Look at the examples of some common statements about geometric figures and the supporting reasons.
Based on the figure below and that M is the midpoint of AT, we can say that that ΔARM≅ΔTRM. Why can we say this?
Choose from the facts, definitions, postulates, or theorems below, to justify the statement.
RM≅RM  Interactive button. Assistance may be required. Check Your Answer Correct, a segment is congruent to itself. 
AM = ½AT  Interactive button. Assistance may be required. Check Your Answer This is a correct statement but does not help to justify the statement. 
AM≅MT  Interactive button. Assistance may be required. Check Your Answer Correct, this names a pair of congruent sides. 
SAS  Interactive button. Assistance may be required. Check Your Answer Incorrect because we haven't shown any congruent angles. 
AAS  Interactive button. Assistance may be required. Check Your Answer Incorrect because we haven't shown any congruent angles. 
SSS  Interactive button. Assistance may be required. Check Your Answer Yes, we have 3 pairs of congruent sides. 
Based on the information below, what polygon is being described?
Choose the figure which best fits the description.
Click on the links below to reveal some of the most commonly used justifications for geometry proofs.
Congruent Triangles Some common reasons used to justify statements about congruent triangles are shown. 

Quadrilaterals Some common reasons used to justify statements about quadrilaterals are shown. 

Parallel and Perpendicular Lines Some common reasons used to justify statements about parallel and perpendicular lines are shown. 

Similar Polygons Some common reasons used to justify statements about similar polygons are shown. 
Sometimes, logical reasoning is used just for fun. In this case, you can describe how you arrived at your conjecture or solution very informally.
A basket contains five apples. Do you know how to divide them among five kids so that each one has an apple and one apple stays in the basket? 