Side-Angle-Side Congruence by basic rigid motions

A geometric realization of a proof in H. Wu’s “Teaching Geometry According

to the Common Core Standards”

Given two triangles, ABC and A0B0C0.Assume two pairs of equal corresponding sides with the angle between them equal.

We want to prove the triangles are congruent.

AB

C

A0

C0

B0

side

side

angle

angl

eside

side

angl

e

angle

In other words, given ABC and A0B0C0,

AB

C

A0

C0

B0

A = A0, |AB| = |A0B0|,

and |AC| = |A0C0|,

we must give a composition of basic rigid motions that maps ABC exactly onto A0B0C0.

side

side

side

side

with

We first move vertex A to A0 by a translation along the vector from A to A0

AB

C

A0

C0

B0

translates all points in the plane. Original positions are shown with dashed lines and new positions in red.

Then we use a rotation to bring the horizontal side of the red triangle (which is the translated image of AB by ) to A0B0.

AB

C

A0

C0

B0

AB

C

A0

C0

B0

maps the translated image of AB exactly onto A0B0 because |AB| = |A0B0| and translations preserve length.

Now we have two of the red triangle’s vertices coinciding with A0 and B0 of A0B0C0.

AB

C

A0

C0

B0

After a reflection of the red triangle across A0B0, the third vertex will exactly coincide with C0.

Can we be sure this composition of basic rigid motions

AB

C

A0

C0

B0

takes C to C0 — and the red triangle with it?

(the reflection of the rotation of thetranslation of theimage of ABC)

Yes! The two marked angles at A0 are equal since basic rigid motions preserve degrees of angles,

AB

C

A0

C0

B0and CAB = C0A0B0

is given by hypothesis.

A reflection across A0B0 does take C to C0

— and the red triangle with it!

AB

C

A0

C0

B0

Since basic rigid motions preserve length and since |AC| = |A0C0|,

by Lemma 8, the red triangle coincides with A0B0C0.

after a reflection across A0B0,

The triangles are congruent. Our proof is complete.

Given two triangles with two pairs of equal sides and an included equal angle,

maps the image of one triangle onto the other.Therefore, the triangles are congruent.

basic rigid motions

AB

C

A0

C0

B0

A0

C0

B0

a composition of

(translation, rotation, and reflection)