1. **State the problem:** Given an isosceles trapezoid ROMA, prove that the diagonals RM and AO are congruent. 2. **Recall definitions and properties:** - An isosceles trapezoid has one pair of parallel sides and the non-parallel sides (legs) are congruent. - The diagonals of an isosceles trapezoid are congruent. 3. **Given:** Isosceles trapezoid ROMA. 4. **To prove:** $\overline{RM} \cong \overline{AO}$. 5. **Proof steps:** **Step 1:** Given. **Step 2:** $\overline{OR} \cong \overline{MA}$ because the legs of the isosceles trapezoid are congruent. **Step 3:** $\angle ROM \cong \angle AMO$ because these are base angles opposite the congruent legs. **Step 4:** $\overline{OM} \cong \overline{MO}$ trivially, as they are the same segment. **Step 5:** By the SAS (Side-Angle-Side) Congruence Postulate, triangles $\triangle ROM$ and $\triangle AMO$ are congruent. **Step 6:** Therefore, corresponding parts of congruent triangles are congruent, so $\overline{RM} \cong \overline{AO}$. This completes the proof that the diagonals of an isosceles trapezoid are congruent.