In the figure, AD and CE are bisectors of angle A and angle C respectively. If angle ABC= 90°. Find ∠ADC+ ∠AEC.

We have

Angle sum property of triangles states that the sum of angles in a triangle is always 180°.

In ∆ABC,

By Angle Sum Property of triangles, we get

∠ABC + ∠BCA + ∠CAB = 180°

⇒ 90° + ∠BCA + ∠CAB = 180° [∵ it is given that, ∠ABC = 90°]

⇒ ∠BCA + ∠CAB = 180° - 90°

⇒ ∠BCA + ∠CAB = 90° …(i)

In ∆ADC,

By Angle Sum Property of triangles, we get

∠ADC + ∠ACD + ∠CAD = 180° …(ii)

In ∆ACE,

By Angle Sum Property of triangles, we get

∠ACE + ∠CAE + ∠AEC = 180° …(iii)

Adding equations (ii) and (iii), we get

(∠ADC + ∠ACD + ∠CAD) + (∠ACE + ∠EAC + ∠AEC) = 180° + 180°

[∵ AD and CE are the bisectors of angles A and C respectively]

⇒ ∠ADC + ∠ACE + (3 × 45°) = 360°

⇒ ∠ADC + ∠ACE + 135° = 360°

⇒ ∠ADC + ∠ACE = 360° - 135°

⇒ ∠ADC + ∠ACE = 225°

Thus, ∠ADC + ∠ACE = 225°