1. **Problem Statement:** Given an isosceles triangle with vertex $O$ and base points $E$ and $V$, where the two sides meeting at $O$ are congruent and the segment $EV$ is split into two equal parts at $E$, determine the postulate that makes the triangles congruent. 2. **Understanding the Setup:** The triangle has two sides $OE$ and $OV$ marked with two tick marks each, indicating $OE \cong OV$. 3. The segment $EV$ is divided into two equal parts at $E$, meaning $E$ is the midpoint of $EV$, so $EE_1 \cong EV_1$ (where $E_1$ and $V_1$ are points on $EV$). 4. **Postulate to Use:** The Side-Side-Side (SSS) postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. 5. **Applying SSS:** Since $OE \cong OV$, and $EE_1 \cong EV_1$, and $EV$ is common to both triangles, all three sides correspond and are congruent. 6. **Conclusion:** By the SSS postulate, the two triangles are congruent. **Final answer:** The triangles are congruent by the **SSS** postulate.