## Congruent Triangles Postulates

### Timothy Mark DillehayLee County School District

#### Description

Students enjoy this engaging activity in discovering three lettered postulates that prove triangles congruent. Students have a great insight to the workings and reasoning behind ASA, SAS, SSS, and AAS.

#### Objectives

Uses properties and relationships of geometric shapes to construct formal and informal proofs.

#### Materials

-Three blank typing pages for each student in the class.
-Ruler (1 for each student)
-Protractor (1 for each student)
-“Congruent Triangle – Student” sheet (1 for each student)
-“Congruent Triangle – Teacher” sheet (only 1 needed)

#### Preparations

1. Copy the “Congruent Triangles – Student” one per student.
2. Copy ONE “Congruent Triangles – Teacher”.
4. Have paper (preferred color typing paper), protractors, and rulers ready (one per student).

#### Procedures

2. Teacher says, ”We are going to imagine we are camping out. One of our tasks for the weekend is to build a campfire. While we are collecting fire logs we notice some geometry logic. Derrick (example) has collected three pieces of wood. One measuring 2 feet, another 3 feet, and another 3 feet. If you have also collected three logs, but all we know about your log is that one of them is 2 feet, what do we know?” Accept all answers, not giving a correct or incorrect answer, but mentioning that we will be investigating these thoughts or conjectures in the lab activity today.

3. Pass out three blanks sheets of typing paper, a ruler, and a protractor to each student.

4. Read number one in the packet with the students. Prompt students to compare their triangle with others by overlapping the two sheets of paper. The student who wrote darker, or with a pen should place their paper on the bottom. Mention that rotation and flipping the papers may be needed to find them exactly the same.

5. Have student volunteers answer the first question. If correct restate the answer to the group, if incorrect, probe the students to the correct answer.

6. Complete step 4 and 5 until you have reached the story portion. Mention that you will need to give more time on Number three. This is the hardest triangle of all to draw, since the measurements after drawing a 3-inch and 4 inch, may not leave you with five inches.

7. Have a volunteer read the following story out loud to the class. Instruct the student to retell the story (summarized) in the their own words.

“To show that AAA does not work in proving triangles congruent to one another listen to this story.
If you look at the front corner of your room it measures 90 degrees. If you are standing in the Gymnasium the front corner also measures 90 degrees. This is one “A,” a pair of angles.
If you continue this process you will end with having 4 pairs of congruent angles, “AAAA”. Although the Gymnasium and the classroom have “AAAA” the gym is much larger. So AAA may help in proving similar, but not to prove triangles congruent.

8. Have several students read what they have written for the summary.

9. Continue reading the practice portion of the lesson to the third page.
Hint: make clear that two triangles may be congruent, even if you have traveled clockwise about one, and counter clockwise about the other.

10. Give students a time limit to work on the third page of the notes.

12. Give students a time limit to work on the fourth page of the notes.

14. Collect rulers and protractors from each student.

15. Have each student staple his open exercise (drawing papers) to the back of his notes.

#### Assessments

Use completed “Congruent Triangles - Student” (associated file) to formatively assess the student’s ability in determining triangles congruent and for what reason. Acceptable work in all areas:
1. at least 80% of the notes completed;
2. 90% correct responses to which three lettered postulates prove pairs of triangles congruent;
3. 90% correct additional drawings to prove triangles congruent by the stated postulate.
4. participation in the opening exercise of creating and comparing triangles with classmates.

#### Extensions

Have student create and compare "Hypotenuse Leg" and "Leg Leg" postulates when dealing with right triangles. Point out that one "A" angle is already established in all right triangles.

#### Attached Files

Congruent Triangles -Student     File Extension: pdf

Congruent Triangles - Teacher      File Extension: pdf