Question

use the diagram below for #10. 10. if (mangle10 = 77^{circ}), (mangle7 = 47^{circ}) and (mangle16 = 139^{circ}), find the measure of each missing angle. a. (mangle1=) f. (mangle6=) k. (mangle13=) b. (mangle2=) g. (mangle8=) l. (mangle14=) c. (mangle3=) h. (mangle9=) m. (mangle15=) d. (mangle4=) i. (mangle11=) n. (mangle17=) e. (mangle5=) j. (mangle12=) o. (mangle18=)

Explanation:

Step1: Vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle10$ are vertical angles, so $m\angle1 = m\angle10=77^{\circ}$.

Step2: Linear - pair property

$\angle1$ and $\angle2$ form a linear - pair. Since the sum of angles in a linear - pair is $180^{\circ}$, $m\angle2=180^{\circ}-m\angle1 = 180 - 77=103^{\circ}$.

Step3: Vertical - angle property

$\angle7$ and $\angle3$ are vertical angles, so $m\angle3 = m\angle7 = 47^{\circ}$.

Step4: Linear - pair property

$\angle3$ and $\angle4$ form a linear - pair. So $m\angle4=180^{\circ}-m\angle3=180 - 47 = 133^{\circ}$.

Step5: Linear - pair property with $\angle16$

$\angle16$ and $\angle15$ form a linear - pair. So $m\angle15=180^{\circ}-m\angle16=180 - 139 = 41^{\circ}$.

Step6: Vertical - angle property

$\angle15$ and $\angle5$ are vertical angles, so $m\angle5 = m\angle15 = 41^{\circ}$.

Step7: Angle - sum property of a triangle

In the triangle formed by angles $\angle5$, $\angle6$, $\angle7$, using the angle - sum property of a triangle ($m\angle5 + m\angle6+m\angle7 = 180^{\circ}$), we substitute the known values: $41^{\circ}+m\angle6 + 47^{\circ}=180^{\circ}$. Then $m\angle6=180-(41 + 47)=92^{\circ}$.

Step8: Vertical - angle property

$\angle6$ and $\angle8$ are vertical angles, so $m\angle8 = m\angle6 = 92^{\circ}$.

Step9: Vertical - angle property

$\angle1$ and $\angle9$ are vertical angles, so $m\angle9 = m\angle1 = 77^{\circ}$.

Step10: Linear - pair property

$\angle10$ and $\angle14$ are vertical angles, so $m\angle14 = m\angle10 = 77^{\circ}$.

Step11: Linear - pair property

$\angle14$ and $\angle13$ form a linear - pair. So $m\angle13=180^{\circ}-m\angle14=180 - 77 = 103^{\circ}$.

Step12: Vertical - angle property

$\angle11$ and $\angle7$ are vertical angles, so $m\angle11 = m\angle7 = 47^{\circ}$.

Step13: Linear - pair property

$\angle11$ and $\angle12$ form a linear - pair. So $m\angle12=180^{\circ}-m\angle11=180 - 47 = 133^{\circ}$.

Step14: Vertical - angle property

$\angle16$ and $\angle18$ are vertical angles, so $m\angle18 = m\angle16 = 139^{\circ}$.

Step15: Linear - pair property

$\angle18$ and $\angle17$ form a linear - pair. So $m\angle17=180^{\circ}-m\angle18=180 - 139 = 41^{\circ}$.

Answer:

a. $m\angle1 = 77^{\circ}$
b. $m\angle2 = 103^{\circ}$
c. $m\angle3 = 47^{\circ}$
d. $m\angle4 = 133^{\circ}$
e. $m\angle5 = 41^{\circ}$
f. $m\angle6 = 92^{\circ}$
g. $m\angle8 = 92^{\circ}$
h. $m\angle9 = 77^{\circ}$
i. $m\angle11 = 47^{\circ}$
j. $m\angle12 = 133^{\circ}$
k. $m\angle13 = 103^{\circ}$
l. $m\angle14 = 77^{\circ}$
m. $m\angle15 = 41^{\circ}$
n. $m\angle17 = 41^{\circ}$
o. $m\angle18 = 139^{\circ}$