Step 1: The two wave types.
Earthquakes send out P waves and S waves. P waves can pass through both solids and liquids, but S waves can pass through solids only.
Step 2: Why S waves are the key test.
Since S waves need a solid to travel, finding them in a layer proves that layer is solid. They simply stop in any liquid.
Step 3: Apply it to the mantle.
Observations show that S waves do travel through the whole mantle. That can only happen if the mantle behaves as a solid.
Step 4: Compare with the P wave options.
P waves passing through, or being bent in the mantle, would happen whether the mantle were solid or liquid. So those facts do not by themselves prove a solid mantle.
Step 5: Final choice.
The observation that fits a solid mantle is that S waves do propagate through it.
\[ \boxed{\text{S waves propagate through the mantle}} \]