Geometry students are all subjected to triangle congruence theorems: SSS, SAS, AAS, and ASA. What are the implications of easing the hypotheses of these theorems?
Consider the Side-Side-Side congruence theorem, which can be stated "If the three sides of a triangle are specified (and satisfy the triangle inequality), then the triangle is uniquely determined". What if only two sides of a triangle are fixed and the third side (and hence the included angle between the two given sides) is allowed to vary?
As you can see, given two sides of a triangle without specifying the included angle or the opposite side, there is little restriction on the properties of a triangle. Some geometry textbooks discuss this situation as the Hinge Theorem: Given two sides of a triangle, as the included angle increases, so does the opposite side.
If two angles of a triangle are given without specifying a length of any side, what, if anything can we conclude about the triangle?
As you can see, the shape of the triangle remains the same even thought the size changes. Perhaps we should have anticipated this, since the given information matches the hypotheses of the AA similarity theorem: A triangle with two fixed angles is similar to any other triangle with those same angles.
What if one side and one angle of a triangle are given? If the angle is adjacent to the given side, we have the following situation.
The other possibility is that the given angle is opposite the given side of the triangle.
In GSP did you get a familiar looking figure? We have a (perhaps) unexpected connection to a form of the converse of the Inscribed Angle Theorem: If a the sides of a fixed angle intercept a fixed segment, then the vertex of the angle lies on the circle determined by the vertex and the end points of the segment.
This brief activity investigated the triangle congruence theorems
SSS, SAS, ASA, and AAS by omitting one of the hypotheses for each
theorem to determine the effect on the shape or size of the triangle.
With GSP one could explore properties other than those considered
here. As any of the above triangles varies, what happens to the
incenter, the circumcenter, the orthocenter, the Euler line? How
do the medial triangle, the orthic triangle, or the pedal triangles
Several GSP scripts are available here to aid in constructing some of these special points or triangles associated with a given triangle.
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