The perpendicular bisector of side AB of ∆ABC intersects side AB at point D and BC at point E. If m∠CAB = 82° and m∠C = 68°, find m∠CAE.

Solution:In Δ ABC , perpendicular bisector of side AB of ∆ABC intersects side AB at point D and BC at point E. Also, m∠CAB = 82° and m∠C = 68°.Join A E.In Δ ABC∠A+∠B+∠C=180°→→→Angle sum property of triangle.82°+∠B+68°=180°∠B= 180°-150°∠B=30°In Δ A DE and Δ B DEAD=B D→→DE is a perpendicular bisector.∠ADE=∠BDE=90°DE is common.Δ A DE ≅ Δ E DB→→[S AS]∠DEB=∠D A E→→[CPCT]----(1)In Δ DBE∠EDB + ∠DBE+∠BED=180°→→Angle sum property of Triangle.90° +30°+∠BED=180°∠BED=180°-120°∠BED=60°So, ∠BAE=∠BED=60°------[using (1)]∠CAE=∠CAB - ∠BAE     = 82°-60°     = 22°