Geometry - Triangle Congruence (ASA, AAS)and does the how
Up until now, sides have been hogging the spotlight they're really hammy. If we give angles the floor for a bit, maybe they'll make some interesting discoveries for us. Imagine pointing a laser off of each end at the desired angle. There's only one point where these laser lines will intersect. In other words, specifying two angles and the length of the line segment between them can give us only one triangle. That means we can determine whether two triangles are congruent by knowing two angles and the included side. Let's take a look at what we have.
We've just studied two postulates that will help us prove congruence between triangles. However, these postulates were quite reliant on the use of congruent sides. In this section, we will get introduced to two postulates that involve the angles of triangles much more than the SSS Postulate and the SAS Postulate did. Understanding these four postulates and being able to apply them in the correct situations will help us tremendously as we continue our study of geometry. Let's take a look at our next postulate. If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. In a sense, this is basically the opposite of the SAS Postulate.
But we don't have to know all three sides and all three angles SSS stands for "side, side, side" and means that we have two triangles with all three sides equal. If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.
The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape be unique , thus insuring congruency. If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent. If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G , creating two non-congruent triangles using SSA.
Triangle Congruence - ASA and AAS
Congruence criteria of triangles(ASA and AAS).
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