by basic rigid motions A geometric realization of a proof in H Wus Teaching Geometry According to the Common Core Standards Given two triangles ABC and A 0 B 0 C 0 Assume two pairs of equal corresponding sides with the angle between them equal ID: 246158
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Slide1
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”Slide2
Given two triangles, ABC and A0
B
0C0.
Assume two pairs of equal corresponding sides with the angle between them equal.
We want to prove the triangles are congruent.
A
B
C
A
0
C
0
B
0
side
side
angle
angle
side
sideSlide3
angle
angle
In other words, given
ABC and
A
0
B
0
C0
, A
B
C
A
0
C
0B0 A = A
0,
|AB| = |
A
0
B
0
|,
and |AC| = |
A
0
C
0
|,
we must give a composition of basic rigid motions that maps
ABC
exactly
onto
A
0
B
0
C
0
.
side
side
side
side
withSlide4
We first move vertex A to A
0
by a translation
along the vector from A to A
0
A
B
C
A
0
C
0
B
0
translates all points in the plane.
O
riginal positions are shown with dashed lines and
new positions in red.
Slide5
Then we use a rotation
to bring the horizontal side of the red triangle (which is the translated image of AB by
) to A
0
B
0
.
A
B
C
A
0
C
0
B
0Slide6
A
B
C
A
0
C
0
B
0
maps the translated image of AB exactly onto A
0
B
0
because
|AB| = |
A
0
B
0
| and translations preserve length.
Slide7
Now we have two of the red triangle’s vertices coinciding with
A
0 and B0 of
A0B0C0
.
A
B
C
A
0
C
0
B
0
After a reflection of the red triangle across A
0
B
0
, the third vertex will exactly coincide with C
0
.Slide8
Can we be sure
this composition of basic rigid motions
A
B
C
A
0
C
0
B
0
takes C to
C
0
— and the red triangle with it?
(the
reflection
of
the
rotation
of the
translation
of
the
image
of
ABC)Slide9
Yes! The two marked angles at A
0
are equal since basic rigid motions preserve degrees of angles,
A
B
C
A
0
C
0
B
0
and
CAB =
C
0
A
0
B
0
is given by hypothesis.
A reflection across
A
0
B
0
does take
C to
C
0
— and the red triangle with it!Slide10
A
B
C
A
0
C
0
B
0
Since basic rigid motions preserve length
and since |AC| = |A
0
C
0
|,
by Lemma 8, the red triangle coincides with
A0B
0C0.
after a reflection across A0
B0,
The triangles are congruent. Our proof is complete.Slide11
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
A
B
C
A
0
C
0
B
0
A
0
C
0
B
0
a
composition of
(
translation
,
rotation
,
and
reflection
)