Find the area of a triangle if the base is 2, the angles at the base are 30 and 45.

Let’s designate a triangle: ABC.
Angle B = 180 ° – (30 ° + 45 °) = 105 °.
Let’s define BC and AB.
AC / Sin B = BC / Sin A;
BC = AC * Sin A / Sin B;
BC = 2 * 0.7071 / 0.9659;
BC = 1.47 cm.
AC / Sin B = AB / Sin C;
AB = AC * Sin C / Sin B;
AB = 2 * 0.5 / 0.9659;
AB = 1.04 cm.
Let’s omit the height BO. Consider a triangle ABO.
Angle ABO = 180 ° – (90 ° + 45 °) = 45 °. Angle ABO = Angle BAO – isosceles triangle.
AB = AO = 1.04 cm.
Let’s define the area.
S = AC * BO / 2 = 2 * 1.04: 2 = 1.04 cm2.
Answer: the area of the triangle is 1.04 cm2.