[ Isosceles, inscribed, and circumscribed trapezoids ]

[Sum of the angles of a triangle. Theorem of the exterior angle.]

The smaller base of an isosceles trapezoid is equal to the lateral side, and the diagonal is perpendicular to the lateral side. Find the angles of the trapezoid.
Prove that the diagonal is the bisector of the angle at the larger base of the trapezoid.

## Solution

Let $A D$ be the larger base of trapezoid $A B C D, A B=B C=C D$ and $\angle A C D=90^{\circ}$. Let $\angle A D C=\alpha$. Then $\angle C A D=90^{\circ}-\alpha$.

Since triangle $A B C$ is isosceles, $\angle B A C=\angle B C A=\angle C A D=90^{\circ}-\alpha$, and since $\angle B A D=\angle C D A$ $=\alpha$, then $90^{\circ}-\alpha+90^{\circ}-\alpha=\alpha$.

From this, $3 \alpha=180^{\circ}$. Therefore, $\alpha=60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-24.jpg?height=280&width=546&top_left_y=317&top_left_x=756)

## Answer

$60^{\circ}, 60^{\circ}, 120^{\circ}, 120^{\circ}$.