In the right-angled triangle ABC (angle C of the straight line), the height CB is drawn.

In the right-angled triangle ABC (angle C of the straight line), the height CB is drawn. Prove that if the angle CBA = 30 degrees, then AB: AD = 4: 1.

Since in a right-angled triangle, by condition, the angle CBA = 30, the length of the leg lying opposite the angle 30 is equal to half the length of the hypotenuse AB.

AC = AB / 2.

The sum of the acute angles of a right-angled triangle is 90, then the angle BAC = (90 – CBA) = (90 – 30) = 60.

Then in the triangle ACD the angle ACD = (90 – CAD) = (90 – 60) = 30.

In a right-angled triangle ACD, leg AD lies opposite angle 30, then AD = AC / 2 = (AB / 2) / 2 = AB / 4.

AB / AD = 4/1, as required.