In triangle ABC, angle A is 90 degrees, angle C is 15 degrees, and angle of DBC is 15

In triangle ABC, angle A is 90 degrees, angle C is 15 degrees, and angle of DBC is 15 degrees (D is on the AC side). Prove that ВD = 2AB, ВС is less than 4АВ

In a right-angled triangle ABC, we determine the value of the angle ABC.

Angle ABC = (180 – 90 – 15) = 75.

Then the AВD angle = (75 – 15) = 60.

In a right-angled triangle AВD, the angle ADВ = (18 – 90 – 60) = 30, then the leg AB is equal to half of the hypotenuse of the ВD, and ВD = 2 * AB, which was required to be proved.

The ВСD triangle is isosceles since its angles at the base of the ВС are equal to 150, then СD = ВD = 2 * AB. Then, in the ВСD triangle, the side ВD = СD = 2 * AB, and since BC cannot be equal to the sum of the lengths of ВD and СD, then BC <4 * AB, which was required to be proved.