We are given a quadrilateral with two right angles, one angle of $130^\circ$, and two equal sides in the triangle portion of the quadrilateral. Our goal is to find the measure of angle $k$.

1. Problem Description

We are given a quadrilateral with two right angles, one angle of $130^\circ$, and two equal sides in the triangle portion of the quadrilateral. Our goal is to find the measure of angle $k$.

2. Solution Steps

First, let's analyze the quadrilateral. The sum of the interior angles of a quadrilateral is $360^\circ$. Let's call the top left angle of the quadrilateral $x$. The other angles are $90^\circ$, $90^\circ$, and $130^\circ$. So we have

$x + 90^\circ + 90^\circ + 130^\circ = 360^\circ$.

$x + 310^\circ = 360^\circ$.

$x = 360^\circ - 310^\circ$.

$x = 50^\circ$.

Now consider the triangle. We are given that two of its sides are equal, which means it is an isosceles triangle. Therefore, the angles opposite those sides are also equal. Let's call these angles $k$.

The sum of the angles in a triangle is $180^\circ$. We know that the top angle of the triangle is $x=50^\circ$.

Therefore, $k + k + 50^\circ = 180^\circ$.

$2k + 50^\circ = 180^\circ$.

$2k = 180^\circ - 50^\circ$.

$2k = 130^\circ$.

$k = \frac{130^\circ}{2}$.

$k = 65^\circ$.

3. Final Answer

$k = 65^\circ$