Writing geometric proofs
A geometric proof is a logical argument showing that a statement must be true. Organising statements and reasons carefully makes even long arguments clear.
Structure of a proof
A proof starts from the given information and ends with the statement to prove. Each step is a statement supported by a reason: given information, a definition, a postulate, a property, or a previously proved theorem. Drawing and labelling a diagram first, and marking the given facts on it, is the best way to plan.
Two-column proofs
A two-column proof lists statements on the left and reasons on the right. For example, given that angle 1 and angle 2 are a linear pair and angle 1 measures 40 degrees: m angle 1 + m angle 2 = 180 (linear pair postulate); 40 + m angle 2 = 180 (substitution); m angle 2 = 140 (subtraction property of equality).
Paragraph and flow proofs
A paragraph proof writes the same reasoning in sentences. A flowchart proof uses boxes and arrows to show how statements lead to one another. All formats must justify every step; the format is a matter of clarity, not correctness.
Proving vertical angles congruent
Given lines crossing to form vertical angles 1 and 3, with angle 2 between them. Angles 1 and 2 form a linear pair, so they are supplementary; angles 2 and 3 also form a linear pair, so they are supplementary. Since m angle 1 = 180 - m angle 2 and m angle 3 = 180 - m angle 2, by substitution m angle 1 = m angle 3.
Proving the triangle angle sum
Draw a line through one vertex of a triangle parallel to the opposite side. The three angles at that vertex lie along a straight line, so they sum to 180 degrees. The two outer ones are alternate interior angles to the triangle's other two angles, so they are congruent to them. Therefore the triangle's three angles sum to 180 degrees.
Indirect proof
An indirect proof, or proof by contradiction, assumes the opposite of what you want to prove and shows this leads to something impossible. To prove a triangle has at most one right angle: suppose it had two; their sum would already be 180 degrees, leaving 0 degrees for the third angle, which is impossible.
Tips for proofs
Work backwards from the goal: ask what you would need to know to reach it. Look for congruent triangles, parallel lines, and angle pairs in the diagram. Never assume something because it looks true in a drawing; diagrams can mislead. Each reason must be a definition, postulate, property, or proven theorem.
Worked example
Given that M is the midpoint of segment AB and AM = 3x + 2 and MB = 5x - 6. By the definition of midpoint, AM = MB, so 3x + 2 = 5x - 6 (substitution). Then 8 = 2x (subtraction property) and x = 4 (division property). So AM = 14, MB = 14, and AB = 28 (segment addition).
Key takeaways
A proof moves from given facts to a conclusion, justifying every statement with a definition, postulate, property, or proven theorem. Two-column, paragraph, and flowchart proofs present the same logic in different formats. Vertical angles are congruent because both are supplementary to the same angle. A triangle's angles sum to 180 degrees, proved with a parallel line and alternate interior angles. Indirect proof assumes the opposite and reaches a contradiction. Work backwards from the goal, and never trust a diagram's appearance alone.
Flashcard
Logical argument that a statement is true
Check your understanding
In an indirect proof, you begin by: