In isosceles ∆DEK with base DK , EF is the angle bisector of ∠E, m∠DEF = 43°, and DK = 16cm. Find: KF, m∠DEK, m∠EFD.

Answer:Step-by-step explanation:The isosceles triangle, DEK with base DK is shown in the attached photo. Since DK is the base and triangle DEK is an isosceles triangle, the other two sides, DE and KE are equal. If EF is the angle bisector of ∠E, then line EF bisects angle DEK into 2 equal angles. So if m∠DEF = 43°, thenm∠KEF = 43°m∠DEK = m∠DEF + m∠KEF = 43° + 43° = 86°Also, bisector EF divides line DK into 2 equal parts. The means that Line DK = Line DF + Line KF and Line DF = Line KFIf Line DK = 16Line KF = Line DK/2 = 16/2 = 8 cmAlso if m∠DEK = 86°, then,m∠DEK + m∠EDF + m∠EKF = 180°This is because the sum of the angles in a triangle is 180Recallm∠EDF = m∠EKF This is because the triangle is an isosceles triangle and the base angles are equal. Therefore, m∠EDF = m∠EKF = (180 - 86)/2= 47°Also m∠EDF + m∠DEF + m∠EFD = 180°m∠EFD = 180° - (47° + 43°)m∠EFD = 90°