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Similarity and Congruency of Planar Figures

Congruent Figures

Two polygons are congruent if they are the same size and shape - that is, if their corresponding angles and sides are equal.

(congruent
Figures or angles that have the same size and shape
)

Congruency of Triangles

Triangles are more complex when attempting to determine the congruency between two triangles. Instead of having all their corresponding angles and sides equal, triangles have many different conditions. Here, we would make a list of the most common factors of congruency.

Here, A=angle, S=side, whereas H=Hypotenuse

Let's start:

There are 8=23 different possibilities: SSS, SSA, SAS, SAA, ASS, ASA, AAS, AAA

SSS is more formally known as the Side-Side-Side Congruence Theorem (or maybe Edge-Edge-Edge Congruence Theorem).

 

 
If in two triangles the three sides are pairwise congruent, then the triangles are congruent.

 

SAS is more formally known as the Side-Angle-Side Congruence Theorem. Be sure the angle you are using is BETWEEN the two sides you are using. Thus if sides AB and BC are used, angle B is the included angle. Order is important and is implied by the order the letters are specified.

 

 

 

 
If in two triangles two sides and included angle are pairwise congruent, then the triangles are congruent.

 

AAS is more formally known as the Angle-Angle-Side Congruence Theorem. The side used here is opposite the first angle.

 

If in 2 triangles 2 angles and a non-included side are pairwise congruent, then the triangles are congruent.

 

ASA is more formally known as the Angle-Side-Angle Congruence Theorem. The side used here is BETWEEN the two angles you are using. Thus if angle A and angle B are used, side AB is the included side.

 
If in 2 triangles 2 angles and included sides are pairwise congruent, then the triangles are congruent.

 
ASA Triangle AAS Triangle

There is a fundamental difference between ASA and AAS which may not be readily apparent to the beginning geometry student. Consider the two triangles given above. Notice how the given side is between the two angles in the ASA triangle, whereas the given side is opposite one of the angles in the AAS triangle. 

Triangle Non-congruences: AAA, and SSA=ASS

There is no Angle-Angle-Angle Congruence theorem. You should perhaps draw various equilateral triangles like those to the right to convince yourself of this. There is, however, an Angle-Angle-Angle Similarity Theorem. In fact, since if you know two angles, the third is fixed as 180-their measure, it is known as the AA Similarity Theorem. 

 

There is also one more congruency condition which we seldom see around. It is the RHS, known as the Right-angle -Hypotenuse-side - angle Congruence Theorem

 

 

Similar Figures

Figures that are the same shape but not necessarily the same size are called similar figures. You encountered similar figures in the topic about ratios and proportions. The triangles at the right are similar.

In a pair of similar figures, the measures of corresponding angles are equal, and the corresponding sides are in proportion.

Likewise, similar triangles also have special conditions to determine similarity. BUT, since congruency is a stricter condition than similarity, we can then loosen the conditions a bit as compared to conditions for congruency

IN this situation, take a=angle and s=side

Case 1

The most basic condition is AAA, known as the Angle-Angle-Angle postulate

 

 

Case 2

Another postulate would be the SSS.

Here, SSS means that the sides are in ratio ,instead of being equal, as in the case for congruency test.

 

 

Case 3

The last postulate we'll be introducing would be the SAS, in which two pairs of corresponding sides are in ration & their included angles are equal. (here, k, the ratio of the corresponding sides is known as the scale factor)