If ∠1 ≅ ∠2 ≅ ∠3, ∠4 ≅ ∠5, and m∠4 = m∠3 + 10°, What is m∠5?
(exterior angles)

m∠5  =60°

Step-by-step explanation:

As we can see, the shape is a hexagon, so that the total measures of its 6 interior angles are 720 degree.

We assume the interior angles of the hexagon are 1', 2', 3', 4', 5' and 6' as the image attached.

As all angles 1, 2, 3, 4, 5, 6 are exterior angles of the hexagon, so that we have 6 pairs of supplementary angles: 1 and 1'; 2 and 2'; 3 and 3'; 4 and 4'; 5 and 5'; 6 and 6'.

=> m∠1 + m∠1' = m∠2 + m∠2'=m∠3+m∠3' = m∠4 + m∠4' = m∠5 + m∠5' = m∠6 + m∠6' = 180 degree

=> m∠1 + m∠1' + m∠2 + m∠2' + m∠3+m∠3' + m∠4 + m∠4' + m∠5 + m∠5' + m∠6 + m∠6' = 180 × 6 = 1080

=> (m∠1 + m∠2 + m∠3 + m∠4 + m∠5+m∠6) + (m∠1' + m∠2' + m∠3' + m∠4' + m∠5' + m∠6') = 1080

=> (m∠1 + m∠2 + m∠3 + m∠4 + m∠5+m∠6) + 720 = 1080

=> (m∠1 + m∠2 + m∠3 + m∠4 + m∠5+m∠6)  = 360 (1)

As given:

+) m∠6 = 90

+) ∠1 ≅ ∠2 ≅ ∠3

+) ∠4 ≅ ∠5

Replace ∠1, ∠2 by ∠3; ∠5 by ∠4; ∠6 = 90 in the equation (1), we have

m∠3 + m∠3 + m∠3 + m∠4 + m∠4 + 90 = 360

=> 3m∠3 + 2m∠4 =270 (2)

As given: m∠4 = m∠3 + 10°

Replace in the equation (2), we have:

3m∠3 + 2 × (m∠3 + 10) =270

=> 3m∠3 + 2m∠3 + 20 = 270

=> 5m∠3 = 270 -20 = 250

=> m∠3 = 250/5 = 50°

=> m∠4 = m∠3 + 10° = 50 + 10 = 60°

As ∠4 ≅ ∠5 => m∠5 ≅ m ∠4 =60°

step-by-step explanation: