Geometry Proofs - CPCTC, Two-Column Proofs, FlowChart Proofs, Proof By Contradiction

In these lessons, we will learn learn geometry proofs - how to use CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction.

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Two Column Proofs
Geometry Lessons


CPCTC is an acronym for Corresponding Parts of Congruent Triangles are Congruent. CPCTC is commonly used at or near the end of a proof which asks the student to show that two angles or two sides are congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be congruent.

How to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC?

This video offers a look at two triangle proofs that involve the CPCTC theorem.

  1. Write the 3 congruent parts.
  2. Write the statement for triangle congruency. (SSS, SAS, HL, ASA, or AAS)
  3. Prove the statement.

How do we use CPCTC?

Two Column Proofs

Two column proofs are organized into statement and reason columns. Each statement must be justified in the reason column. Before beginning a two column proof, start by working backwards from the “prove” or “show” statement. The reason column will typically include “given”, vocabulary definitions, conjectures, and theorems.

How to organize a two column proof.

This video introduces the structure of 2 column proofs and works through three examples.

Practice drawing diagrams and completing two column proofs from word problems

Example 1:
Two circle intersect at two points. Prove that the segment joining the centers of the circles bisects the segment joining the points of intersection.

Example 2:
If each pair of opposite sides of a four sided figure are congruent, then the segments joining opposite vertices bisect each other.

Example 3:
If a point on the base of an isosceles triangle is equidistant from the midpoints of the legs, then that point is the midpoint of the base.

Flowchart Proofs

Flowchart proofs are organized with boxes and arrows; each “statement” is inside the box and each “reason” is underneath each box. Each statement in a proof allows another subsequent statement to be made. In flowchart proofs, this progression is shown through arrows. Flowchart proofs are useful because it allows the reader to see how each statement leads to the conclusion.

How to outline a flowchart proof?

Using flow charts to do proofs.

Using flowcharts in proofs for Geometry

How to create a flowchart proof?
Step 1: Understand why something is true by trying different ideas until you come up with an argument that leads to what you’re trying to show using conjectures, definitions, and properties that we’ve already learned.

Step 2: Put the idea into a paragraph or flowchart where every statement is followed by a reason that explains how to get from the previous statement to this new one.

Proof by Contradiction

How to write an Indirect Proof or Proof by Contradiction?

  1. Assume the negation of the “Prove” or “Conclusion”.
  2. Write a proof until you reach a contradiction of either the given info or a theorem, definition, or other known fact.
  3. Write a summary that states the assumption was false and the original “prove” is true.

CA Geometry: Proof by Contradiction

Theorem: A triangle has at most one obtuse angle.
Eduardo is proving the theorem above by contradiction. He began by assuming that in ⃤ABC, ∠A and ∠B are both obtuse. Which theorem will Eduardo us to reach a contradiction?

CA Geometry: More Proofs

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
Mathway Calculator Widget

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