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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13 thoughts on “Is there an SSA Congruence?

  1. Great post Guillermo. I approach the question of AAA as a method to proving triangles congruent, by addressing the definition of congruent figures. The definition of congruent figures states the shapes must be the same shape AND size. I then draw two triangles with the same angle measure, but different sizes (similar to your figure 2).

    Coupling the visual representation with the definition really helps many of my students understand this misunderstood topic in high school geometry.

    Regards,

    Mr. Pi

  2. Thanks Mr. Pi.

    In my experience, students will realize that triangles are congruent, if they copy a triangle using compass and straigtedge given certain conditions. The only thing that we need to teach them is how to copy segments and angles using compass and straightedge.

    Regards,

    Guillermo

  3. Good day.

    I allow my students to explore SSA to determine which case(s) don’t work. My students explore SSA where the angle is acute, right, and obtuse. Also they explore the side lengths, by considering 3 cases – Ss, sS, and SS. SsA represents the side closest to the given angle is shorter. sSA represents the side closest to the angle is longer. And ssA represents the case where both sides are congruent.

    SsA always constructs 1 triangle, which includes the Hypotenuse-Leg Thm. Students need to be careful of the sSA and the ssA cases.

  4. May i ask some question about SSA ?
    There is 2 triangle ,they are Triangle ABC and Triangle DEF
    AB = DE (S), AC = DF (S), angle C = angle F (A),and both angle B and E is either
    (acute or obtuse) are them congruence ?

    • This reply is very late. I don’t usually open my twitter account. Anyway, as we have discussed above, it’s possible that they are congruent and it’s also possible that they are not.

  5. you have given a nice information about the definition of congruence of triangles but it will be nice if youi could have given a figure of this congruence

      • Hi meenurk. I don’t know if I understand you right, but are you referring to “Your comment is waiting moderation?” when you comment? It’s automatically done by the blog software to prevent spam. My apologies. 🙂

  6. i have used this information to teach my students which i have used in the smartclass and my students had got a brilliant idea about the congruence of ssacongruence criteria

  7. i have a question to you we know thatthe perimeter of a rectangle is 2(l+b).i will give you a proof.
    2(l+b=p,2(ell+b)=p.2(bone+b),taking b common,2b(one+1)p=4b=p.if you are a brilliant could you please find the meaning of the statement.otherwise you are afool and i will never open any of your document in web.

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