We will reason to find the values of angles 1 through 6. To do so, we will use a key fact of triangles which is:
the sum of the angles of a triangle is 180°.
So, we will start by finding the value of angle 1. Note that angle 1 is in the triangle XYZ, whose other angles are 58° and 65°. Then, we have the following equation
[tex]\text{Angle 1 + 58\degree+65\degree=180\degree}[/tex]Since 58+65 = 123 then we have
[tex]\text{Angle 1 + 123 =180}[/tex]By subtracting 123 on both sides, we get that
[tex]\text{Angle 1 =180-123 = 57\degree}[/tex]So angle 1 measures 57°.
We can see that angles 1 and 2 are supplementary. That is, their measures add up to 180°. So, we have the following equation
[tex]\text{Angle 1 + Angle 2 =180}[/tex]Since angle 1 = 77° we have that
[tex]77\text{ + Angle 2 = 180}[/tex]which implies that angle 2 measures 123°. Using the same principle we can find the value of angle 5, since we have
[tex]\text{Angle 2 + Angle 5 = 180}[/tex]since angle 2 measures 123, we have that
[tex]123+\text{ Angle 5 = 180}[/tex]which implies that angle 5 measures 57°. Now, we see that angle 6 is in triangle VXW, so we can find the value of angle 6 as follows
[tex]\text{Angle 6 + Angle 5 + 67 = 180}[/tex]Then, since angle 5 measures 57° we have
[tex]\text{Angle 6 + 57\degree+67\degree=180\degree}[/tex]Since 57+67=124. Then , we have
[tex]\text{Angle 6 + 124 = 180 }[/tex]Subtracting 124 on both sides, we get
[tex]\text{Angle 6 = 180-124 = 56}[/tex]Now, we are missing to find the values of angles 3 and 4. To do so, first notice that
[tex]\text{Angle 2 + Angle 3 +Angle 4=180}[/tex]since these are the angles of triangle WXZ. We already know the measure of the angle 2 (123), so we have
[tex]\text{Angle 3 + Angle 4 =}180\text{ -123 = 57}[/tex]Unfortunately, the question doesn't give any more details on the triangles, so there are multiple solutions of values of angles 3 and 4 such that the equation holds