Download Semester 1 Closure

Semester 1 Closure
Geometer:
CPM Chapters 1-6
Period:
DEAL
 Take time to review the notes we have taken in class so far and previous closure packets. Look for
concepts you feel very comfortable with and topics you need more help with. Look for connections
between ideas as well as connections with material you learned previously.
Chapter 1: Shapes and Transformations
Chapter 2: Angles and Measurements
Chapter 3: Justification and Similarity
Chapter 4: Trigonometry and Probability
Chapter 5: Completing the Triangle Toolkit
Chapter 6: Congruent Triangles
For questions 1-6, use the figure to the right.
1) What is different name for plane R. (answers will vary) plane ABC
2) What is different name for  1. _  BAD
3) What is different name for line m. ⃡AE or ⃡AD or ⃡DE
4) Name a pair of congruent segments. BD and DC
5) Name three collinear points. Points A, D, E or Points B, D, C
6) Name a pair of supplementary angles. (answers will vary)  BDE or
 EDC
7) Name the transformation(s) that are not isometric. Justify your answer.
A dilation is not isometric because the side lengths are not the same- does not preserve size.
(Figures are isometric if the pre-image and images are congruent- same shape AND size.)
8) Using transformations show that a||b.
Translate  1 down along the line c making  1 and  2
supplementary. Since  1 and  2 were same side interior
angles in the original picture. Therefore, if same side interior angles
are supplementary, then the lines a and b are parallel.
9) Use the following regular hexagon to answer the following questions.
A) Name the image of B after a 240 clockwise rotation about Z. point D
B) Name the image of A reflected over 𝐹𝐶. Point E
10) Find the measure of the numbered angles.
m∠1 = 110°
m∠1 = 112°
m∠1 = 138°
11) Given a || b cut by a transversal c, which of the following statements is NOT true.
A)
B)
C)
D)
 1 and  3 are corresponding angles
 2 and  3 are same-side int. angles
 3 and  4 are vertical angles
 1 and  4 are alternate int. angles
12) Refer to the diagram to answer the following questions:
Assume the lines are parallel.
A) Name a pair of same-side interior angles.
Answers will vary (ex:  2 and  3)
B) Name a pair of corresponding angles.
Answers will vary (ex:  2 and  4)
C) Name a pair of alternate interior angles.
Answers will vary (ex:  2 and  6)
13) Write the coordinates of the vertices of the image ABCD for each transformation.
A) Translation (x, y)  (x + 4, y – 2)
A’(7, 2), B’(11, 2), C’(10, -1), D’(6, -1)
B) reflection across x = -1
A’(-5, 4), B’(-9, 4), C’(-8, 1), D’(-4, 1)
C) rotation of 180o about the origin
A(-3, -4), B’(-7, -4), C’(-6, -1), D’(-2, -1)
D) Dilation with scale factor of 2 about the origin
A(6, 8), B(14, 8), C(12, 2), D(4, 2)
m∠1 = 80°
m∠2 = 80°
m∠3 = 100°
m∠4 = 80°
14) Complete the transformations below and then answer the corresponding questions.
a. Reflect JKLM across the y-axis to form J’K’L’M’.
Then rotate J’K’L’M’ 90o counterclockwiseabout the origin to form J’’K’’L’’M’’.
b. What single rigid transformation transforms JKLM to J’’K’’L’’M’’? 180o rotation about the origin
K”
J”
L”
M”
L’
M’
K’
J’
15) How many lines of symmetry does a square have? Hint: Draw a diagram.
4 lines of symmetry
16) Classify the triangle by its sides.
a. ___equilateral______
b. ________isosceles_________
17) Classify the triangle by its angles.
obtuse triangle
18) Find the value of x.
x = 7°
19) Find the value of x if AC = 20.
X = 5 units
20) Find the value of x if B is the midpoint of 𝑨𝑪.
X = 28° m∠𝑻𝑸𝑴 =44°
21) Find mTQM .
22) Find the value of x.
X = 6 units
x = 14°
23) Find the value of x. x = 54°
24) Find the measure of the sides of equilateral ∆JKL if JK = 5x – 7 and JL = 2x + 5.
The measure of each side is 13 units.
25) 𝑸𝑿 bisects  PQR. If m 1 = 4x -12 and m 2 = 2x + 6, find m XQR.
m XQR = 24°
26) Find the value of x.
x = 16°
27) Find the value of x and y. State geometric relationships used in solving.
y = 79° triangle angle sum theorem, supplementary angle pair
x = 47° alternate interior angles of parallel lines are congruent
28) Find the value of x and y.
x = 𝟑√𝟐 units
y = 3 units
Multiply each expression below.
29) (2𝑑 − 3)(𝑑 + 4)
30) 2𝑥(𝑥 + 3)
2𝑑2 + 5𝑑 − 12
2𝑥 2 + 6𝑥
32) Find the intersection of MN and LO .
Point R
45-45-90 ratios
31) (𝑟 − 5)2
𝑟 2 − 10𝑟 + 25
33) Find the slope of a line perpendicular to 3y + 2x = 4.
𝒎=
𝟑
𝟐
34) What is the slope of the line that contains (6, -4) and (-4, 1)?
𝒎=−
𝟏
𝟐
35) Which pair of lines is perpendicular?
a. y = 3x + 5
b. y = 4x – 5
c. y = 5x – 3
y = -3x – 8
y = 4x + 2
y = 5x + 3
b. y = -2x – 1
2x + y = 4
c. y = 3x + 8
3x + y = 6
1
2
d. y = − 3 𝑥 − 11
3
y = 2x – 5
36) Which pair of lines is parallel?
a. 𝑦 = 5𝑥 + 3
5x + y = 10
37) a. Graph a line parallel to y = 3x – 1 that contains (2, 3).
b.Write the equation of the new line in point slope form.
𝐲 − 𝟑 = 𝟑(𝐱 − 𝟐)
1
2
38) a. Graph a line perpendicular to y = x – 5 that contains (-2, 2).
b. Write the equation of the new line in slope-intercept form.
𝐲 = −𝟐𝐱 − 𝟐
d. x = 3
y=3
39) What is the distance between (-1, 6) and (5, -2)?
10 units
40) The two rectangles are similar. Which is a correct proportion for corresponding sides?
4m
8m
12 m
x
a.
b.
c.
d.
41) The Sears Tower in Chicago is 1450 feet high. A model of the tower is 24 inches tall. What is the ratio of the
height of the model to the height of the actual Sears Tower?
1
725
42) Solve the proportion.
m = 25
43) The figures are similar. Give the ratio of the perimeters and the ratio of the areas of the first figure to the second.
The figures are not drawn to scale.
5
Ratios of the perimeters = 6
25
Ratios of the Areas = 36
15
44) Find the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle.
5 ft
3 ft
9 ft
11 ft
𝟓
𝟑
18
45) You want to produce a scale drawing of your living room, which is 24 ft by 15 ft. If you use a scale of 4 in. = 6 ft,
what will be the dimensions of your scale drawing?
10 in x 16 in
46) Figure
. Name a pair of corresponding sides? 𝐓𝐐 𝐚𝐧𝐝 𝐁𝐂
47) ABCD ~ WXYZ. AD = 6, DC = 3, and WZ = 59. Find YZ. The figures are not drawn to scale.
X
B
W
A
Zy
C
D
Z
29.5 units
48) Triangles ABC and DEF are similar. Find the lengths of AB and EF.
A
D
5x
5
E
B
4
x F
C
AB = 10 units and EF = 2units
49) State whether the triangles are similar. If so, write a similarity statement and the postulate or theorem you used. If
not similar, write not similar.
a. ∆ABD~∆CBD by SAS~
b. ∆CAB~∆ZXY by AA~
50) Campsites F and G are on opposite sides of a lake. A survey crew made the measurements shown on the
diagram. What is the distance between the two campsites? The diagram is not to scale.
42.3 m
51) Find the length of the missing sides. Then determine the perimeter of the triangle.
5’
52) Find the length of the missing side. The triangle is not drawn to scale.
5√3’
25
24
7 units
53) You traveled 10 miles east and then 8 miles south. Exactly how far are you from your starting point?
x = 2√41 miles
54) Two students were asked to find the value of x in the figure. The equations they used are below. Decide whether
each student is correct or incorrect. If the student is incorrect, explain what they did wrong and then show the
correct equation. If the student was correct, write correct.
Lee’s equation:
𝑥
sin 57 = 15
Jamila’s equation:
15
cos 33 = 𝑥
Incorrect.
The hypotenuse is x and the opposite
side is 15. Therefore the equation
15
should be sin 57 = 𝑥 .
Correct
55) Find the length of the leg. If your answer is not an integer, leave it in simplest radical form.
Leg = 8√2
16
45°
Not drawn to scale
56) Find the value of the variable(s). If your answer is not an integer, leave it in simplest radical form.
x
y
30°
20
x = 30 units, y = 10√3 𝑢𝑛𝑖𝑡𝑠
57) Write the tangent ratios for
20
𝑇𝑎𝑛 𝑃 = 21
and
.
P
21
𝑇𝑎𝑛 𝑄 = 20
tangent ratios of complementary angles are reciprocals!
29
21
R
Q
20
Not drawn to scale
58) A large totem pole in the state of Washington is 100 feet tall. At a particular time of day, the totem pole casts a
249-foot-long shadow. Find the angle of elevation to the nearest degree. Draw a diagram to support your work!
x = 22°
59) Find the value of x. Round to the nearest tenth.
10
x = 8.1 units

x
Not drawn to scale
60) A spotlight is mounted on a wall 17.4 feet above the ground in an office building. It is used to light an entrance
door 9.3 feet away. To the nearest degree, what is the angle of depression from the spotlight to the entrance door?
𝜽 =62o
61) State whether the triangles are congruent. If so, write a congruence statement and the postulate or theorem you
used. If not congruent, write not congruent.
a. ∆𝐴𝐷𝐵 ≅ ∆𝐶𝐷𝐵 by SAS≅
b. triangles are not congruent
(no AA for congruent triangles)
62) If ∆𝑫𝑳𝑸 ≅ ∆𝑬𝑴𝑹, then which of the following are NOT true?
a. LDQ  MRE
b. DLQ  EMR
c. RME  QLD
63) ∆𝑿𝒀𝒁 ≅ ∆𝑨𝑩𝑪. What angle is congruent to Y ?
d. MRE  LQD
∠𝑌 ≅ ∠𝐵
64) Write the converse of the conditional statement: If it is raining, then the ground is wet.
If the ground is wet, then it is raining.
65) Is the following a statement a good definition? Explain.
A square has four right angles.
No, a rectangle also has 4 right angles. A better defition would include the fact that a square also has 4 congruent sides.
66) Find a counterexample for the statement: If two shapes are similar, then they are congruent.
An equilateral triangle with side lengths 4 units and an equilateral triangle with side lengths 8 units.
67) State the postulate or theorem you can use to prove the triangles congruent. If the triangles cannot be proven
congruent, write not possible.
a) _ ASA≅ or AAS≅ ______
b) ______ SSS≅ ______
c) ___not possible____
d) ____ ASA≅ or AAS≅ _
e) ____not possible____
f) ______HL≅_______
g) _______ HL≅_______
h)SAS≅ or HL≅ or SSS≅
i) _____ SAS≅_________
j) ASA≅ or AAS≅
k) __ AAS≅ or ASA≅
i) ___not possible___
68) Name a pair of overlapping congruent triangles. State whether the triangles are congruent by SSS, SAS, ASA,
AAS, or HL.
ΔWZX ≅ ΔXYW by AAS ≅
69) Which two sides must be congruent to use ASA≅ to prove DMX  JCX ?
̅̅̅̅ ≅ 𝐽𝑋
̅̅̅
𝐷𝑋
70) Which two angles must be congruent to use SAS≅ to prove ADX  CBX ?
∠𝐷 ≅ ∠𝐵
71) Are the triangles congruent? If so, write a congruence statement and justify your reasoning. If not, write not
congruent.
yes! ΔABC ≅ ΔEDC by HL ≅
72) Given: 𝑨𝑩 || 𝑫𝑪
𝑨𝑫 || 𝑩𝑪
Prove: ∆𝑨𝑫𝑪 ≅ ∆𝑪𝑩𝑨
Statements
Reasons
1) 𝐴𝐵 || 𝐷𝐶
1) Given
2) 𝐴𝐷 || 𝐵𝐶
3) BAC  DCA
4) BCA  DAC
2) Given
3) If lines parallel, then alternate interior angles congruent.
4) If lines parallel, then alternate interior angles congruent.
5) 𝐴𝐶 ≅ 𝐴𝐶
6) ∆𝐴𝐷𝐶 ≅ ∆𝐶𝐵𝐴
5) Reflexive Property of Congruence
6) ASA≅
73) Given: 𝑬𝑰 and 𝑯𝑭 bisect each other at G.
Prove:  E ≅  I
Statements
Reasons
1) 𝐸𝐼 and 𝐻𝐹 bisect each other at G.
1) Given
2) 𝐸𝐺 ≅ 𝐺𝐼
2) Definition of segment bisector
3) 𝐻𝐺 ≅ 𝐺𝐹
4) EGF  HGI
5) ∆𝐸𝐺𝐹 ≅ ∆𝐼𝐺𝐻
6)  E ≅  I
3) Definition of segment bisector
4) Vertical angles are congruent.
5) SAS≅
6) CPCTC
74) Given: MJ  JK
KL  ML
JK  ML
Prove: JM  KL
Statements
Reasons
1) MJ  JK
1) Given
2) KL  ML
2) Given
3) JK  ML
4) J and L are right angles
3) Given
4) Definfition of perpendicular
5) ∆MJK and ∆KLM are right triangles
5) Definition of right triangles
6) 𝑀𝐾 ≅ 𝑀𝐾
7) ∆𝑀𝐽𝐾 ≅ ∆𝐾𝐿𝑀
6) Reflexive Propety of Congruence
7) HL≅
8) JM  KL
8) CPCTC
75) Given: ∆AED is isosceles
AB  CD
Prove: ∆BEC is isosceles
Statements
1)∆AED is isosceles with vertex ∠E
Reasons
1) Given
2) AB  CD
2) Given
3) 𝐴𝐸 ≅ 𝐸𝐷
4) ∠𝐴 ≅ ∠𝐷
5) ∆𝐸𝐴𝐵 ≅ ∆𝐸𝐷𝐶
3) Definition of isosceles triangle
4) Base Angles of Isosceles Triangle Congruent
5) SAS≅
6) 𝐵𝐸 ≅ 𝐶𝐸
7) ∆BEC is isosceles
6) CPCTC
7) Definition of isosceles triangle
76) Find the area and perimeter of ΔABC.
1
1
Area = [2 (4)(4)] + [2 (4√3)(4)] = 8 + 8√3 𝑢𝑛𝑖𝑡𝑠
Perimeter = 8 + 4√2 + 4 + 4√3 𝑢𝑛𝑖𝑡𝑠 = 12 + 4√2 + 4√3 𝑢𝑛𝑖𝑡𝑠