# Find the angle ABC of an isosceles trapezoid ABCD, if the diagonal AC makes

**Find the angle ABC of an isosceles trapezoid ABCD, if the diagonal AC makes angles equal to 30 and 80 degrees with the base AD and the lateral side CD.**

Let’s consider different solutions.

Option 1.

In the triangle ACD, two angles are known by condition, we find the third:

∠ D = 180 ° – (∠ CAD + ∠ CDA) = 180 ° – (30 ° + 80 °) = 70 °.

The trapezoid is isosceles by condition, ∠ D = ∠ A = 70 °.

The angles adjacent to one side add up to 180 °, find the angle B:

∠ B = 180 ° – ∠ A = 110 °.

Option 2.

In the triangle ACD, two angles are known by condition, we find the third:

∠ D = 180 ° – (∠ CAD + ∠ CDA) = 180 ° – (30 ° + 80 °) = 70 °.

BC || AD, AC – secant.

∠CAD = ∠ BCA = 30 ° (lying crosswise).

∠ С = ∠ BCA + ∠ CDA = 30 ° + 80 ° = 110 °.

The trapezoid is isosceles by condition, ∠ С = ∠ В = 110 °.

Answer: the angle ABC is 110 °.