Almost-congruent triangles
Weakening the restrictions on congruent triangles

by

D. Hembree

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?

The lengths of the two red sides are set by the sliders AB and AC. The green side is allowed to vary in length, which also changes the angle included between the red sides.
Click anywhere on the figure to open Geometer's Sketchpad and investigate this situation.

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?

In the figure above, angles A and B are fixed and side AB is allowed to vary in length. Click anywhere on the figure to open Geometer's Sketchpad and explore this situation.

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.

In the figure above, side AB and angle A are given and the other sides and angles are allowed to vary. Click anywhere on the figure to open Geometer's Sketchpad and explore this situation.

The other possibility is that the given angle is opposite the given side of the triangle.

In the figure above, the length of side AB and the measurement of angle C are fixed. The other measurements of the triangle are allowed to vary. Can you make any predictions about the location of point C?  Click anywhere on the figure to open Geometer's Sketchpad and explore this situation.

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.

Suggested Extensions

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 change?
Several GSP scripts are available here to aid in constructing some of these special points or triangles associated with a given triangle.