Find the side AB of trapezoid ABCD if the angles ABC and BCD are 30 ° and 120 °, respectively, and CD is 25 cm.

ABCD is a trapezoid.

∠ABC = 30 °.

∠BCD = 120 °.

CD = 25 cm.

AB -?

1. Since the base of the trapezoid BC and DA are parallel, the sum of the angles at the sides of the trapezoid is 180 °.

∠ABC + ∠BAC = 180 °.

∠BАC = 180 ° – ∠ABC.

∠BАC = 180 ° – 30 ° = 150 °.

∠BCD + ∠CDA = 180 °.

∠CDA = 180 ° – ∠BCD.

∠CDA = 180 ° – 120 ° = 60 °.

2. Lower from the peaks A and C heights to the bases of the trapezoid AM and CH.

Consider a right-angled triangle ΔCНD, the side of the trapezoid CD is the hypotenuse of a right-angled triangle ΔCНD.

sin∠CDA = CH / CD.

CH = CD * sin∠CDA.

3. Consider a right-angled triangle ΔADM, the lateral side of the trapezoid AB is the hypotenuse of the right-angled triangle ΔABM.

AM = CH.

sin∠ABS = CH / AB.

AB = CH / sin∠ABS = CD * sin∠CDA / sin∠ABS.

AB = 25 * sin∠60 ° / sin30 ° = 25 * √3 * 2/2 = 25 * √3 cm.

Answer: AB = 25 * √3 cm.