Explanation:
1. ABCD is a rectangle and Q is the midpoint of CD.
Given
2. DQ ≅ QC
Since Q is the midpoint of CD, then by definition of midpoint, DQ ≅ QC.
3. AD ≅ BC
In a rectangle, opposite sides are congruent. AD and BC are opposite sides of the rectangle.
4. ∠D ≅ ∠C
The angles of a rectangle are right angles, and right angles are congruent.
5. ΔADQ ≅ ΔBCQ
Since we have two triangles with two pairs of congruent sides and one pair of congruent angles between them, the triangles are congruent by SAS.
6. AQ ≅ BQ
Corresponding parts of congruent triangles are congruent. Since the triangles are congruent, the hypotenuses AQ and BQ are also congruent.