To determine what can be proved using a diagram, we need to consider the properties and theorems related to the elements depicted in the diagram. The following points can be proved based on typical geometric diagrams:
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The Pythagorean Theorem: This can be proved if the diagram involves a right triangle, demonstrating that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
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The Triangle Angle Sum Theorem: This can be demonstrated in any triangle, showing that the sum of the interior angles of a triangle equals 180 degrees.
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Alternate Interior Angles are Congruent: This can be proved if the diagram shows two parallel lines cut by a transversal, indicating that the angles formed on opposite sides of the transversal are equal.
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Two Lines are Parallel if They Have the Same Slope: While this statement is true in coordinate geometry, proving it would require a diagram that includes lines with defined slopes.
Without seeing the specific diagram, we cannot definitively choose which theorem can be proved. However, if the diagram involves a triangle, the Triangle Angle Sum Theorem is often a likely candidate. If it depicts parallel lines and a transversal, then proving that alternate interior angles are congruent could be possible.
In summary, depending on the diagram, any of the provided options could potentially be proven: the Pythagorean Theorem, the Triangle Angle Sum Theorem, alternate interior angles being congruent, or the parallelicity of lines based on slopes.
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