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Common Core Math 2 Unit 1A Modeling with Geometry
1
Name:____________________________ Period: _____ Estimated Test Date: ___________
Unit 1B
Modeling with Geometry:
Congruency
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Math 2 Unit 1A Modeling with Geometry
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Main Concepts Page #
Study Guide 4
Vocabulary 5-7
Similarity 8-14
Congruency 15=19
Triangle Theorems 20-23
CPCTC 24-26
Cross-Sections 27-29
Partitioning a Line Segment 30-33
Unit Review 34-36
HW Answers 37-28
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Standards
G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G-STR.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and
leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Unit Description In this unit, students study the concept of congruency and explore sufficient conditions for congruent triangles. Students are also introduced to proof, informally proving the Triangle Angle Sum Theorem and the Midsegment Theorem. Students apply geometry concepts to modeling situations to solve problems.
Essential Questions By the end of this unit, I will be able to answer the following questions:
Why are only four types of transformations needed to describe the motion of a figure?
How do the SSS, SAS, and ASA congruence criteria follow from the definition of congruence in terms of rigid motions?
Unit Skills I can Congruent Figures: Use the definition of congruence in terms of rigid motions to decide if two figures are congruent. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.* Show that the AAA and SSA criteria are not sufficient to prove that two triangles are congruent.* Triangles: Prove the Triangle Angle Sum Theorem. Draw the midsegment of a triangle. Prove the Midsegment Theorem. Applications: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Find the point on a directed line segment between two given points that partitions the segment in a given
ratio. Modeling: Identify the shape of a cross-section of a three dimensional object. Identify the three dimensional figure generated by the rotation of a two-dimensional shape. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk
or a human torso as a cylinder). Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Common Core Math 2 Unit 1A Modeling with Geometry
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Vocabulary: Define each word and give examples and notes that will help you remember the word/phrase.
AA Similarity
Example and Notes to help YOU remember:
AAS Theorem
Example and Notes to help YOU remember:
ASA Postulate
Example and Notes to help YOU remember:
Congruent
Example and Notes to help YOU remember:
CPCTC
Example and Notes to help YOU remember:
Cross-Section
Example and Notes to help YOU remember:
Common Core Math 2 Unit 1A Modeling with Geometry
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HL Theorem
Example and Notes to help YOU remember:
Isosceles Base Angle Theorem
Example and Notes to help YOU remember:
Midsegment Theorem
Example and Notes to help YOU remember:
PAP Similarity
Example and Notes to help YOU remember:
PPP Similarity
Example and Notes to help YOU remember:
SAS Postulate
Example and Notes to help YOU remember:
Common Core Math 2 Unit 1A Modeling with Geometry
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Similarity
Example and Notes to help YOU remember:
SSS Postulate
Example and Notes to help YOU remember:
Triangle Sum Theorem
Example and Notes to help YOU remember:
Common Core Math 2 Unit 1A Modeling with Geometry
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Properties of Dilations/Similarity
Dilate ∆𝐴𝐵𝐶 about the origin with a scale factor of 2. Graph the new triangle; label the vertices 𝐴′, 𝐵′, & 𝐶′. Complete the following using your dilation.
1. Using a protractor, measure the angles of ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. What do you notice?
2. Using the distance formula, calculate the lengths of 𝐴𝐵, 𝐴′𝐵′, 𝐴𝐶, 𝐴′𝐶′, 𝐵𝐶, 𝑎𝑛𝑑 𝐵′𝐶′. What do you
notice?
3. Dilations create similar figures. Based on your observations from 1 and 2, what can we say about similar
figures?
4. What do you notice about 𝐴𝐵̅̅ ̅̅ 𝑎𝑛𝑑 𝐴′𝐵′̅̅ ̅̅ ̅̅ ? 𝐴𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝐴′𝐶′̅̅ ̅̅ ̅̅ ? Note that A and A’ lie on the origin. What
conclusion can you make about the segments of an image when the corresponding segments of the
preimage pass through the center of dilation?
5. Using the slope formula, calculate the slopes of 𝐵𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝐵′𝐶′̅̅ ̅̅ ̅̅ . What do you notice? What conclusion
can you make about the segments of an image when the corresponding segments of the preimage do
not pass through the center of dilation?
A
B C
Common Core Math 2 Unit 1A Modeling with Geometry
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Similarity
Similar Triangles that have the same _____________ but not necessarily the same _____________.
If
____ ____ , ____ ____ , ____ ____
and
= =
Prove Similarity by
________ Similarity Theorem: 3 pairs of ______________ ________
________ Similarity Theorem: _____ pairs of proportional sides and __________________ angle between
them
ABC DEF
A
B C
E
F D
Common Core Math 2 Unit 1A Modeling with Geometry
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________ Similarity Theorem: 2 pairs of ____________ __________
Definition: Triangles are _________________ () if the corresponding ______________ are ____________ and
the __________________ of the lengths of the corresponding sides are __________________.
Practice
Given that the triangles are similar, determine the scale factor and solve for the missing lengths.
a. ∆𝑉𝑈𝑊~∆𝐾𝑈𝐿 b. ∆𝐿𝐵𝑀~∆𝐶𝐵𝐷 c. ∆𝑅𝑄𝑃~∆𝐷𝐶𝐵
G
H I
L
J K
M
N O
Q
P R
Common Core Math 2 Unit 1A Modeling with Geometry
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Similarity HW (©Kuta Software LLC. All rights reserved.)
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Math 2 Unit 1A Modeling with Geometry
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CW/HW Directions: Answer the following problems deal with congruency and rigid motion.
1. In the diagram at the right, a transformation has occurred on ABC.
a) Describe a transformation that created image ABC from ABC.
b) Is ABC congruent to ABC? Explain.
2. The vertices of MAP are M(-8, 4), A(-6, 8) and P(-2, 7).
The vertices of MAP are M(8, -4), A(6, -8) and P(2, -7).
a) Plot MAP.
b) Verify that the sides of the triangles are congruent by using the distance formula.
c) Describe a rigid motion that can be used to MAP
3. Given PQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent
to STR with S(2, -4), T(6, 2) and R(0, 0).
a) Plot STR.
b) Describe a rigid motion which can be used to verify the triangles are congruent.
Common Core Math 2 Unit 1A Modeling with Geometry
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4. Given RST with R(1, 1), S(4, 5) and T(7, 5).
a) Plot the reflection of RST in the y-axis and
label it RST.
b) Is RST congruent to RST? Explain.
c) Plot the image of RSTunder the
translation (x, y) (x + 4, y – 8). Label the image of RST.
d) Is RST congruent to RST? Explain.
e) Is RST congruent to RST? Explain.
5. Given DFE with D(1, -1), F(9, 6) and E(5,7) and BAT with B(1, 1), A(-6, 9) and T(-7, 5).
a) Describe a transformation that will yield BAT as
the image of DFE.
b) Is BAT congruent to DFE? Explain.
6. Given CAP with C(-4, -2), A(2, 4) and P(4, 0) and SUN with S(-8, -4), U(4, 8) and N(8, 0).
a) Describe a transformation that will
yield SUN as the image of CAP.
c) Is CAP congruent to SUN? Explain.
Common Core Math 2 Unit 1A Modeling with Geometry
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Congruent Triangles Investigation What does it mean to say two triangles are congruent?
Part 1: List the ways to justify that triangles are similar.
Examine the triangles with all side lengths labeled.
a. Are they similar? Why?
b. What is the scale factor? _______ : _______
c. What do we know about the corresponding angles of similar triangles?
d. What does this tell us about the pair of triangles?
Part 2: Examine the triangles with two sides lengths and an included angle labeled.
a. Are they similar? Why?
b. What is the scale factor? _______ : _______
c. Since the triangles are similar, what do we know about P and H? What do we know about I and
O?
d. Use the scale factor you gave in part b to determine the length of OH.
e. What does this tell us about the pair of triangles?
3 cm2 cm
4 cm4 cm
3 cm2 cm
T
R
SB
C
A
1.5 cm
5 cm
4 cm
4 cm
5 cm H
O
G
I
P
Common Core Math 2 Unit 1A Modeling with Geometry
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Part 3: Examine the triangles with two angle pairs marked congruent.
a. Are they similar? Why?
b. What is the scale factor? _______ : _______
c. Since the triangles are similar, what do we know about K and Y?
d. Use the scale factor you gave in part c to determine the lengths of KL and YZ.
e. What does this tell us about the pair of triangles?
CONGRUENCY POSTULATES AND THEOREMS
Think back to the three situations we examined.
In Part 1, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle.
The triangles are congruent by the ___________, __________, __________ (SSS) Postulate.
In Part 2, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of
sides and an included angle of another triangle. The triangles are congruent by the
___________, __________, __________ (SAS) Postulate.
4 cm
2.15 cm
4 cm
3.2 cm
32523252
K
L M ZX
Y
3 cm2 cm
4 cm4 cm
3 cm2 cm
T
R
SB
C
A
1.5 cm
5 cm
4 cm
4 cm
5 cm H
O
G
I
P
Common Core Math 2 Unit 1A Modeling with Geometry
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In Part 3, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of
angles and an included side of another triangle. The triangles are congruent by the
___________, __________, __________ (ASA) Postulate.
Two other Postulates
Angle, Angle, Side Theorem
Which Postulate is used to create this theorem?
Hypotenuse, Leg Theorem
By Pythagorean Theorem, if you know the hypotenuse and 1 leg, you can calculate the 2nd leg by
_________________ Theorem and prove congruence by ____________ postulate.
Note:
Postulate – Statement which is taken to be true without proof.
Theorem – Statement that can be demonstrated to be true by accepted mathematical operations and arguments.
4 cm
2.15 cm
4 cm
3.2 cm
32523252
K
L M ZX
Y
5 cm 5 cm 10 cm 10 cm
Common Core Math 2 Unit 1A Modeling with Geometry
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HW SSS, SAS, ASA, AAS, and HL Congruence (©2011 Kuta Software LLC. All rights reserved.)
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Math 2 Unit 1A Modeling with Geometry
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Isosceles Triangle Investigation 1. In the box, draw an angle and label the
vertex C. This will be your vertex angle.
Measure C.
2. Using point C as center, swing an arc that
intersects both sides of C
3. Label the points of intersection A and B. Construct side AB. You have constructed isosceles ΔABC with base AB.
4. Measure sides AC and BC. What is the relationship between AC and BC?
5. Use your protractor to measure the base
angles (A and B) of isosceles ΔABC.
6. Compare your results with the rest of the class. What relationship do you notice about the base angles of each isosceles triangle?
Midsegment investigation
1. Draw triangle JKL with points J (-2, 3), K (4, 5) and L (6, -1)
2. Find the midpoint M on JL and N on KL using the midpoint
formula.
3. Find the slope of lines JK and MN. What is the relationship
of the slopes? What does this prove about JK and MN?
4. Find the length of JK and MN using the distance formula.
What is the relationship between JK and MN?
5. What do you think you can generalize about the midsegment of a triangle?
Triangle Sum Theorem investigation
Common Core Math 2 Unit 1A Modeling with Geometry
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1. Using a protractor, measure each of the interior angles in both triangles. Record your findings below:
mMBP = _______ mAGO = _______
mBPM = _______ mGOA = _______
mPMB = _______ mOAG = _______
2. What is the sum of the three interior angles of each triangle?
3. What can you generalize about all triangles?
Isoceles Base Angle Theorem
Isosceles Base Angle Theorem Converse
Triangle Midsegment Theorem
Triangle Angle Sum Theorem
Common Core Math 2 Unit 1A Modeling with Geometry
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HW Triangle Sum Theorem, Isosceles Triangle Theorem & Converse, Midsegments Find the values of the variables. You must show all work to receive full credit. Figures are not drawn to scale. 1. x = _____ y = _____ z = _____ 2. x = _____ 3. x = _____ 4. x = _____ 5. x = _____ 6. x = _____ 7. x = _____ y = _____ 8. x = _____ y = _____
9. Find the perimeter of ABC.
x y
z 10
20
8
x 6
6
4
4
13
10
x
5
x + 1
2x
31 5
60°
x
6x + 11
x + 23 3y - 9
2y + 6
3x - 6
x 2x + 1
y
26
12
23 A
B
C
Common Core Math 2 Unit 1A Modeling with Geometry
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50
x
x 54
63
11
10x
2x + 75x - 8
4040
F G
4x - 6
18
16
F
G H
x
98
x
74
12
10
10
A D
E
B C
(10x+6)
(5x -10)
(4x + 20)
10. One side of the Rock and Roll Hall of Fame is an isosceles triangle made up of smaller triangles based on
mid-segments. The length of the base of the building is 229.5 feet. What would the base of the bold
triangle be?
Find the value of x. 11. 12. 13. 14. x = ________ x = ________ x = ________ x = ________ 15. 16. 17. 18. x = ________ x = ________ x = ________ x = ________ 19. 20. 21. 22. x = ________ x = ________ x = ________ x = ________ Complete the following using the diagram to the right.
23. a. If 𝐸𝐴̅̅ ̅̅ ≅ 𝐸𝐷̅̅ ̅̅ , then _______ _______.
b. If 𝐸𝐵̅̅ ̅̅ ≅ 𝐸𝐶̅̅̅̅ , then _______ _______.
c. If ΔEAD is an isosceles right triangle with right angle AED, then the measure of A is ________.
229.5 ft
10x+5 7x+20
x
Common Core Math 2 Unit 1A Modeling with Geometry
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Corresponding Parts of Congruent Triangles are Congruent
_________________ is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are
Congruent.”
SSS, SAS, ASA, AAS, and HL use corresponding parts to prove ______________ are congruent.
CPCTC uses congruent triangles to prove ______________________ ____________ are congruent.
Examples: A and B are on the edges of a ravine. What is AB?
Given: YW bisects XZ, XY YZ
Prove: XYW ZYW
Given: PR bisects QPS and QRS.
Prove: PQ PS
Given: NO || MP, N P Prove: MN || OP
Given: J is the midpoint of KM and NL. Prove: KL || MN
Given: Isosceles ∆PQR, base QR, PA PB
Prove: AR BQ
Z
Common Core Math 2 Unit 1A Modeling with Geometry
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T
CB
A
C
Y
X
B
A
C
A
B
D
E
Unit 1 HW: CPCTC I. Draw and label a diagram. Then solve for the variable and the missing measure or length.
1. If BATDOG, and mB=14, mG=29, and mO=10x+7 find the value of x and mO. x = ___________
mO = _________
2. If COWPIG, and CO=25, CW=18, IG=23 and PG=7x-17 find the value of x and PG.
x = ___________
PG=___________
3. If DEFPQR and DE=3x-10, QR=4x-23, and PQ=2x+7, find the value of x and EF.
x = ___________
EF = __________
II. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are
able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in the blanks.
4. EFD_________________ 5. ABC_________________ 6. LKM _________________ by ________ by ________ by ________
7. ABC_________________ 8. ABC_________________
by ________ by ________
H
G
E
FD
J
MK
L
Common Core Math 2 Unit 1A Modeling with Geometry
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A
B
CD
G
E
F
B
A
43
21
D
C
B
A
43
21
D
C
B
A
III. Use the given information to mark the diagram and any additional congruence you can determine from the
diagram. Then complete the triangle congruence statement and give the reason for triangle congruence. 9. 10.
Given: 1 3, 2 4 Given: ABD CBD, ADB CDB
ABC_________________ by __________ ABD_________________ by __________ 11. 12.
Given: G is the midpoint of FB and EA Given: 1 3, CDAB
ABG_________________ by __________ ABC_________________ by __________ IV. Use the given information and triangle congruence statement to complete the following.
13. ABC GEO, AB = 4, BC = 6, and AC = 8. What is the length of GO? How do you know?
14. BAD LUK, mD=52°, mB=48° and mA=80°.
a. What is the largest angle of LUK?
b. What is the smallest angle of LUK?
15. HOT SUN. SUN is isosceles. Is there enough information to determine if HOT is isosceles? Explain why
or why not.
Common Core Math 2 Unit 1A Modeling with Geometry
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Cross-Sections of Solids The picture above shows the cross-section created when a knife slices an apple. A cross-section of a solid is formed when a plane passes through the solid. Below are some examples of cross-sections that can be used in various applications to explain the internal components of real-world solids. We will examine cross-sections of geometric solids. Notice the 2 cross-sections formed using a cylinder below.
If the plane is parallel to the bases, the cross-section is a circle.
If the plane passes through the cylinder vertically as shown, the cross-section is a rectangle.
Determine the shape of the cross section for each of the figures below under the given condition. Be as specific as possible. 1. Triangular Prism
a. Plane passes through the interior vertically parallel to the triangular base
b. Plane passes through the interior horizontally
2. Square Prism (not a cube and not sitting on the square base … See it pictured here -->)
a. Plane passes vertically through the center of the interior of the prism so that it is
parallel to the square base
b. Plane passes vertically through the center of the interior of the prism so that it is
not parallel to the base
c. Plane passes horizontally through the center of the interior of the prism
Source: http://images.tutorvista.com/cms/images/67/cylinder-cross-section.png
Common Core Math 2 Unit 1A Modeling with Geometry
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3. Square Pyramid
a. Plane passes horizontally through the pyramid
b. Plane passes vertically through the pyramid so that it passes through the center
c. Plane passes diagonally through the pyramid so that it is parallel to one of the sides
4. Cylinder
a. Plane passes horizontally through the cylinder
b. Plane passes diagonally through the cylinder (angled slightly from horizontal)
5. Sphere
a. Plane passes horizontally through the center
b. Plane passes horizontally through the sphere, but not through the center
c. Plane passes vertically through the center
d. Plane passes vertically through the sphere, but not through the center
e. Plane passes diagonally through the sphere
6. Which of the cross-sections in #5 above are guaranteed to be congruent?
From Core-Plus Mathematics Course I: p. 453 (Unit 6) 7.
(Additional problems can be found at http://www.glencoe.com/sites/pdfs/impact_math/ls1_c2_cross_sections.pdf)
Common Core Math 2 Unit 1A Modeling with Geometry
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HW: Cross-sections (pictures are not to scale) 1) In the figure, the shaded region is a planar cross-section of the
rectangular solid. What is the area of the cross-section to the nearest inch?
2) A right circular cone with a diameter of base 8 centimeters and height 12 centimeters is shown. What is the radius of the cross-seciton that occurs 6 centimeters from the vertex, parallel to the base?
3) In the figure, the shaded areas show the intersection of two planar cross-sections that divide the large rectangular solid into four identical retangular solids. IF the four smaller rectangular solids formed by the cross-section are pulled apart, what is the total surface area of the four solids?
4) The shaded area in the figure below is a planar cross-section of the pyramid. The pyramid’sedges are all 16 centimeters long and the base of the pyramid is a square.
5) In the figure, the shaded area is a planar cross-section that is perpendicular to the bases of the right cylinder. What is the area of the cross-section?
Common Core Math 2 Unit 1A Modeling with Geometry
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Partitioning a Line Segment
Example 1) Segment AB has coordinates A(-7,4) and B(8,9). Find the coordinates of a point C that divides
segment AB in a ratio of 3:2.
a. Draw 𝐴𝐵 with point 𝐶 between points 𝐴 and 𝐵. Label the coordinates of 𝐴, 𝐵, and 𝐶, and label the lengths
of AC and CB.
b. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑥 direction.
c. Solve your proportion to find the 𝑥 coordinate of 𝐶.
d. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑦 direction.
e. Solve your proportion to find the 𝑦 coordinate of 𝐶.
f. Write 𝐶 as an ordered pair.
g. Check your answer by using the distance formula to verify that AC:CB is 3:2.
Common Core Math 2 Unit 1A Modeling with Geometry
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Example 2) Find point C on 𝐴𝐵 such that 𝐴𝐶: 𝐶𝐵 = 2: 1, when 𝐴 = (1, 1) and 𝐵 = (4, 1).
a. Draw 𝐴𝐵 with 𝐶 between 𝐴 and 𝐵. Label the coordinates of 𝐴, 𝐵, and 𝐶, and label the lengths of AB and BC.
b. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑥 direction.
c. Solve your proportion to find the 𝑥 coordinate of 𝐶.
d. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑦 direction.
e. Solve your proportion to find the 𝑦 coordinate of 𝐶.
f. Write 𝐶 as an ordered pair.
g. Check your answer by using the distance formula to verify that 𝐴𝐶 is twice as long as 𝐶𝐵.
Common Core Math 2 Unit 1A Modeling with Geometry
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HW: Partitioning a Line Segment
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Math 2 Unit 1A Modeling with Geometry
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Unit Review
1. Given BED BUG, find x and BU if BE = 6x+16, UG=10x-2 and ED = 6x+6.
x=__________ BU=________________
2. In ∆ABC, Ab, AC = 6x-5, BC = 3x+13, and AB = 4x+7. Find x and the length of the base.
x ___________
base = ___________
3. In an isosceles triangle, a vertex angle measures 36⁰. What is the measure of each base angle?
____________ 4. Solve for the value of x in parts a and b. Solve for the measure of angle 1 in part c. a. b. c. 5. Decide whether it is possible to prove the triangles are congruent. If yes, then state which congruence
postulate you would use.
a. b. c. d.
6. If ABCPQR and C=2x+2, R=3x-18, find the value of x.
7. ∆XYZ ∆JKL. IfY=14-x, K=2x+50 and L=-4x find the Zm .
Common Core Math 2 Unit 1A Modeling with Geometry
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8. IfABCLMN, and AB=4x-y, LM =2x-2y, BC = x-37 and MN =21 find the value of x and y. 9. ∆PQR ∆XYZ. List three pairs of congruent angles.
____________________ _____________________ ____________________ 10. Suppose ∆ABC ∆EFG. For each of the following, name the corresponding part.
A : ______________ BCA : ______________ AC : _______________
11. In ∆EFG, E = 6x – 8, F = 7x + 3, and G = 3x – 7. Find the measure of each angle.
12. Two angles of a triangle measure 18 o and 77 o. Find the measure of the third angle.
A. 19o B. 36 o C. 59 o D. 64 o E. 85 o
13.
A. PMN
B. NPM
C. NMP
D. MNP
14. In the given triangles, ABCXYZ Which two statements identify corresponding congruent parts for these triangles?
A. AB XY and CY
B. AB YZ and CX
C. BC XY and AY
D. BC YZ and AX
15. BOY DEA. YB = 4x+8, AD = 60, AE = 80, AND DE = 40. Find x.
16. EXC STP. C = 3x+12, T = 81, and X = 12x – 15 . Find m C .
Common Core Math 2 Unit 1A Modeling with Geometry
36
50
75
40
C
A
B
Y
Z
67.545
B
A
C Y
X
Z
12
18
9
6
C
B
A
E
D
105
3 6F
G
R
N
16
128
10
15
20
A
B C M
U
K
17. MNODEF. Calculate the value for x, y, and z. 𝑚∠𝑀 = 50 x = ________ 𝑚∠𝑁 = 60 𝑚∠𝑂 = 70 y = ________ 𝑚∠𝐷 = (2𝑥 − 20)
𝑚∠𝐸 = (1
2𝑦 + 10) z = ________
𝑚∠𝐹 = (10 + 𝑧)
18. The figure shows . ABCEDC. C is the midpoint of BD and AE. What reason would you use to prove the triangles are congruent, if any?
If the triangles are congruent, complete the statement: A _____
19. If ∆JKM ∆RST, how do you know JK RS? A. Definition of a line segment C. SSS Postulate B. CPCTC D. SAS Postulate
20. ΔWHY ~ ΔGEO. mW = 55, mE = 60. Find mY. 21. ΔMAD ~ ΔCOW. MA = 8, AD = 5, CO = 6. Find the scale factor of ΔMAD to ΔCOW and the length of OW.
Scale factor = ________ OW = _________
22. If two triangles are congruent, what is the scale factor? Scale factor = ____ : ____ Determine whether the following triangles are similar. If so, complete the similarity statement and a state a specific reason. If not, write “not similar”. (Figures are not drawn to scale!) 23. 24. 25. 26. 27. 28.
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Homework Answers:
Similar Triangles: Page 1) Not similar
2) Similar, SSS, FGH
3) Similar, SAS, VLM
4) Similar, AA, TUV 5) Not similar
6) Not similar
7) Similar SSS, QRS 8) Not similar
9) Similar, AA, HTS
10) Similar, SAS, UVW
11) Similar, SSS, FRS
12) Similar, SAS, CDE 13) 22 14) 54 15) 9
16) 11 17) 8 18) 9 19) 7 20) 11
Congruency and Rigid Motion: Pages
1 a) Translation: (x, y) (x + 6, y – 7) b) A translation is a rigid motion which preserves
length, thus creating a figure with the same size and shape as the pre-image.
2 b) MA = M’A’ = √20
AP = A’P’ = √17
MP = M’P’ = √45 c) A counterclockwise rotation of 180° 3 b) Reflection across the line y = x 4 b) Yes. A reflection is a rigid motion which
preserves length, thus creating a figure with the same size and shape as the pre-image.
d) Yes. A translation is a rigid motion which preserves length, thus creating a figure with the same size and shape as the pre-image.
e) Yes. Both triangles are congruent to RST, therefore by the transitive property of congruence, they are congruent to each other. Or rigid motion congruence is transitive.
5 a) Counterclockwise rotation of 90o around the origin. b) Yes. A rotation is a rigid motion which preserves
length, thus creating a figure with the same size and shape as the pre-image.
6 b) A dilation centered at (0,0) with a scale factor 2. c) While these triangles are the same shape, they
are not the same size. A dilation is NOT a rigid motion and does not preserve the congruency of figures. These triangles are similar, but not congruent.
SSS, SAS, ASA, AAS, and HL Congruence: Pages 1) Not congruent 2) HL 3) Not congruent 4) SAS 5) SSS 6) SAS
7) SAS 8) AAS 9) HL 10) ASA 11) ASA 12) ASA
13) SSS 14) Not congruent 15) AAS 16) SAS
17) I C
18) SQR CRQ
19) UVW EWV or
VWU WVE
20) 𝑌𝑋̅̅ ̅̅ 𝑌𝐾̅̅ ̅̅ 21) G T
22) 𝐸𝐶̅̅̅̅ 𝑁𝐿̅̅ ̅̅
Triangle Sum Theorem, Isosceles Triangle Theorem & Converse, Midsegments: Pages 1) x = 8 y = 10 z =
10 2) x = 6.5 3) x = 20 4) x = 9 5) x = 31 6) x = 10
7) x = 35/4 y = 15 8) x = 3 y = 7 9) 36.5 10) 57.275 11) 80 12) 45 13) 53
14) 11 15) 5 16) 6 17) 41 18) 8 19) 6 20) 5
21) 55 22) 8.63
23) a D A_.
b. ECB EBC. c. 45°.
Common Core Math 2 Unit 1A Modeling with Geometry
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CPCTC and Naming Congruent Triangles: Pages 1) x = 13
𝑚∠𝑂= 137 deg 2) x = 5
PG= 18 3) x = 17
EF = 45
4) GHD; SSS
5) TBC; ASA
6) JKM; SSS, ASA or SAS
7) not congruent
8) EDC; AAS
9) DCB; ASA
10) CBD; ASA
11) EFG; SAS
12) DCB; SAS 13) GO = 8
corresponds to AC
14) a) U
b) L
15) Yes, they are congruent, so
HOT has the exact same size and shape and
SUN.
Cross-sections: Page 1) 3,225 in2 2) 2 cm 3) 1,536 cm2 4) 54.7 cm2 5) 720 cm2 Partitioning a Line Segment: Pages 1) (1, 5) 2) (1, 2)
3) (1
5, −
8
5)
4) (−1, −1
3)
(1,4
3)
5) (13, 12)
6) (7,5) 7) (4,10) 8) (3,13) 9) (7,11)
Test Review: Pages 1) x=2, BU=28 2) x=6, base=31 3) 72 4) a) -15
b) 1 c) 74°
5) a) SSS b) Not c) SSS d) SAS
6) 20 7) -48 8) x=3, y=-6
9) PX
QY
RZ 10) E, FGE, EG 11) 64, 87, 29 12) E
13) D 14) D 15) 13 16) 36 17) x=35
y=100 z=60
18) SAS, E 19) B 20) 65
21) 4:3 15/4
22) 1:1 23) PPP 24) AA 25) Not 26) PAP 27) Not 28) PAP
Common Core Math 2 Unit 1A Modeling with Geometry
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Common Core Math 2 Unit 1A Modeling with Geometry
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