Question

1. if (mangle2 = 98^{circ}), (mangle3 = 23^{circ}) and (mangle8 = 70^{circ}), find each measure.

a. (mangle1=)
b. (mangle4=)
c. (mangle5=)
d. (mangle6=)
e. (mangle7=)
f. (mangle9=)
g. (mangle10=)

Explanation:

Step1: Find $\angle1$

In a triangle, the sum of interior angles is $180^{\circ}$. Consider the triangle with $\angle2 = 98^{\circ}$ and $\angle3=23^{\circ}$. Let the third - angle be $\angle1$. Then $m\angle1=180^{\circ}-(m\angle2 + m\angle3)=180-(98 + 23)=59^{\circ}$.

Step2: Find $\angle4$

$\angle4$ and $\angle2$ are vertical angles. Vertical angles are equal. So $m\angle4 = m\angle2=98^{\circ}$.

Step3: Find $\angle5$

$\angle5$ and $\angle3$ are vertical angles. Vertical angles are equal. So $m\angle5 = m\angle3=23^{\circ}$.

Step4: Find $\angle6$

$\angle6$ and $\angle8$ are supplementary (linear - pair). If $m\angle8 = 70^{\circ}$, then $m\angle6=180 - 70=110^{\circ}$.

Step5: Find $\angle7$

Consider the triangle with $\angle6 = 110^{\circ}$ and $\angle1 = 59^{\circ}$. Let the third - angle be $\angle7$. Then $m\angle7=180-(m\angle6 + m\angle1)=180-(110 + 59)=11^{\circ}$.

Step6: Find $\angle9$

$\angle9$ and $\angle7$ are vertical angles. So $m\angle9 = m\angle7=11^{\circ}$.

Step7: Find $\angle10$

$\angle10$ and $\angle8$ are vertical angles. So $m\angle10 = m\angle8=70^{\circ}$.

Answer:

a. $m\angle1 = 59^{\circ}$
b. $m\angle4 = 98^{\circ}$
c. $m\angle5 = 23^{\circ}$
d. $m\angle6 = 110^{\circ}$
e. $m\angle7 = 11^{\circ}$
f. $m\angle9 = 11^{\circ}$
g. $m\angle10 = 70^{\circ}$