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TEST 1 MATH 236 Feb. 10, 2003 ANSWERS
M. B. & P. 12.1 -12.5, Van Hiele Article, Activities: Tangrams, Geoboard, Reflection & Rotation Symmetry,
Quad., Angles, Poly. Tiling, Polyhedron. 100 points
l is parallel to m
2
1
6
A
5
l
A
3
D
4
Figure 1
B
Figure 2
C
2
1
3
4
5
m
Figure 3
6
7
8
Figure 4
Polygon B
is a
regular
pentagon
B
7
8
1. (24 pts—2 pts each) Fill in the blanks:
a. Refer to figure 1. m(∠1) + m(∠2) + m(∠3) + m(∠4) + m(∠5) + m(∠6) = (6 − 2)180=720°
Divide Polygon A into 4 triangles. The sum of the measure of the vertex angles of a triangle is 180°
b. Refer to Figure 2. ∠ABD and ∠DBC are both acute angles; they are also ADJACENT angles.
Two angles are adjacent if they share a common side. Both angles contain ray BD.
c. Refer to figure 3. ∠7 and ∠8 are not congruent but they are SUPPLEMENTARY.
Two angles are supplementary if the sum of their measures is 180°. Note supplementary angles do not
have to be adjacent.
d . Refer to Figure 3. ∠1 and ∠3 are congruent and are a pair of VERTICAL ANGLES.
e. Refer to Figure 3
∠4 and ∠5 are congruent and are a pair of ALTERNATE INTERIOR ANGLES.
f. Refer to figure 4. m(∠7)=
(5 − 2)180
= 108 °
5
g. Refer to figure 4. m(∠8) =
360
= 72 °
5
f. The regular pentagon can be divided up into three triangles (5 −2) and the sum of the measures of all the
vertex angles is 3 x 180° = 540° . Since the pentagon is regular each vertex angle has the same measure and
there are five of them. g. The sum of the measures of all of the exterior angles of any polygon is 360°, since
this is a regular pentagon all five of the exterior angles have the same measure.
g. A geometric figure is A SET OF POINTS IN THE PLANE OR IN THREE-DIMENSIONAL SPACE.
h There are FIVE regular polyhedra. The dodecahedron has faces that are PENTAGONAL regions.
i. Refer to figure 5. Triangle ABD is a scalene triangle.
What is the most complete name of the polyhedron
OBLIQUE TRIANGULAR PYRAMID
I
A
Figure 6
E
J
H
C
j. Refer to figure 6. Polygon EHGF is congruent to
polygon IJKL, NOTE THESE ARE THE BASES AND ARE
D
B
TRAPEZOIDS.. Polygons IJHE, HJKG, FGKL, and EILF
Figure 5
have all of their vertex angles of measure 90° . NOTE THESE
ARE THE LATERIAL FACES. What is the most complete name of the polyhedron .
RIGHT TRAPEZOIDAL PRISM.
F
k. Refer to figure 7. This is a right pentagonal prism. How many lateral faces
does this prism have? Answer: FIVE
In this case, the top and bottom are the bases. All faces that are not the bases are
Figure 7
called lateral faces. In the case of a prism they are always bounded by parallelograms.
If the lateral faces are all rectangles then the prism is a right prism (and not tipped over).
l. A child functioning at van Hiele level 0 level recognizes shapes holistically. The child
can pick out a square from a collection of blocks but does not recognize it has four right angles
The name of this van Hiele level is Visualization or Recognition.
K
L
G
Page 2 of 3
Math 236 Spring 03 Test 1
↔
↔ ↔
1
B
2. (2 pts) If BA , FC , and DE are parallel and m(∠ABC) = 140°
and m(∠EDC) = 155° find m(∠BCD).
F
2
C
ANSWER:
m(∠BCD) = 65 degrees.
4
(2 pts) Explain how you arrived at your answer:
1. ∠ABC and ∠1 are supplementary so m(∠1) = 40°
3
2. ∠1 and ∠2 are alternate interior angles and so have the same
D
measure. Hence m(∠2) = 40° .
3. ∠EDC and ∠3 are supplementary so m(∠3) = 25° .
4. ∠3 and ∠4 are alternate interior angles and so have the same
measure. Hence m(∠4) = 25° .
The vertex angle of a regular pentagon has
5. m(∠BCD) = m(∠4) + m(∠2) = 25° + 40° = 65°
measure of 108°. 3 x 108° = 324° ≠360°. So
Note: There are other possible explanations.
it does not tile the plane.
3. (6 pts) Which polygonal regions
can be used to tile the plane with
a monohedral, edge-to-edge tiling
that has translation symmetry.
A
B
E
F
D
C
A
E
CIRCLE YOUR ANSWERS:
A
B
C
D
E
F
Note: B and F are regular polygons.
4. (8 pts) Following are some quadrilaterals. Congruent sides are indicated. Below each figure circle all the
numbers corresponding to names that describe the figure.
Names:
1. rhombus
2. rectangle
3. trapezoid
4. parallelogram
5. kite
3
5
1 4 5
1 2 4 5
6. none of the above
5. (8 pts ) Following are some triangles. Congruent sides are indicated. Below each figure circle all the
numbers corresponding to names that describe the figure.
Names:
1. scalene triangle
2. isosceles triangle
3. equilateral triangle
4. acute triangle
5. obtuse triangle
6. right triangle
1 4
2 3 4
1 5
2 6
7. none of the above
6. (8 pts) On the dot paper draw each of the following if possible. The vertices of polygons should be on the
dots and please use a straightedge.
(a) A trapezoid
(b) a hexagon that
(c) a quadrilateral that
(d) a parallelogram
that is
is concave
is both a kite and a
that is not a square
isosceles
parallelogram
or a rectangle
3 faces meet
4 faces meet
7. (3 pts) The faces of all the polyhedra are regular polygonal
regions. Which of the polyhedra are regular polyhedrons?
Answer. B
B
A is not because the same number of faces do not meet at each vertex.
C is not because all of the faces are not congruent, some are triangular regions and others are square regions.
B is regular and is an octahedron.
C
A
Page 3 of 3
Math 236 Spring 03 Test 1
8. (4 pts) Find the measure of x and y.
Answer:
y
x
m(∠x) = 40 °
m(∠y) = 50 °
9. (4 pts) Circle the letters of those
polygons that have rotation symmetry.
Answer:
Draw in all the lines of symmetry.
A
40°
Angle x is vertical to an angle of measure 40 degrees and
hence they have the same measure. The sum of the
measure of the angles of a triangle is 180 degrees. Since
one of the angle is a right angle of measure 90 degrees ,
the measure of angle y must be 50 degrees.
POLYGON A IS A SQUARE
POLYGON B
10. A student’s explanation of the sum of the measures of the vertex angles of the hexagon is given below:
“The hexagon can be divided up into six triangles hence the sum of the measure of the vertex angles of
the hexagon is equal to the sum of the measure of the angles of six triangles or
C
6 x 180 = 1080 degrees,
D
B
(1 pts) Is this correct? Circle your answer: NO
(2 pts) Explain your answer: The six triangles include more than just the
vertex angles of the polygon. The following angles are not part of the
G
E
vertex angles of the polygon ∠BGC, ∠CGD, ∠DGE, ∠EGF, ∠FGA,
∠AGB. The sum of the measures of these angles is 360 degrees. So if
A
the student was going to use this method, he or she needs to subtract 360
F
degrees from 1080 degrees. The answer is 1080 − 360 = 720 degrees.
11.
(28 pts—2 pts each)
Classify each of the following as true or false. Circle T if it is true; circle F if it is false.
F (a) There exist trapezoids with all vertex angles being acute. If all the angles were acute then the
sum of the measures of the angles would be less than 360 degrees.
F (b) If V = number of vertices, F = number of faces, and E = number of edges.Then for the polyhedron
shown to the right V + F − E ≠ 2 . For all polyhedra V + F − E = 2
F (c) Every cylinder is a prism. The lateral face on a cylinder is curved so it is not a polyhedron. A
prism is a polyhedron.
T (d) Every pyramid is a polyhedron.
F
(e) The bases of a prism lie in perpendicular planes. The bases lie in parallel planes
F
(f) A cone is a polyhedron. The lateral face on a cone is curved so it is not a polyhedron.
F
(g) Any polygon with congruent sides must have congruent angles. Consider the rhombus:
T (h) If two angles of a scalene triangle are complementary then the triangle is a right triangle. The sum of
the measures of the vertex angles of a triangle is 180 degrees. So if two of the angles are
complementary that means the sum of their measures is 90 degrees. Hence the measure of the third
angle must be 90 degrees and so it is a right triangle.
T (i) If a student is at van Hiele level 2, the student does not comprehend the significance of
deduction as a whole or the role of axioms. Empirically obtained results are often used in
conjunction with deduction techniques.
T
(j) In a regular polyhedron the same number of faces meet at each vertex. See problem 7 explanation.
T
(k) If all the sides of a quadrilateral are congruent, the quadrilateral is a rhombus.
T
(l). If a student is at van Hiele Level 3, Deduction, the student can construct, not just
memorize proofs; the possibility of developing a proof in more than one way is seen.
F (m) The geometric figure shown to the right is a prism. Does not satisfy the definition.
F
(n) Two lines are parallel if they do not intersect. Two skew lines in
three-dimensional space do not intersect but they are not parallel.