1. **Problem Statement:** Given that AD and BC are equal in length and both are perpendicular to the line segment AB, prove that the line segment CD bisects AB. 2. **Understanding the problem:** We have a line segment AB with points A and B. From points A and B, perpendicular segments AD and BC are drawn such that AD = BC. 3. **Goal:** Show that the line segment CD intersects AB at its midpoint O, meaning AO = OB. 4. **Key properties and formulas:** - Since AD and BC are perpendicular to AB, angles at A and B are right angles. - AD = BC (given). - To prove CD bisects AB, we need to show AO = OB. 5. **Step-by-step proof:** - Let the length of AB be $x$. - Since AD and BC are perpendicular to AB and equal in length, triangles ADO and BCO are right triangles with AD = BC. - Consider triangles ADO and BCO: - AD = BC (given) - Angles at A and B are right angles (perpendicularity) - AO and BO are parts of AB - By the RHS (Right angle-Hypotenuse-Side) congruence criterion, triangles ADO and BCO are congruent. - Therefore, AO = BO. 6. **Conclusion:** Since AO = BO, point O is the midpoint of AB, so CD bisects AB. **Final answer:** The line segment CD bisects AB because triangles ADO and BCO are congruent, implying AO = OB.