Question

m∠1 = m∠2 = m∠3 = m∠4 = m∠5 = m∠6 = m∠7 =

Explanation:

Step1: Use vertical - angle property

$\angle2$ and the $115^{\circ}$ angle are vertical angles, so $m\angle2 = 115^{\circ}$.

Step2: Use linear - pair property

$\angle2$ and $\angle5$ form a linear pair. Since the sum of angles in a linear pair is $180^{\circ}$, $m\angle5=180 - 115=65^{\circ}$.

Step3: Use linear - pair property for $\angle4$

$\angle4$ and the $115^{\circ}$ angle form a linear pair. So $m\angle4 = 180 - 115=65^{\circ}$.

Step4: In the right - triangle with $\angle7$

In the right - triangle at the bottom, one angle is $90^{\circ}$ and another is $61^{\circ}$. Using the angle - sum property of a triangle ($m\angle A+m\angle B + m\angle C=180^{\circ}$), $m\angle7=180-(90 + 61)=29^{\circ}$.

Step5: Use angle - sum property in another triangle

In the triangle with $\angle3$, $\angle4$, and $\angle6$, we know $\angle4 = 65^{\circ}$ and $\angle7 = 29^{\circ}$. Since $\angle3$ and $\angle7$ are alternate interior angles (assuming parallel - like relationships based on the figure structure), $m\angle3 = 29^{\circ}$. Then, using the angle - sum property of a triangle ($m\angle3+m\angle4+m\angle6 = 180^{\circ}$), $m\angle6=180-(29 + 65)=86^{\circ}$.

Step6: In the large triangle with $\angle1$

In the large triangle, we know one angle is $90^{\circ}$ and $\angle6 = 86^{\circ}$. Using the angle - sum property of a triangle, $m\angle1=180-(90 + 86)=4^{\circ}$.

Answer:

$m\angle1 = 4^{\circ}$
$m\angle2 = 115^{\circ}$
$m\angle3 = 29^{\circ}$
$m\angle4 = 65^{\circ}$
$m\angle5 = 65^{\circ}$
$m\angle6 = 86^{\circ}$
$m\angle7 = 29^{\circ}$