# In a right-angled triangle ABC with a right angle C, the outer angle at the vertex A is 120, AC + AB = 18cm. Find AC and AB.

Outside angle at vertex A – angle BAD = 120 degrees. The angles BAD and BAC (angle A) are adjacent, together they make up an unfolded angle, which is 180 degrees. Then:

angle BAD + angle BAC = 180 degrees;

120 + angle BAC = 180;

angle BAC = 180 – 120;

angle BAC = 60 degrees.

In triangle ABC, angle C = 90 degrees (by condition), angle A = angle BAC = 60 degrees. Then, by the theorem on the sum of the angles of a triangle, the angle B is equal to:

angle A + angle B + angle C = 180 degrees;

60 + angle B + 90 = 180;

angle B = 180 – 150;

angle B = 30 degrees.

The AC leg lies opposite an angle of 30 degrees, therefore AC is equal to half the hypotenuse:

AC = AB / 2.

Let’s compose a system of equations:

AC = AB / 2;

AC + AB = 18.

Substitute the AC value from the first equation into the second:

AB / 2 + AB = 18;

(AB + 2AB) / 2 = 18;

3AB / 2 = 18;

3AB = 36;

AB = 36/3;

AB = 12 cm.

We substitute the obtained value AB into the first equation of the system of equations:

AC = AB / 2;

AC = 12/2;

AC = 6 cm.

Answer: AC = 6 cm, AB = 12 cm.