The angles in a triangle are such that one angle is 78 degrees more than the smallest angle, while the third angle is 4 times as large as the smallest angle. Find the measures of all three angles.

Let the angles are x, y and z.

Let the smallest angle is x

one angle is 78 degrees more than the smallest angle.⇒ y = x + 78 → (1)

the third angle is 4 times as large as the smallest angle. ⇒ z = 4 x → (2)

we should know that the sum of the angles of the triangle = 180°

∴ x + y + z = 180 → (3)

substitute from (1) and (2) at (3)

∴ x + (x + 78) + (4x) = 180

∴ 6x = 180 - 78 = 102

∴ x = 102 : 6 = 17°

∴ y = 17 + 78 = 95°

∴ z = 4x = 68°

So, the measures of all three angles are 17°, 95° and 68°