Given: KLMN is a parallelogram, LF is perpendicular to KN, LD is perpendicular to NM, angle FLD = 35. Find angles of KLMN

The question is four parts, It is required the angles of parallelogram KLMN

So, It is required ⇒ ∠K  , ∠L  , ∠M  and ∠N

See the attached figure which represents the explanation of the problem

Part (1): Find  ∠N:

in the shape FLDN

∵ LF ⊥ KN  ⇒⇒⇒ ∴ ∠LFN = 90°

∵ LD ⊥ NM ⇒⇒⇒ ∴ ∠LDN = 90°

∵ The sum of all angels of FLDN = 360°

∵ ∠FLD = 35°

∴ ∠FLD + ∠LDN + ∠DNF + ∠LFN = 360°

∴ 35° + 90° + ∠DNF + 90° = 360°

∴ ∠DNF = 360° - ( 90° + 90° + 35°) = 360° - 215° = 145°

∴ The measure of angle N = 145°

Part (2): Find  ∠M:

∵ KLMN is parallelogram , ∠N = 145°

∴ The angles N and M are supplementary angles ⇒ property of the parallelogram

∴ ∠M + ∠N = 180°

∴ ∠M = 180° - ∠N = 180° - 145° = 35°

∴ The measure of angle M = 35°

Part (3): Find  ∠L:

∵ KLMN is parallelogram , ∠M = 35°

∴ The angles N and M are supplementary angles ⇒ property of the parallelogram

∴ ∠M + ∠KLM = 180°

∴ ∠KLM = 180° - ∠M = 180° - 35° = 145°

OR ∠KLM = ∠N = 145° ⇒⇒⇒  property of the parallelogram

∴ The measure of angle KLM = 145°

Part (4): Find  ∠K:

∵ KLMN is parallelogram , ∠N = 35°

∴ The angles N and M are supplementary angles ⇒ property of the parallelogram

∴ ∠K + ∠N = 180°

∴ ∠K = 180° - ∠N = 180° - 145° = 35°

OR ∠K = ∠M = 35° ⇒⇒⇒  property of the parallelogram

∴ The measure of angle K = 35°