Euclidean geometry/Triangle congruence and similarity

Notice: Incomplete

Summary of Chapter

This chapter will explain what makes triangles congruent, as well as ways to determine that two triangles are congruent. It will also explain what makes triangles similar. To see the textbook related to this chapter, see the Wikibook chapter.

Triangle Congruency

Definition: Two or more triangles are congruent if all three of their corresponding angles and all three of their corresponding line segments are congruent.

Possible Conditions

a) Specifying nothing. This is insufficient. Demonstrate this by drawing two obviously incongruent triangles.

b) Specifying only one side. This is insufficient. Demonstrate this by drawing two obviously incongruent triangles using the same side.

c) Specifying only one angle. This is insufficient. Demonstrate this by drawing two rays with the same starting point and connecting them in two different ways, giving incongruent triangles.

d) Specifying two sides. This is insufficient. Ask students to build hinges with straight and rigid components, and an elastic band connecting exposed ends. Point out that it's possible to get incongruent triangles with the setup by changing the angle.

e) Specifying two angles. This is insufficient. Demonstrate this by drawing two rays from a given starting point, then sliding a wedge (attached to a ruler) along one ray, using the side not attached. The ruler should intersect the second ray. Point out that the two angles are always identical, yet the distance between them varies.

f) Specifying a side and angle on it. This is insufficient. Demonstrate this by drawing a line segment and a ray (with the same starting point) and connecting the exposed end of the line segment to the ray in two different ways. Point out that the angles on the formerly exposed end are different, resulting in incongruent triangles.

g) Specifying a side and angle not on it. This is insufficient. Demonstrate this by...

h) Specifying three sides. This is sufficient (see below)

i) Specifying three angles. This is insufficient on a flat plane. Demonstrate this by drawing two rays from a given starting point and two parallel lines, each intersecting both rays. Point out that both angles on a given ray (formed by parallel lines) are corresponding, and thus congruent (on a flat plane)

j) Specifying two sides and the included angle. This is sufficient (see below)

k) Specifying two sides and a non-included angle. This is only sometimes sufficient (see below)

l) Specifying two angles and a side on both. This is sufficient (see below)

m) Specifying two angles and a side not on both. This is sufficient (see below)

n) Specifying three sides and an angle. This is sufficient as three sides is enough.

o) Specifying three angles and a side. This is sufficient as the side and the two angles on it is enough.

p) Specifying two sides and both excluded angles. This is sufficient on a flat plane (or any surface as long as the sides aren't from poles on a sphere and the excluded angles are on the equator) as...

q) Specifying two sides, the included angle, and an excluded angle. This is sufficient as two sides and the included angle is enough.

r) Specifying three sides and two angles. This is sufficient as three sides is enough.

s) Specifying three angles and two sides. This is sufficient as a side and the two angles on it are enough.

t) Specifying everything. This is sufficient as only specifying the sides is enough.

Triangle Congruency Postulates

1) The SSS Triangle Congruency Postulate

Postulate: Every SSS (Side-Side-Side) correspondence is a congruence.

Activity: Supply a bunch of things which are both straight and rigid. Ask students to collect any triplet which can be glued into a triangle and to do so. Point out that the triangles created are completely rigid.

2) The SAS Triangle Congruency Postulate

Postulate: Every SAS (Side-Angle-Side) correspondence is a congruence.

Point out that the elastic-band connected hinge would be rigid if you glued the rigid sides to a wedge.

Note: It is important that the angle is included; that is, it is adjacent to the two congruent sides.

3) The ASA Triangle Congruency Postulate

Postulate: Every ASA (Angle-Side-Angle) correspondence is a congruence.

Demonstrate this by asking a student to give a side length, another student to give an angle, and a third student to give another angle such that the provided angles add to less than $\pi$ or $180^{\circ }$, and try to construct triangles with both angles on the side length. Only two triangles can be made, corresponding to which end of the side you put the angles on, and those triangles are mirror images of each other.

Note: The congruence condition is only called that when the side is included.

4) The AAS Triangle Congruency Postulate

Postulate: Every AAS (Angle-Angle-Side) correspondence is a congruence.

Demonstrate this by asking a student to give a side length, another student to give an angle (which will be on the side), and a third student to give another angle such that the provided angles add to less than $\pi$ or $180^{\circ }$ (which won't be on the side). Draw a line segment with the provided length and a ray with the first provided angle on one end of said line segment. Slide your ruler along the ray, always preserving the second provided angle. Point out that there is only one instance of the ruler meeting the line segment on the other end.

Congruency vs. Similarity

Congruency is, simply put, when two things have the same size and the same shape. Similarity, however, has only part of that definition: two things are similar if they have the same shape, but not necessarily the same size. Any two circles, for example, are similar, because they have the same shape, but one circle can be huge and another small. The same goes for all squares and equilateral triangles.

Triangle Similarity

Definition: Two or more triangles are similar if their angles are congruent and their sides are proportional.

Triangle Similarity Theorems

1) The AA Triangle Similarity Theorem

Theorem: Every AA (Angle-Angle) correspondence is a similarity.

2) All congruences are also similarities.