4. if m∠3 = 54°, find each measure.\na. m∠1=\nb. m∠2=\nc. m∠4=\nd. m∠5=\ne. m∠6=\nf. m∠7=\ng. m∠8=\nh…

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle3$ and $\angle1$ are vertical angles. Since $m\angle3 = 54^{\circ}$, then $m\angle1=54^{\circ}$.

Step2: Identify complementary - angle relationship

$\angle1$ and $\angle2$ are complementary (because they form a right - angle). So $m\angle2 = 90^{\circ}-m\angle1$. Substituting $m\angle1 = 54^{\circ}$, we get $m\angle2=90 - 54=36^{\circ}$.

Step3: Identify vertical - angle relationship for $\angle4$

$\angle4$ and $\angle2$ are vertical angles, so $m\angle4 = m\angle2=36^{\circ}$.

Step4: Identify vertical - angle relationship for $\angle5$

$\angle5$ and $\angle3$ are vertical angles, so $m\angle5 = m\angle3=54^{\circ}$.

Step5: Identify supplementary - angle relationship for $\angle6$

$\angle5$ and $\angle6$ are supplementary (a linear pair). So $m\angle6 = 180^{\circ}-m\angle5$. Substituting $m\angle5 = 54^{\circ}$, we get $m\angle6 = 180 - 54=126^{\circ}$.

Step6: Identify vertical - angle relationship for $\angle7$

$\angle7$ and $\angle6$ are vertical angles, so $m\angle7 = m\angle6=126^{\circ}$.

Step7: Identify vertical - angle relationship for $\angle8$

$\angle8$ and $\angle5$ are vertical angles, so $m\angle8 = m\angle5=54^{\circ}$.

Step8: Identify vertical - angle relationship for $\angle9$

$\angle9$ and $\angle4$ are vertical angles, so $m\angle9 = m\angle4=36^{\circ}$.

Step9: Identify vertical - angle relationship for $\angle10$

$\angle10$ and $\angle9$ are vertical angles, so $m\angle10 = m\angle9=36^{\circ}$.

Step10: Identify vertical - angle relationship for $\angle11$

$\angle11$ and $\angle8$ are vertical angles, so $m\angle11 = m\angle8=54^{\circ}$.

Step11: Identify vertical - angle relationship for $\angle12$

$\angle12$ and $\angle11$ are vertical angles, so $m\angle12 = m\angle11=54^{\circ}$.

Step12: Identify vertical - angle relationship for $\angle13$

$\angle13$ and $\angle10$ are vertical angles, so $m\angle13 = m\angle10=36^{\circ}$.

Step13: Identify vertical - angle relationship for $\angle14$

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

Answer:

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