Question

1. if m∠2 = 98°, m∠3 = 23° and m∠8 = 70°, find each measure.

a. m∠1=
b. m∠4=
c. m∠5=
d. m∠6=
e. m∠7=
f. m∠9=
g. m∠10=

Explanation:

Step1: Find $\angle1$

$\angle1$ and $\angle2$ are supplementary (linear - pair). So $m\angle1 = 180^{\circ}-m\angle2$. Given $m\angle2 = 98^{\circ}$, then $m\angle1=180 - 98=82^{\circ}$.

Step2: Find $\angle4$

In the triangle with $\angle2$ and $\angle3$, using the angle - sum property of a triangle ($m\angle2 + m\angle3+m\angle4 = 180^{\circ}$). Given $m\angle2 = 98^{\circ}$ and $m\angle3 = 23^{\circ}$, then $m\angle4=180-(98 + 23)=59^{\circ}$.

Step3: Find $\angle5$

$\angle4$ and $\angle5$ are supplementary (linear - pair). So $m\angle5 = 180^{\circ}-m\angle4$. Then $m\angle5=180 - 59 = 121^{\circ}$.

Step4: Find $\angle6$

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

Step5: Find $\angle7$

In the triangle with $\angle6$ and $\angle8$, using the angle - sum property of a triangle ($m\angle6 + m\angle7+m\angle8 = 180^{\circ}$). Substitute $m\angle6 = 110^{\circ}$ and $m\angle8 = 70^{\circ}$, then $m\angle7=180-(110 + 70)=0^{\circ}$ (this is wrong, assume there is a non - zero angle at $\angle7$ and we use another approach. Let's consider the large triangle formed by $\angle3,\angle4,\angle7,\angle8$). The sum of angles in a triangle is $180^{\circ}$. Let's find $\angle7$ from the fact that the sum of angles around a point is $360^{\circ}$. First, find the third angle in the small triangle with $\angle2$ and $\angle3$ which is $180-(98 + 23)=59^{\circ}$. Then, considering the angles related to $\angle7$, we know that $\angle7$ and the non - named angle in the small triangle with $\angle2$ and $\angle3$ and $\angle8$ are related. In the large triangle formed by the outer lines, we know that the sum of angles in a triangle is $180^{\circ}$. Let's use the fact that $\angle7$ and $\angle8$ and the angle adjacent to $\angle4$ (in the large triangle) sum to $180^{\circ}$. The angle adjacent to $\angle4$ in the large triangle is $180 - \angle4=121^{\circ}$. So $m\angle7=180-(121 + 70)= - 11^{\circ}$ (wrong, there is a mis - understanding of the figure, let's start over).
In the triangle with $\angle2$ and $\angle3$, the third angle is $180-(98 + 23)=59^{\circ}$.
Since $\angle7$ and the non - named angle in the $\angle2,\angle3$ triangle and $\angle8$ are part of a larger geometric relationship. We know that in the triangle formed by the outer lines, if we consider the angles around the intersection points.
The correct way:
In the triangle with $\angle2$ and $\angle3$, the third angle $x = 180-(98 + 23)=59^{\circ}$.
$\angle7$ and $\angle8$ and the angle adjacent to $\angle4$ in the large - scale triangle relationship.
We know that $\angle7=180-(59 + 70)=51^{\circ}$

Step6: Find $\angle9$

In the large triangle formed by the outer lines, we know that one angle is $\angle3 = 23^{\circ}$, another is $\angle8 = 70^{\circ}$, and $\angle9$ is the third angle. So $m\angle9=180-(23 + 70)=87^{\circ}$.

Step7: Find $\angle10$

$\angle9$ and $\angle10$ are supplementary (linear - pair). So $m\angle10 = 180^{\circ}-m\angle9$. Then $m\angle10=180 - 87 = 93^{\circ}$.

Answer:

a. $m\angle1 = 82^{\circ}$
b. $m\angle4 = 59^{\circ}$
c. $m\angle5 = 121^{\circ}$
d. $m\angle6 = 110^{\circ}$
e. $m\angle7 = 51^{\circ}$
f. $m\angle9 = 87^{\circ}$
g. $m\angle10 = 93^{\circ}$