1. **Problem Statement:** Compare the measures of angles $m\angle U$ and $m\angle E$ using the Hinge Theorem. 2. **Hinge Theorem:** If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. 3. **Converse of Hinge Theorem:** If two sides of a triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the second, then the included angle in the first triangle is greater than the included angle in the second triangle. 4. **Step-by-step analysis:** - For each pair of angles or sides given, identify the two triangles involved. - Check if two sides of one triangle are congruent to two sides of the other. - Compare the included angles or third sides accordingly. - Use $>$, $<$, or $=$ based on the Hinge Theorem or its converse. 5. **Examples:** - For $m\angle 1$ and $m\angle 2$ where two sides are equal but $m\angle 1 > m\angle 2$, then the side opposite $m\angle 1$ is longer. - If the third side opposite $m\angle 1$ is longer than that opposite $m\angle 2$, then $m\angle 1 > m\angle 2$. 6. **Applying to given problems:** - $m\angle U$ and $m\angle E$: If the two sides around $\angle U$ are congruent to those around $\angle E$ and $m\angle U > m\angle E$, then the side opposite $\angle U$ is longer. - For $m\angle 1$ and $m\angle 2$, compare sides opposite these angles using the theorem. - For segments like $MS$ and $LS$, compare lengths based on the included angles. - Repeat this logic for all pairs: $KD$ vs $CP$, $XB$ vs $ZB$, $HJ$ vs $KP$, $FH$ vs $GE$, $KP$ vs $KG$, $TP$ vs $AG$. 7. **Summary:** - Use the Hinge Theorem to relate angle sizes to opposite side lengths. - Use the converse to relate side lengths to included angle sizes. **Final answers depend on the specific measurements given in the diagrams, but the relationships are:** - If $m\angle 1 > m\angle 2$, then side opposite $m\angle 1 >$ side opposite $m\angle 2$. - If side opposite $m\angle 1 >$ side opposite $m\angle 2$, then $m\angle 1 > m\angle 2$. This logic applies to all the listed comparisons.