Geometer’s Sketchpad Essentials 4 – The SSS Triangle Congruence

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. » Read more

The Infinitude of Pythagorean Triples

In the Understanding the Fermat’s Last Theorem post, I have mentioned about Pythagorean Triples.  In this post, we will show that there are infinitely many of them. We will use intuitive reasoning to prove the theorem.

For the 100th time (kidding), recall that the Pythagorean Theorem states that in a right triangle with side lengths a, b and c, where c is the hypotenuse, the equation c^2 = a^2 + b^2  is satisfied. For example, if we have a triangle with side lengths 2 and 3 units, then the hypotenuse is \sqrt{13}. The converse of the Pythagorean theorem is also true: If you have side lengths, a, b and c, which satisfies the equation above, we are sure that the angle opposite to the longest side is a right angle.

We are familiar with right triangles with integral sides. The triangle with sides (3, 4, 5) units, for instance, is a right triangle.  This is also the same with (5, 12, 13) and (8, 15, 17).  We will call this triples, the Pythagorean triples ,or geometrically, right triangles having integral side lengths. » Read more

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.

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