7. The quadrilateral ABCD in the illustration, which has been enlarged below, has the following properties. Edge AB is equal to edge BC in length. Angles ADC and ABC are equal in measure and both equal 90 degrees. The edge BE is perpendicular to AD and measures 5 cm. Our task is to find the area of this quadrilateral. In order to do so, I will draw in several helping lines by extending the edge CD and making a right triangle here, and then extending the edge BE and drawing a line emanating from A parallel to BC. So, this line here is parallel to this edge here. So, now by alternate interior angles, we know that this angle here is the same as this angle here, and consequently, also the same as this angle here, and the last angle in this triangle that we have created is the same as this angle. So, since the edges AB and BC are equal, we have a side angle congruence showing us that the triangle here we have created that I'm shading in red is the same triangle as the one that's part of the figure, and what that tells us is that this is a edge of length 5, and what we have is a square that has the same area as the area of the quadrilateral, and that is exactly 25, so the answer is C.

quadrilateral

geometry

area

right triangle

congruence