What type of thinking is Mr. Gaines promoting with his challenge about geometric shapes?

Mr. Gaines is promoting deductive reasoning with his challenge about geometric shapes. Deductive reasoning involves starting with general principles or known facts and using logical steps to arrive at specific conclusions. In the context of geometric shapes, Mr. Gaines might be encouraging students to use established theorems, definitions, and properties of geometry to derive specific conclusions about shapes or solve problems related to them.

For example, if students know that the sum of the angles in a triangle is always 180 degrees, they can apply that knowledge to deduce the measures of angles in a particular triangle, given certain information. This type of logical reasoning demonstrates how students can progress from general rules to specific cases, fully engaging with the principles of geometry through deduction.