# In the parallelogram ABCD, the perpendicular BK is drawn to the straight line AD, and point K lies on the side of AD.

**In the parallelogram ABCD, the perpendicular BK is drawn to the straight line AD, and point K lies on the side of AD. Find the sides and angles of the parallelogram if it is known that AK = 3cm, KD = 5cm, angle ABK = 30 degrees.**

The height, lowered from the top of the parallelogram to the opposite side, forms a right angle with the opposite side, in which the side AK, KB are the legs, AB is the hypotenuse. Since point K divides the side AD into two parts of the known lengths, we find the length of AD:

5 + 3 = 8 (cm).

In a parallelogram, two opposite sides are equal. Side AD is equal to the side BC, so BC = 8 cm.

Let’s find the side AB through the ratio of the opposite leg to the hypotenuse:

sin30 ° = 3 / AB;

½ = 3 / AB;

AB = 3: ½;

AB = 6 (cm);

AB = CD = 6 cm.

∠BAD = 180 ° – 30 ° – 90 ° = 60 °.

∠BAD = ∠BCD = 60 °.

∠ABC = ∠ADC = 30 ° + 90 ° = 120 °.

ANSWER: AB = 6 cm, BC = 8 cm, CD = 6 cm, AD = 8 cm, ∠ABC = 120 °, ∠BCA = 60 °, ∠CDA = 120 °, ∠DAB = 60 °.