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Transcript
```A C E
Investigation 2
Applications
1. Since these figures appear to be
8. Certainly congruent because the
parallelograms, there are two possible
correspondences of vertices that will pair
congruent sides and angles:
A → L, B → K, C → N, and D → M or
A → N, B → M, C → L, and D → K.
Pythagorean Theorem guarantees that the
third sides are the same length.
9. There is not enough information to
determine congruence.
2. The correspondence of vertices that seems
right here is: E → R, F → P, and G → Q.
3. The correspondence of vertices that seems
right here is: S → X, T → Z, and U → Y.
10. The two figures do not appear congruent,
and the information marked on the
drawing does not provide enough
justification for invoking any of the
standard congruence criteria.
4. The correspondence of vertices that seems
right here is: A → R, B → S, C → T, D → P,
and E → Q.
11. The two figures may or may not be
5. There are many pairs of congruent
triangles in the given figure:
∆JNM  ∆LNK, ∆JNM  ∆KNL,
∆JNK  ∆MNL, ∆JNK  ∆LNM, and
∆JLM  ∆KML  ∆LJK  ∆MKJ.
Note: We’ve listed what are essentially the
same triangles several times with vertices
in different order since there is symmetry
in the design that allows more than one
correspondence of vertices consistent with
congruence.
12. The two figures are not congruent. Three
congruent. The corresponding sides share
an angle of different measures.
corresponding angles do not guarantee
congruence.
13. In this figure it appears that ∆ABC 
∆EBD, a congruence that could be
established by a line reflection in the
bisector of ABE or by a combination of a
counterclockwise rotation of 90° centered
at B and reflection in the line that contains
points B and D.
14. In this figure it appears that ∆PQR 
6. There are several different combinations
of congruent triangles that can be paired
up in response to this question. We give
several responses to indicate the types
of answers that could be given. There
are essentially only two transformations
needed—line reflections and half-turn
∆UTS, a congruence that could be
established by translation that carries Q to
T (also P to U and R to S).
15. Measurement evidence should suggest
that the given triangles are not congruent.
16. In this figure it appears that ∆ABC 
∆EDF, a congruence that could be
established most easily by a line
reflection in the perpendicular bisector
of BD .
∆JNM  ∆LNK, ∆KML  ∆MKJ, can
be shown by half-turn rotation centered
at N. That point must be the midpoint of
each diagonal due to the symmetry of the
rectangle.
17. In this figure it appears that ∆JKL  ∆TSR,
a congruence that could be established in
several sequences of transformations. One
such sequence would slide L to R, rotate
that image of ∆JKL counterclockwise by
90° centered at R, and reflection in the line
that contains RT .
∆JNM  ∆KNL, ∆JLM  ∆KML,
can be shown by reflection in a vertical
line through point N, again due to the
symmetry of the rectangle.
7. Certainly congruent by the Side-Side-Side
criterion.
Butterflies, Pinwheels, and Wallpaper
1
Investigation 2
A C E
Investigation 2
18. In this figure it appears that ∆JKL 
∆MNO, a congruence that could be
established by a line reflection in JL and
translation to the right. Of course, other
sequences of transformations are possible.
Note: For Exercises 19–22, the instructions
do not imply that the parts of triangle ABC
necessarily correspond to parts of triangle
DEF by the order of the vertex labels;
in other words, triangle ABC could be
congruent to triangle DFE.
19. It is possible, but not certain, that the two
triangles are congruent. One could be
larger than the other. Three corresponding
angles do not guarantee congruence.
20. It is not possible for the two triangles to be
congruent. If they were, then DE = 3 cm
and ∆DEF would have to be an isosceles
triangle with congruent base angles. ∆ABC
cannot be isosceles, because B has
measure of 30°.
21. The two triangles must be congruent
because the Pythagorean Theorem implies
that the third sides of each must be
15.62 – 8.22 ≈ 13.3 cm.
22. These two triangles are congruent by the
Side-Angle-Side criterion.
23. Without any information about side lengths,
the directions are unlikely to produce a
congruent copy of the given figure.
24. Although the given information is not in
one of the standard congruence criterion
forms, the recipient could deduce that
the measure of A has to be 35° and
thus he/she will have Angle-Side-Angle
data to construct a triangle that must be
congruent to Figure 2.
25. Definitely congruent by the Angle-SideAngle criterion.
26. Again, this information satisfies the SideAngle-Side criterion for congruence.
Connections
27. a. The most natural basic design element
would be a 120° sector bounded by
two ‘radii’ of the hexagon, though
there are several ways to choose the
sector boundaries.
30. a. Rectangle perimeter is 30 cm and area
is 36 cm2; parallelogram perimeter is
34 cm and area is 36 cm 2.
b. Congruence, area, and perimeter.
i. No, you could have a square with
sides 7.5 cm making perimeter of
30 cm and that figure would not be
congruent to the given rectangle.
b. Rotations of 120° and 240° will do
the job.
28. a. The design will appear unmoved after
slides left or right, up or down, or
along slanted lines, all assuming the
appropriate distance of slide is chosen
thoughtfully.
ii. No, the rectangle and parallelogram
given have the same area but are
not congruent.
b. One needs a basic design element
including at least two adjacent fish—one
pointing in each direction.
c. If flips are allowed, one fish can be
replicated to fill in each spot of the
wallpaper design.
29. The circles are congruent because one can
imagine sliding or reflecting one center
onto the other. Since the circles have
the same radius and diameter, they are
identical size (and, of course, shape).
Butterflies, Pinwheels, and Wallpaper
iii. No, the rectangle shown and the
square with sides 7.5 cm have the
same perimeter, but the rectangle
area is 36 cm2 and the square area
would be 56.25 cm2.
iv. No, the given rectangle and
parallelogram have the same area
but not the same perimeter.
2
Investigation 2
A C E
Investigation 2
31. a. No
are congruent by the Angle-Side-Angle
criterion (angles at A and D are right
angles, sides AC = DC, and BCA 
ECD because they are vertical
or opposite angles formed by
intersecting lines).
b. This is sufficient information because
opposite sides and opposite angles
of a parallelogram are congruent and
consecutive angles are supplementary.
b. DE = AB.
33. a. This congruence is true for a variety of
c. There are several different combinations
of sufficient information to determine
the shape and size of a quadrilateral.
Side-Angle-Side-Angle-Side is one;
Angle-Side-Angle-Side-Angle is another.
Less information won’t work.
reasons. For instance, the angles at O
are both right angles by construction of
the coordinate grid, OB = OC because
both are length 3, and AO = AO.
b. AC  AB , OC  OB , OAC 
OAB , and ACO  ABO .
34. a. PS and QR are hypotenuses of right
triangles that have legs 2 and 3 units or
one could reason that reflection in the
y-axis would ‘move’ point R to S and
point Q to P.
b. The angles are congruent because they
are images under the reflection in the
y-axis.
35. a. To build a physical triangle you have
to make sides of definite lengths.
Those lengths determine exactly one
triangular shape, so unless the anchors
at each vertex break, the figure cannot
be distorted into any other shape.
b. Bracing a quadrilateral by connecting
32. a. It makes sense to locate point D on AC
so that AC = DC. Then locate point E
on a line that is perpendicular to CD
so that points B, C, and E are collinear.
This will mean that the two triangles
opposite vertices with a diagonal
essentially creates two triangles that
are rigid figures (joined together by the
common side).
Extensions
36. Two sides and an angle of one triangle
congruent to two sides and an angle of
another triangle guarantee congruence only
if the angle is include between the sides.
37. Yes; proof of this fact is beyond the level of
this course, but some students might get
hooked trying to find a counterexample!
This question is equivalent to asking, “If
you know the sum and the product of
two numbers, do you know the individual
numbers?”
Butterflies, Pinwheels, and Wallpaper
38. a. All three angles are the same size.
b. All three sides are the same length.
39. a. The angle measures of the pentagon
must all be 108° because each triangle
has one 72° and two others that add
to 108°.
b. The side lengths must all be the same.
c. The segments from C to the vertices
must all be the same length.
3
Investigation 2
A C E
Investigation 2
40. In addition to the sides and angles given
to be congruent, AD is a side of both
triangles, and it sets up a Side-Angle-Side
congruence condition for ∆ABD  ∆DEA.
This means that angle B  angle E because
these are corresponding angles in the
two triangles just shown to be congruent.
And we know that angle BCA  angle ECD
because these are vertical angles. Since we
have two pairs of congruent corresponding
angles and one pair of congruent
corresponding sides in ∆ABC and ∆DEC,
∆ABC  ∆DEC (using the Angle-AngleSide condition).
Butterflies, Pinwheels, and Wallpaper
4