Download F E I J G H L K

Test Two Review Math 236 Sp. 07
1. Find the measures of the missing angles:
y
x
w
2. . Line j is parallel to line k. Segment AB is parallel to segment CE. Find
the measures of angles 1 – 6. If you used a computation to find an angle
measure –SHOW IT.
If you used an angle property to find an angle measure –
STATE THE PROPERTY.)
(do not use a protractor)
Every answer should have a computation or a property listed.
k
List the angle measures in the order you find them.
z
40°
B
50 °
D
1
3
2
5
4
50 °
80 °
6
A
C
3. Be able to prove that the sum of the angles of any triangle is 180 degrees. Also explain
one way to do it using inductive logic and using paper.
4. a. Write down a formula for finding the sum of the measures of the interior angles for any
polygon. Explain what the numbers and letters in your formula represent. Be able to show
two ways on a drawing of any polygon how we can use the fact about sum of interior angles
of a triangle to find the sum of the interior angles of any polygon. You also need to explain the
thinking that goes along with each method to develop a formula for finding the sum of interior
angles of any polygon.
b. This figure is a regular polygon. Find the measures of the following angles
(be sure to show all your work, do not use a protractor).
M
I
i. m (∠MLN) =
ii. m(∠IJK)=
J
L
c. What is the sum of the exterior angles for any polygon?
N
Be able to do problems like those from 12.4 –using the various formulas to
find measures of different angles.
d. The sizes of five interior angles of a hexagon are 65°, 72°, 124°, 116°, and 150°. Can we
find the measure of the last angle? If so how, if not, why not.
e. If the hexagon had been a regular hexagon instead, could we find the measure of one
interior angle. If so how, if not why not.
f. Demonstrate using methods shown in class how to find the number of diagonals of any
polygon-so you are developing the general formula here! Explain what the letters and
numbers in the formula represent. Draw a polygon with 6 or more sides and illustrate how to
use the formula to find the number of diagonals.
K
4. For the polyhedron shown. Find the number of vertices, faces,
and edges. What does V + F – E = for any polyhedra?
I
5. a. This is a right trapezoidal prism. How many lateral faces does this
prism have? How many bases does it have? How many total faces?How
many edges, and how many vertices.
b. Find the following: a prism has bases that are 20-gons
How many total faces does it have? How many lateral faces?
How many edges? How many vertices?
E
J
H
K
L
F
G
6. a. How many bases does this pyramid have, how many lateral faces? How
many total faces? How many edges and how many vertices?
b. If a pyramid has a base that is a 20-gon. How many total faces does it have?
How many lateral faces? How many edges? How many vertices?
7. Give the most complete name of the following geometric figures.
a.
b.
c.
d.
e.
A
C
D
B
8. How many regular polyhedra are there? Why can only one be formed using
square polygonal regions as faces? Why can 3 be formed with equilateral triangles as faces?
Why can only one be formed using regular pentagonal regions as faces? Why can’t one be
formed using regular hexagons for faces?
9. The faces of all the polyhedra are regular polygonal regions. Which of the polyhedra are
regular polyhedrons?
C
A
B
Circle your answer. A B C
10. Answer true or false, provide an example if a statement is false.
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
(a) Every cylinder is a prism.
(b) Every pyramid is a polyhedron.
(c) The bases of a prism lie in perpendicular planes.
(d) A cone is a polyhedron.
(e) Any polygon with congruent sides must have congruent angles.
(f) In a regular polyhedron the same number of faces meet at each vertex.
(g) If all the sides of a quadrilateral are congruent, the quadrilateral is a rhombus.
(h) The geometric figure shown to the right is a prism.
11. A rectangle can be defined as a quadrilateral with four right angles. But a rectangle is also
a parallelogram. For any lines with letters a, b, c etc. provide a reason that justifies that step.
To show that a rectangle is a parallelogram.
Given rectangle RUST
Angles 1, 2, 3, and 4 are right angles.
a. m(∠2) = 90°
b. m(∠5) = 90°
Then m(∠ 5) = m(∠ 4) since they are both right angles
c. RU || ST
Then m(∠ 5) = m(∠ 1) since they are both right angles.
d. RS ||UT
R
S
U
1
2
3
4
5
T
12. . i. List all combinations of corresponding sides and angle measurements
where congruence of two triangles is guaranteed.
ii. . List all combinations of corresponding sides and angle measurements
where congruence of two triangles is not guaranteed.
13. a. Given ΔRST ≅ ΔJKL , complete the following statement ΔSRT ≅ __________
b. You are given ΔRST and ΔXYZ with ∠S ≅ ∠Y . To show that ΔRST ≅ ΔXYZ by the
ASA (Angle Side Angle) congruence property, what more would you need to know?
D
14. The figure to the right can be used to show Side Side Angle is not a
a condition of congruence.
a. Name the two triangles that satisfy SSA (Side Side Angle) but are not congruent.
b. Mark on the figure the sides that are congruent and angles that are congruent.
E
A
B
15. Matt constructs a triangle with one side of length 6 cm and one side of length 8 cm and
included angle of measure 40° and Esmeralda also constructs a triangle with one side of length
6 cm and one side of length 8 cm and included angle of measure 40°. Must the two triangles
must be congruent. WHY OR WHY NOT?
16. Refer to the figure. Using only the indicated information, can it be shown
that the two triangles are congruent? WHY OR WHY NOT?
17. For each of the situations below, draw two triangles, ABC and MLN, and mark
them and state why they are congruent (just use abbreviations for the congruence
properties.)
i. ∠A ≅ ∠M , ∠B ≅ ∠L, AB ≅ ML
ii. AB ≅ ML; BC ≅ LN ; AC ≅ MN
A
P
18. Which two triangles to the right are
congruent?
Be careful not to rely on your perception.
a. State the congruence correctly
_______ ≅ _________
N
28°
28°
4
3.8
3.8
4
2
28 °
L
4
M
Q
U
2
C
B
b. State the property you used to determine that the two are congruent.
c. Now explain why you can not state conclusively that the third triangle is congruent to the
other two.
19. Prove that any point on the perpendicular bisector of a line segment is equidistant from the
endpoints of the line segment.
What make up our line segments are the sides of the triangle!
C
Provide the following reasons for the proof we need:
1. AD = BD and ∠ADC ≅ ∠BDC by construction
A
D
2. CD = CD equal to itself
3. ADC ≅ BDC
(Give reason here!)
4. AC = BC
(give reason here!)
B
20. Provide justifications in the proof below
Given: DA ≅ AB ≅ BC ≅ CD
(If the sides of a quadrilateral are
congruent, then it is a
parallelogram)
Prove. Quad ABCD parallelogram
A
D
2
1
4
3
B
C
STATEMENTS
1. DA ≅ AB ≅ BC ≅ CD .
2. DB ≅ DB
3.
ΔDAB ≅ ΔBCD
4. ∠1 ≅ ∠ 4 and ∠ 2 ≅ ∠ 3
REASONS
1. Given
2.
3.
4.
5.
6.
5.
6.
↔ ↔
↔ ↔
AB ll CD and AD ll CB
Quad ABCD is a
parallelogram
21. You will want to look over the other proofs we did in class:
In any parallelogram, opposite angles are congruent
In any parallelogram, diagonals bisect each other
In any parallelogram, opposite sides are congruent