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B0
C
side
B
side
A
C0
Angle-Side-Angle Congruence
by basic rigid motions A
0
A geometric realization of a proof in
H. Wu’s “Teaching Geometry According
to the Common Core Standards”
A composition of basic rigid motions
maps ABC onto A0B0C0 and proves ASA:
B0
But let’s go through it
carefully, step by step….
C
C0
A
B
A0
a translation, a rotation, and a reflection.
ASA criterion for congruence:
If two triangles have two pairs of equal angles
and the common side of the angles in one
triangle is equal to the corresponding side in the
other triangle, then the triangles are congruent.
Given two triangles, ABC and A0B0C0.
Assume two pairs of equal
corresponding angles
with the sides between
them equal. C
A
B0
C0
B
A0
We want to prove the triangles are congruent.
In other words, given ABC and A0B0C0,
with  A =  A0,  B =  B0, and |AB| = |A0B0|,
B0
C
side
C0
A
side
B
A0
we must find a composition of basic rigid
motions that maps ABC exactly onto A0B0C0.
Step 1: Bring vertex A to A0 by a translation 𝑇
along the vector from A to A0.
B0
C
A
C0
B
A0
𝑇 translates all points in the plane. Original positions
are shown with dashed lines and new positions in red.
Step 2: Bring sides AB and A0B0 together.
Use rotation ℛ around point A0 to bring the
translated image of AB by 𝑇 to A0B0.
B0
C
C0
A
B
A0
ℛ maps the translated image of AB exactly onto A0B0
because |AB| = |A0B0| and rotations preserve length.
Step 3: Bring vertices C and C0 together.
B0
C
A
C0
B
A0
We want to get the third vertex C to coincide with
C0 so we reflect the image of ABC across A0B0.
Can we be sure this reflection takes the image of
C exactly to C0 and not to some other place?
B0
C
C0
A
B
A0
?
Yes, we can. Basic rigid motions preserve degrees
of angles.
B0
C
C0
A
B
A0
Therefore the reflected angle CAB =  C0A0B0
and the reflected angle CBA =  C0B0A0.
Therefore the reflected ray BC coincides with
ray B0C0 and the reflected ray AC coincides with
B0
ray A0C0.
C
C0
A
B
A0
The intersection of the reflected rays BC and AC
(i.e., the reflected C) coincides with the
intersection of ray B0C0 and A0C0 (i.e., C0).
So, a composition of basic rigid motions—
a translation, a rotation, and a reflection—
B0
C
C0
A
B
A0
maps ABC onto A0B0C0 and proves the ASA
criterion for congruence.