - angle relationships for remaining angles

$\angle12$ and $\angle7$ are vertical angles, so $m\angle12 = 126^{\circ}$. $\angle13$ and $\angle4$ are vertical angles, so $m\angle13 = 36^{\circ}$. $\angle14$ and $\angle7$ are vertical angles, so $m\angle14 = 126^{\circ}$.

Step6: Right - angle property

$\angle1$ and $\angle4$ are vertical angles, so $m\angle1 = 36^{\circ}$. $\angle2$ is a right - angle, so $m\angle2 = 90^{\circ}$. $\angle5$ is a right - angle, so $m\angle5 = 90^{\circ}$.

Answer:

a. $m\angle1 = 36^{\circ}$
b. $m\angle2 = 90^{\circ}$
c. $m\angle4 = 36^{\circ}$
d. $m\angle5 = 90^{\circ}$
e. $m\angle6 = 54^{\circ}$
f. $m\angle7 = 126^{\circ}$
g. $m\angle8 = 54^{\circ}$
h. $m\angle9 = 126^{\circ}$
i. $m\angle10 = 54^{\circ}$
j. $m\angle11 = 36^{\circ}$
k. $m\angle12 = 126^{\circ}$
l. $m\angle13 = 36^{\circ}$
m. $m\angle14 = 126^{\circ}$