The following is a list of postulates, properties and definitions used in the proofs in this lesson:
- Addition postulate
- Vertical angles are congruent
- Reflexive property
- Substitution property
- All sides of a square are congruent
- Definition of angle bisector
- Definition of midpoint
- Proving triangles congruent using Side, Side, Side postulate, SSS
- Proving triangles congruent using Side, Angle, Side postulate, SAS
- Proving triangles congruent using Angle, Side, Angle postulate, ASA
Prepare Materials for Class
Print copies of each proof on a different color of paper to help
keep track of what goes together. Cut the proof page into two pieces as
shown, keeping the diagram, Given, and Prove together. Then cut the
statements and reasons into individual strips, separating the
statements from the reasons. Put the statement and reason strips and
the diagram in an envelope for each proof. Make enough of these proof
kits so that each student group will have a copy of each proof.
Divide students into groups and provide each group with a proof kit envelope for Proof 1.
Encourage students to clear a space on their table or cluster of
desks so they have room to spread out all the strips and the diagram
for the proof.
- Determine which strips are statements, and which are reasons.
- Determine which statements are given to start constructing the proof.
- Allow the groups to work through the proof by matching
statements and reasons until they have built a proof that the group
agrees on. On their own paper, each student draws the diagram and
writes out what is given and what is to be proven. Have each student
then write out the proof that their group has agreed on before putting
the strips back in the envelope. The group then produces 1 poster of
their complete proof. They should re-create the diagram, list the given
statements, and re-create the proof on the poster. If each group has
done the same proof, then put a few of the posters up at the front of
the room to compare.
When looking at the posters as a class, as the following questions:
- How accurate is the diagram? Are the segments and angles that look congruent in the diagram actually congruent?
- Are the corresponding parts labeled in the correct order?
- Did any of the groups put markings in the diagram?
- If there are different correct ways to order the statements in the proof, what are they?
- Which statements must come ahead of other statements? Why?
- For which statements or reasons can you predict the ordering?
- How does this predictable order help you write a proof?
After discussion of the first proof, the groups are ready to move on and tackle the other proofs. The proofs are as follows:
Be sure that students remember to write the proof they
generate with the strips before they put the strips back in the
- Proof 1: congr. triangles by ASA and the addition postulate
- Proof 2: congr. triangles by SAS and the addition postulate
- Proof 3: congr. triangles by SAS and the transitive property
- Proof 4: congr. triangles by SAS and the angle bisector
- Proof 5: congr. triangles by SSS and the properties of a square
- Proof 6: congr. triangles by ASA and the conguence of vertical angles
This lesson offers proof kits for straight-forward congruent triangle
proofs. This format can be useful with other proofs, including