5.2: Congruent Triangles
This activity is intended to supplement Geometry, Chapter 4, Lesson 4.
ID: 8817
Time required: 45 minutes
Topic: Triangles & Congruence
- Investigate the SSS, SAS, and ASA sets of conditions for congruent triangles.
Activity Overview
In this activity, students will explore the results when a new triangle is created from an original triangle using the SSS, SAS, and ASA sets of conditions for congruence. In doing so, they will use the Cabri Jr. Compass tool to copy a segment and the Rotation tool to copy an angle.
Teacher Preparation
- This activity is designed to be used in a high school or middle school geometry classroom.
- The Compass tool copies a given length as a radius of a circle to a new location.
- The screenshots on pages 1–7 demonstrate expected student results.
Classroom Management
- This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress. It is recommended that you print out the following pages as a reference for students.
- If desired, have students work in groups with each group focusing on one of the problems.
- The student worksheet helps guide students through the activity and provides a place for students to record their answers and observations.
- The student worksheet includes an optional extension problem to investigate two corresponding sides and the NON-included angle.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
Associated Materials
- Student Worksheet: Congruent Triangles http://www.ck12.org/flexr/chapter/9689, scroll down to the second activity.
- Cabri Jr. Application
Problem 1 – Three Corresponding Sides (SSS)
Step 1: Students should open a new Cabri Jr. file.
They will first construct a scalene triangle using the Triangle tool.
Step 2: Select the Alph-Num tool to label the vertices
Step 3: Have students construct a free point on the screen using the Point tool. Label it point
Step 4: Students will select the Compass tool to copy
- Press ENTER on
AB¯¯¯¯¯¯¯¯ . A dashed circle will appear and follow the pointer. - Press ENTER on point
D . The compass circle is anchored at centerD .
Have students construct a point on the compass circle and label it point
Step 5: Direct students to create
Hide the compass circle with the Hide/Show > Object tool.
Save this file as CongTri. This setup will be used again for Problems 2 and 3.
Step 6: Drag point
Step 7: To copy the other two segments of
Students should copy
Students should copy
Find the intersection point of the two compass circles and hide the circles. Label the intersection point as
Finally, construct segments
Step 8: Students will now investigate whether
Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent. Use the Measure > D. & Length and Measure > Angle tools.
Step 9: Drag vertices of the triangles and observe the results.
Dragging vertices of
Dragging vertices of
If students wish to save their work, use the Save As tool and do not reuse the name CongTri.
Problem 2 – Two Corresponding Sides and the Included Angle (SAS)
Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1.
Recall that
Step 2: Students will select the Rotation tool to copy
- Press ENTER on point
E . This is the center of rotation. - Press ENTER on
DE¯¯¯¯¯¯¯¯ . This is the object to be rotated. - Press ENTER three times on the vertices
A ,B , andC in that order to identify the angle of rotation. A new rotated segment appears.
Step 3: Have students use the Line tool to construct a line from point
Use the Hide/Show > Object tool to hide the endpoint of the rotated segment.
Step 4: Use the Compass tool to copy
Create the intersection point of the compass circle and the line and label it point
Hide the compass circle and the line.
Finally, construct
Step 5: Students will now investigate whether
Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent.
Drag vertices of the triangles and observe the results.
If students wish to save their work, use the Save As tool and do not reuse the name CongTri.
Problem 3 – Two Corresponding Angles and the Included Side (ASA)
Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1.
Recall that
Step 2: Students will select the Rotation tool to copy \begin{align*}\angle{ABC}\end{align*}.
- Press ENTER on point \begin{align*}E\end{align*}. This is the center of rotation.
- Press ENTER on \begin{align*}\overline{DE}\end{align*}. This is the object to be rotated.
Press ENTER three times on the vertices \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} in that order to identify the angle of rotation. A new rotated segment appears.
Step 3: Students will select the Rotation tool to copy \begin{align*}\angle{BAC}\end{align*}.
- Press ENTER on point \begin{align*}D\end{align*}. This is the center of rotation.
- Press ENTER on \begin{align*}\overline{DE}\end{align*}. This is the object to be rotated.
- Press ENTER three times on the vertices \begin{align*}B\end{align*}, \begin{align*}A\end{align*}, and \begin{align*}C\end{align*} in that order to identify the angle of rotation. A new rotated segment appears.
Step 4: Have students use the Line tool to construct lines over the new segments.
Use the Hide/Show > Object tool to hide the endpoints of the rotated segments.
Step 5: Create the intersection point of the two lines and label it point \begin{align*}F\end{align*}. Hide the lines.
Finally, construct \begin{align*}\overline{DF}\end{align*} and \begin{align*}\overline{EF}\end{align*} to complete the new triangle.
Step 6: Students will now investigate whether \begin{align*}\triangle{DEF}\end{align*} is congruent to \begin{align*}\triangle{ABC}\end{align*}.
Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent.
Drag vertices of the triangles and observe the results.
If students wish to save their work, use the Save As tool and do not reuse the name CongTri.