Congruent Triangles Based on Rigid Motions

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SWBAT Use rigid motions to decide if two triangles are congruent.

Big Idea

Students will examine triangles on the coordinate plane and use rigid motions to show two triangles are congruent.

Do Now

5 minutes

As students walk in the classroom, they are given a slip of paper on it. On the graph are several triangles. Some of the triangles are congruent to each other and others are not. Students are asked to identify the congruent triangles and explain why one pair of triangles is not congruent. This Do Now reviews concepts from the previous lesson, Rigid Motions and Congruence.

Students will use the graph from the Do Now during the lesson to connect the previous unit on performing transformations to the concept of congruence. 


15 minutes

To begin the Mini-Lesson, we review the definition of congruence from the previous lesson. If an object can be transformed to occupy the same space as another object, the two objects are congruent. Then we go over the Do Now. I ask, “Which transformations can you use to prove the triangles are congruent?” We go over the specific transformations and their rules, which students investigated in the previous unit. Students use the coordinates of points on the grid to show that corresponding sides of the pre-image and image are the same length, and therefore prove the triangles are congruent (Congruence by Side Side Side will be investigated in a later lesson in the Proofs about Triangles unit).


Since all of the triangles on the grid are right triangles, this can be a good time to briefly review the Pythagorean theorem. At this point in the course, I don’t have the students use the distance formula to find the length of the sides of the triangles. I designed the activity so that the length of two sides of each triangle on the grid can be found by counting boxes and the third side can be found using the Pythagorean theorem (MP7). 


20 minutes

Students work individually to identify pairs of congruent triangles on a grid. For each pair of congruent triangles, students describe the specific transformation that proves the triangles are congruent. They then label the congruent sides of the triangles and write statements of congruence for the triangle pairs and the congruent sides.


After about 12 minutes, I stop the students and we go over the transformations and the congruence statements. I ask for volunteers to show how they labeled the triangles using the document camera.


5 minutes

Exit Ticket:  Explain why a triangle and its image after a reflection over the y-axis will be congruent.

Students often respond that the pre-image and the image have congruent sides. They can prove this using the Pythagorean theorem. Additionally, the pre-image and image are equidistant from the y-axis after a reflection. 

During this lesson, all of the pairs of triangles were proved congruent using Side-Side-Side.  In the next lesson, students will learn about other ways for proving triangles congruent.