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 °.
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