To determine what can be proved using a diagram, we need to consider the nature of each statement listed:
a. Alternate interior angles are congruent: This can generally be proved using a diagram if it represents a transversal intersecting two parallel lines. By showing that the two lines are parallel and using the properties of angles formed by a transversal, you can visually demonstrate that alternate interior angles are congruent.
b. The Pythagorean Theorem: This theorem can be graphically represented using a right triangle and squares constructed on each of its sides. A diagram can visually aid in understanding the relationship between the areas of the squares, providing a proof.
c. Two lines are parallel if they have the same slope: While this can be illustrated with a diagram showing lines on a coordinate plane, the proof of this statement relies on the definitions of slope and parallel lines rather than just a diagram.
d. The Triangle Sum Theorem: This theorem asserts that the sum of the angles in a triangle is 180 degrees. A diagram of a triangle can be used to illustrate this property and can be employed to show the reasoning behind it (e.g., by drawing a line parallel to one side and using alternate interior angles).
In conclusion, statements a, b, and d can be visually supported and proved using diagrams. Statement c may be illustrated but does not inherently require a diagram for proof.