1. **State the problem:** We need to find the size of angle $k$ in a quadrilateral-like shape where two sides are parallel and two segments are equal in length. 2. **Identify given information:** - The left and right slanted sides are parallel. - The bottom base and the diagonal have equal lengths. - The angle between the diagonal and the right slanted side at the bottom-right vertex is $83^\circ$. - The angle adjacent to this, between the bottom base and the diagonal, is $k$. 3. **Use properties of parallel lines and equal lengths:** Since the left and right slanted sides are parallel, alternate interior angles are equal. 4. **Analyze the triangle formed by the bottom base, diagonal, and right slanted side:** - The diagonal and bottom base are equal in length, so this triangle is isosceles. - The base angles opposite these equal sides are equal. 5. **Calculate angle $k$:** - The angle between the diagonal and right slanted side is $83^\circ$. - The two base angles of the isosceles triangle are equal, so let each be $k$. - Sum of angles in a triangle is $180^\circ$. $$k + k + 83 = 180$$ $$2k + 83 = 180$$ $$2k = 180 - 83$$ $$2k = 97$$ $$k = \frac{97}{2}$$ 6. **Simplify:** $$k = 48.5^\circ$$ **Final answer:** $$\boxed{48.5^\circ}$$