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5.2: Congruent Triangles

Difficulty Level: At Grade Created by: CK-12

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.

Associated Materials

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 A\begin{align*}A\end{align*}, B\begin{align*}B\end{align*}, and C\begin{align*}C\end{align*} as shown.

Step 3: Have students construct a free point on the screen using the Point tool. Label it point D\begin{align*}D\end{align*} with the Alph-Num tool.

Step 4: Students will select the Compass tool to copy AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*} to point D\begin{align*}D\end{align*}.

• Press ENTER on AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*}. A dashed circle will appear and follow the pointer.
• Press ENTER on point D\begin{align*}D\end{align*}. The compass circle is anchored at center D\begin{align*}D\end{align*}.

Have students construct a point on the compass circle and label it point E\begin{align*}E\end{align*}.

Step 5: Direct students to create DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*} with the segment tool. This segment is a copy of AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*}.

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 E\begin{align*}E\end{align*}, and observe that the location of DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*} can change but it will not change in length. You must drag either point A\begin{align*}A\end{align*} or point B\begin{align*}B\end{align*} to change the length of DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*}.

Step 7: To copy the other two segments of ABC\begin{align*}\triangle{ABC}\end{align*}, repeat the use of the Compass tool.

Students should copy AC¯¯¯¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} to point D\begin{align*}D\end{align*}.

Students should copy BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} to point E\begin{align*}E\end{align*}.

Find the intersection point of the two compass circles and hide the circles. Label the intersection point as F\begin{align*}F\end{align*}.

Finally, construct segments DF¯¯¯¯¯¯¯¯\begin{align*}\overline{DF}\end{align*} and EF¯¯¯¯¯¯¯¯\begin{align*}\overline{EF}\end{align*} to complete the new triangle.

Step 8: Students will now investigate whether DEF\begin{align*}\triangle{DEF}\end{align*} is congruent to ABC\begin{align*}\triangle{ABC}\end{align*}.

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 ABC\begin{align*}\triangle{ABC}\end{align*} will cause both triangles to change in size and shape.

Dragging vertices of DEF\begin{align*}\triangle{DEF}\end{align*} will cause DEF\begin{align*}\triangle{DEF}\end{align*} to change its location but its size and shape will remain the same as ABC\begin{align*}\triangle{ABC}\end{align*}. Notice that not all vertices of DEF\begin{align*}\triangle{DEF}\end{align*} are available to be dragged, since they have been constructed to be dependent on ABC\begin{align*}\triangle{ABC}\end{align*}.

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 DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*} is a copy of AB¯¯¯¯¯¯¯¯\begin{align*}\overline{AB}\end{align*}.

Step 2: Students will select the Rotation tool to copy ABC\begin{align*}\angle{ABC}\end{align*}.

• Press ENTER on point E\begin{align*}E\end{align*}. This is the center of rotation.
• Press ENTER on DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*}. This is the object to be rotated.
• Press ENTER three times on the vertices A\begin{align*}A\end{align*}, B\begin{align*}B\end{align*}, and C\begin{align*}C\end{align*} 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 E\begin{align*}E\end{align*} through the endpoint of the new segment.

Use the Hide/Show > Object tool to hide the endpoint of the rotated segment.

Step 4: Use the Compass tool to copy BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} to point E\begin{align*}E\end{align*}.

Create the intersection point of the compass circle and the line and label it point F\begin{align*}F\end{align*}.

Hide the compass circle and the line.

Finally, construct DF¯¯¯¯¯¯¯¯\begin{align*}\overline{DF}\end{align*} and EF¯¯¯¯¯¯¯¯\begin{align*}\overline{EF}\end{align*} to complete the new triangle.

Step 5: Students will now investigate whether DEF\begin{align*}\triangle{DEF}\end{align*} is congruent to ABC\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.

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 DE¯¯¯¯¯¯¯¯\begin{align*}\overline{DE}\end{align*} is a copy of \begin{align*}\overline{AB}\end{align*}.

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.

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