Why is aaa not a congruence shortcut

SSA : If two sides and the non-included angle are given, three situations may occur. Two similar triangles are related by a scaling or similarity factor s: if the first triangle has sides a, b, and c, then the second…. Side-Side-Side is a rule used to prove whether a given set of triangles are congruent. The SSS rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. The adjective congruent fits when two shapes are the same in shape and size.

If you lay two congruent triangles on each other, they would match up exactly. Congruent comes from the Latin verb congruere "to come together, correspond with. The Side Angle Side postulate often abbreviated as SAS states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

Side Side Side postulate states that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent. Two triangles are congruent if they have: exactly the same three sides and. Congruent Triangles - Hypotenuse and leg of a right triangle. HL Definition: Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles. In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent.

This is sometimes referred to as the AAA Postulate —which is true in all respects, but two angles are entirely sufficient. The postulate can be better understood by working in reverse order.

Congruent Triangles. When two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position if there is a turn or a flip , but they are there. Given two sides and non-included angle SSA is not enough to prove congruence. So let's take a look at the first one which is side side angle.

Now part of reason why this is the serial type Geometry is because if you switch around the a you get a square root, but I'm not going to give the gratitude of hearing me say that. If we start off with this angle, and a side so I'm going to say this is a fixed angle and this is a side that's rigid notice that I drew a ray here and I'm saying that we need to make a triangle here and I'm going to say that this point right here is the center of the circle, so its going to be about a radius of my marker and I'm going to draw in, in dotted lines and again I am not an artist so if we have this circle that is centered at that point, notice that using a radius I can construct two different lines that are congruent so I'm not changing that third side but these two triangles are definitely not congruent.

To redraw them we have this obtuse triangle here so we have these angles as being congruent we have this side being congruent and we have this third side that I haven't marked so we have 1, 2, 3 so we have side side angle and then this other larger triangle that I was able to draw where we have these two angles being congruent cause I kept that fixed, this side was fixed so these two sides must be congruent and this third side because it's a radius of this circle this side must also be congruent but notice we've created two triangles that are not necessarily congruent which is why side side angle is not a shortcut.

The second shortcut that doesn't work is angle angle angle, couple of different ways to look at this one. One way is to say well if we were to extend that side and if we're to extend this side I can construct a line that is parallel to this side right here and what I've done is I've created corresponding and congruent angles because we have two parallel lines and this is the transversal and this side is also a transversal and this third angle here would have to be congruent to itself, so to redraw this we have two triangles where the 3 angles are corresponding but they're definitely not congruent so we have a little bit of overlap here but the idea is that these two triangles are definitely not congruent but their angles are all corresponding and congruent.

The word that we would use for these is similar. But this is not what we're talking about right now because right now we're saying congruence. These two triangles must be exactly identical so the two shortcuts that don't work angle angle angle because we'll create two triangles that'll have different sizes although they're will have same angles and the second one that doesn't work is the side side angle not only because it's a [IB] but also because we create two different triangles.

Knowing only angle-angle-angle AAA does not work because it can produce similar but not congruent triangles. Triangle Similarity Postulates. If two angles of one triangle are congruent to two angles of another, then the triangles must be similar. If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar.

Therefore, if two corresponding pairs of angles in two triangles are congruent, then the remaining pair of angles is also congruent. So the triangles are congruent.

As an example, if 2 triangles are congruent by SSS, then we also know that the angles of 2 triangles are congruent.