Question

date: hw: angle measures find the missing angle measure.

1. m∠1 =
2. m∠nml =
3. m∠7 =
4. m∠den = 32°. find m∠nef =
5. m∠aht =
6. m∠cde = 171° and m∠zde = 100°. find m∠cdz =
7. if m∠jkl = 78°, then m∠lkm =
8. written response: how do you recognize when you have vertical angles?
9. find the measure of each angle.

a) m∠1 = because
b) m∠2 = because
c) m∠3 = because

Explanation:

Step1: Recall angle - sum property of a full - circle

A full - circle has an angle measure of $360^{\circ}$. For the first problem, we know that the sum of the angles around a point is $360^{\circ}$. Let $m\angle1$ be the unknown angle. Then $m\angle1+112^{\circ}+216^{\circ}=360^{\circ}$.

Step2: Solve for $m\angle1$

$m\angle1=360^{\circ}-(112^{\circ} + 216^{\circ})=360^{\circ}-328^{\circ}=32^{\circ}$

Step3: For $\angle NML$

The sum of angles at a point on a straight - line is $180^{\circ}$. At point $M$, we have $m\angle NML + 102^{\circ}+59^{\circ}=180^{\circ}$. So $m\angle NML=180^{\circ}-(102^{\circ}+59^{\circ})=180^{\circ}-161^{\circ}=19^{\circ}$

Step4: For $\angle7$

Vertical angles are equal. $\angle7$ and the given $35^{\circ}$ angle are vertical angles, so $m\angle7 = 35^{\circ}$

Step5: For $\angle NEF$

Since $\angle DEN$ and $\angle NEF$ are supplementary (form a straight - line), and $m\angle DEN = 32^{\circ}$, then $m\angle NEF=180^{\circ}-32^{\circ}=148^{\circ}$

Step6: For $\angle AHT$

The angle is complementary to the $52^{\circ}$ angle. Complementary angles add up to $90^{\circ}$. So $m\angle AHT=90^{\circ}-52^{\circ}=38^{\circ}$

Step7: For $\angle CDZ$

We know that $m\angle CDE=m\angle CDZ + m\angle ZDE$. Given $m\angle CDE = 171^{\circ}$ and $m\angle ZDE = 100^{\circ}$, then $m\angle CDZ=m\angle CDE - m\angle ZDE=171^{\circ}-100^{\circ}=71^{\circ}$

Step8: For $\angle LKM$

If $m\angle JKL = 78^{\circ}$ and $\angle JKL$ and $\angle LKM$ are complementary (assuming $\angle JKM = 90^{\circ}$), then $m\angle LKM=90^{\circ}-78^{\circ}=12^{\circ}$

Step9: For the angles in the last problem

a) $m\angle1 = 100^{\circ}$ because vertical angles are equal. The angle opposite to the given $100^{\circ}$ angle is $\angle1$.
b) $m\angle2=80^{\circ}$ because $\angle2$ is supplementary to the $100^{\circ}$ angle. $m\angle2 = 180^{\circ}-100^{\circ}=80^{\circ}$
c) $m\angle3 = 100^{\circ}$ because vertical angles are equal. $\angle3$ is vertical to the $100^{\circ}$ angle.

Answer:

1. $m\angle1 = 32^{\circ}$
2. $m\angle NML=19^{\circ}$
3. $m\angle7 = 35^{\circ}$
4. $m\angle NEF=148^{\circ}$
5. $m\angle AHT=38^{\circ}$
6. $m\angle CDZ=71^{\circ}$
7. $m\angle LKM=12^{\circ}$
8. Vertical angles are formed when two lines intersect. They are opposite each other and have equal measures.
9. a) $m\angle1 = 100^{\circ}$, because vertical angles are equal.

b) $m\angle2 = 80^{\circ}$, because it is supplementary to the $100^{\circ}$ angle.
c) $m\angle3 = 100^{\circ}$, because vertical angles are equal.