∠11 and the angle that measures 59° are complementary, then:
m∠11 + 59° = 90°
m∠11 = 90° - 59°
m∠11 = 31°
Angles 9, 6, and the one that measures 59° form an isosceles triangle, this means that:
m∠6 = 59°
m∠6 + 59° + m∠9 = 180°
59° + 59° + m∠9 = 180°
m∠9 = 180° - 2*59°
m∠9 = 62°
∠9 and ∠8 are vertical angles, then:
m∠8 = m∠9
m∠8 = 62°
∠1, ∠3 and ∠8 form an isosceles triangle. ∠1 and ∠3 are congruent. This means that:
m∠1 = m∠3
and:
m∠1 + m∠3 + m∠8 = 180°
m∠1 + m∠1 + m∠8 = 180°
2m∠1 + 62° = 180°
2m∠1 = 180° - 62°
m∠1 = 118°/2
m∠1 = 59°
m∠3 = 59°
∠2, ∠11 and ∠10 form an isosceles triangle. ∠2 and ∠11 are congruent. This means that:
m∠2 = m∠11
m∠2 = 31°
m∠2 + m∠11 + m∠10 = 180°
31° + 31° + m∠10 = 180°
m∠10 = 180° - 2*31°
m∠10 = 118°
∠10 and ∠7 are vertical angles, then:
m∠7 = m∠10
m∠7 = 118°
Similarly to before, ∠4 and ∠5 are congruent with ∠2 and ∠11, then
m∠4 = m∠11
m∠4 = 31°
m∠5 = m∠11
m∠5 = 31°