This is the fourth part of the Geometer’s Sketchpad Essentials Series. In this tutorial, we are going to construct another triangle which is congruent to a given triangle using the concept of the SSS triangle congruence. Recall that the SSS congruence theorem tells us that two triangles are congruent, if their corresponding sides are congruent. In doing the construction, we are going to learn how to use the *Ray tool*, the *Circle tool*, and other commands.

1.) Construct triangle *ABC*.

2.) Next, we construct ray *DE*. To do this, click the **Straightedge** tool box and hold the mouse button to display the other tools. Now, choose the **Ray tool**.

3.) Click two distinct points on the sketch pad and display the names of the two points. Your sketch should look like the first figure.

4.) Next, we will construct a segment *DF* which is congruent to *AC*. To do this, be sure to deselect all the objects by clicking on the vacant part of the sketch pad. Select point *D*, then select segment *AC* (do not select the points!), click the **Construct** menu, and then click **Circle By Center+Radius. **This will produce a circle with center *D* and radius equal to the length of *AC*.

5.) Intersect the circle and the ray by selecting both objects, clicking the **Construct** menu and the selecting **Intersection**. Move point *C* or *A* and observe how the radius of the circle adjusts.

6.) Next we hide the circle. To do this, be sure that the circle (and no other object) is selected, click the **Display** menu from the menu bar, and then click **Hide Circle**.

7.) We now copy *AB*. This means that we will create a circle with center *D* and radius equal to the length of *AB*. To do this, select point *D*, select *AB*, click **Construct**, and then click **Center By Center+Radius**.

8.) Next, we copy the third side *BC*. We construct the circle with center *F* and radius *BC*. To do this, select point *F* and segment *BC*, and click **Construct**, and then click **Center By Center+Radius**. Your sketch should look like the second figure. Note that one of the intersections of the circles is our third vertex. 9. Using the *Construct>Intersection* command will give us two intersections, but we only need one. To do this, click the **Point tool**, hover over the intersection, and click when the two circles are highlighted. Name the intersection *G*.

10.) To complete the triangle, construct segments *DG* and* FG*, and hide the two circles. Hide the ray, point *E*, and connect *DF* to complete the activity. Your sketch should have two congruent triangles.

11. Move the points on both triangles. What do you observe? Explain why your observations are such.