Iven: klmn is a trapezoid m∠k = 90°, m∠n = 45° lk = lm = 10 find: kn, area of klmn

We have a right trapezoid since ∠K=90°. We know that KL=LM, then the triangle KLM is an isosceles and according to the Pythagoras theorem KM = 10√2. 

The sum of all quadrilaterals is 360° and using this fact we can say that ∠LMN=135° and since we know that ΔKLM is an isosceles( ∠LMK=45°), ∠KMN=90° and since ∠MNK=45°, then ∠MKN=45°. It means triangle KMN is a right-isosceles triangle and KM=MN=10√2. According to Pythagoras theorem KN=20

And the area of the trapezoid is