Is there an SSA Congruence?

In the Triangle Congruence post, we discussed about ways to test if two triangles are congruent. The only theorems (or sometimes called postulates) that hold are the SSS, SAS and ASA congruence. We ended our discussion with the question about the AAS (or SAA), AAA and SSA (or ASS) congruence.

Let us try to explore the AAS case.  If we have two triangles (see first pair of in Figure 1), and two pairs of their angles (denoted by the blue and red circles) are congruent the third pair of angles (denoted by the yellow circles in the second pair) are also congruent. Hence, a pair of sides (both included in two pairs of congruent angles) are congruent, which is similar to the ASA congruence. Therefore AAS congruence holds and is equivalent to ASA congruence.

Figure 1 – AAS and ASA congruence postulates are equivalent.

In Figure 2, shown are triangles with three pairs of angles that are congruent. It is clear that the two triangles are not congruent. Therefore, AAA congruence does not hold.

Figure 2 – Triangles having three pairs of congruent angles.

Now, let us try the SSA congruence. Figure 3-A shows triangle ABC with sides and angle marked. We extend AC to the right hand side (see Figure 3-B), then rotate BC about point B (see Figure 3-C). We let C’ be the intersection of BC and the extended segment such that BC is congruent to BC’ (see Figure 3-D).

Figure 3 – Triangels having two pairs of sides and a pair of angles which are congruent.

Looking at Figure 3-A and Figure 3-D, two pairs of their sides and a pair of non-included angles are congruent, but the triangles are not congruent. Therefore, SSA (or ASS) congruence does not hold.

Free Whiteboard Software

If you are using computers during lectures, or fond of screencasting, you would probably need a software where you could scribble.  You would need a blackboard or a whiteboard software. There are several free software that you could use – you could use PaintBrush or Flash Player.

One of my latest internet dig, however, would also be a good choice. It is called  Classic WhiteBoard. It is available in Windows, Linux and Mac platform. Its screenshot is shown below.

What I love about this software is that it is very easy to use (click here to view screencast tutorial) and it really looks like a whiteboard.  It works very much like PaintBrush.  One of the disadvantages, though, is that it has only four colors.

Update: I have uploaded a better whiteboard software added with math tools such as compass, straightedge, triangle and protractor here.

Geonext Tutorial 1 – Constructing an Equilateral Triangle

Geonext is aJava-written interactive and free geometry software. It is developed by the Lehrstuhl für Mathematik und ihre Didaktik (Chair of mathematics and its didactics) of the University of Bayreuth in Germany and released under the GNU General Public License.

Figure 1 - The Geonext Window.

1.) Click the New Board button.
2.) To draw a circle with center A and passing through B, click the Circle tool, click the drawing board to determine the center of the circle, then click another location to determine its radius. Notice that Geonext, automatically names the points in alphabetical order.

3.) With the Circle tool still active, click point B and then click point A to create a circle, with center B passing through point A.

Figure 2 - Circles with centers A and B.

4.) Click the Point tool, and click one of the intersections of the two circles. Notice that a point D was also constructed.
5.) Next, we hide the circles and point D, leaving only points A, B and C on the drawing area. To hide the objects, click Objects menu from the menu bar, click Special Properties and click Hide. Click the two circles.

Figure 3 - The circumference of the two circles are hidden.

Notice, that hidden objects are colored pitch. The figure above is shown when you click point D. That means that Geonext is asking you which object to hide. A circle ca, cb or point D. Since both circles are already hidden, you just have to click point D.

6.) Next, we will use the Polygon tool to draw triangle ABC. To draw the triangle, click the Polygon tool, click point A, click point B, click point C and then click point A to close the triangle.
Q1: Move the vertices of the triangle. What do you observe?
7.)Now, to verify that the triangle is equilateral, we can do two things: reveal the measure of the interior angles or the side lengths.  To reveal the side length, click Texts and Calculations, then click Measure Distance, then click the three sides of the triangle.
8.) You can also measure the angles using the angle tool and using three points. For example, if you want to measure angle B, click the Objects menu, click Texts and Calculations, then click Measure Angle, then click point A, click point B and click point C.
9.  Move the vertices of the triangle. What do you observe?
Q2:.  Explain why ABC is always an equilateral triangle.
10.  Click the File menu and click Save if you want to save your file.

There is also a similar construction here using GeoGebra.

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