Download Secondary II - Cache County School District

Cache County School District 2013-2014
Secondary II
Utah Integrated
Mathematics Core
Student Edition - Honors
Unit 7:
Geometric
Reasoning and
Proof
Secondary II Unit 7 – Geometric Reasoning and Proof: Table of
Contents
Homework Help (QR Codes, links) ...................................................................................
Vocabulary List …………………………………………………………………………...
Section 7.1 – Points, Lines, and Distance, Review Teacher Notes ......................................
Notes, Assignment ......................................................................................................
Section 7.2 – Angles and Other Vocabulary, Review Teacher Notes ..................................
Notes, Assignment .......................................................................................................
Lesson 7.1&7.2 Summary Review/Task.....................…………………………………...
Assignment…………………………………………………………………………..
Measuring and Drawing Angles Review/Task…………………………………………..
Assignment…………………………………………………………………………..
Section 7.3 – Angles and Transversals Part 1, Teacher Notes ............................................
Notes, Assignment .......................................................................................................
Section 7.4 – Angles and Transversals Part 2 Task, Teacher Notes ....................................
Notes, Assignment .......................................................................................................
City Project (High Level Task)……………………………………………………………
Section 7.5 – Introduction to Proof Task, Teacher Notes ....................................................
Notes, Assignment .......................................................................................................
Unit 7 Enrichment Lesson- Teacher Notes, Notes, Assignment …………………………
Using Geometric Relationships……………………………………………………..
Section 7.6 – Classifying Triangles Task, Teacher Notes ....................................................
Notes, Assignment .......................................................................................................
Section 7.7 – Isosceles Triangle Task, Teacher Notes .........................................................
Notes, Assignment .......................................................................................................
Section 7.8 – Exterior Angle Theorem Task, Teacher Notes ...............................................
Task, Assignment.........................................................................................................
Section 7.9 – Mid-Segment of a Triangle Task, Teacher Notes ...........................................
Notes, Assignment .......................................................................................................
Section 7.9b –Medians of a Triangle Task, Teacher Notes, Notes (No Assignment)……..
Section 7.10 – Parallelograms Task, Teacher Notes,
Notes, Assignment .......................................................................................................
Section 7.11 – Rectangles, Rhombi, and Squares Task, Teacher Notes……………..……
Notes, Assignment
Pretzel Activity…… ………………………………………………………………………
Angle Relationships and Equations Matching Activity…………………………………
Secondary II Unit 7 – Geometric Reasoning and Proof:
Homework Helps
Section 7.1
Review Assignment. No additional resources. Google particular topics for help or seek help
from your teacher. Go to www.cachemath2.wordpress.com for selected homework solutions.
Section 7.2
Review Assignment. No additional resources. Google particular topics for help or seek help
from your teacher. Go to www.cachemath2.wordpress.com for selected homework solutions.
Section 7.3
http://goo.gl/V2Y5y
http://goo.gl/01HhY
http://goo.gl/BSgRf
Section 7.4
Video
http://goo.gl/2Sdg4
http://goo.gl/PA6bT
http://goo.gl/hNeUc
Section 7.5
http://goo.gl/Z1yu2
http://goo.gl/1Y8iZ
Section 7.6
http://goo.gl/h9yQb
http://goo.gl/wrJ3O
http://goo.gl/6mekD
Section 7.7
http://goo.gl/0XmDL
http://goo.gl/D7lWK
http://goo.gl/lYqWN
Section 7.8
http://goo.gl/JeOim
Video – scroll through to 20 sec.
http://goo.gl/kfUPu
Video
Section 7.9
http://goo.gl/xb8OU
http://goo.gl/mcllS
http://goo.gl/dVytQ
http://goo.gl/dt6w6
Section 7.10
Resources
http://goo.gl/oxPhb
http://goo.gl/bdv3Z
http://goo.gl/YzMcb
Section 7.11
http://goo.gl/o5dYP
http://goo.gl/Zmgg
http://goo.gl/u4qZV
Chapter 7 – Geometric Reasoning Vocabulary List
Name______________________________
Date ______Hour _____________
Term
Point
Line
Collinear
Plane
Coplanar
Line Segment
Definition (in your own words)
Sketch/Written
Example(when sketch
not possible)
Congruent
Midpoint
Segment
Bisector
Degree
Ray
Opposite Rays
Angle
Sides
Vertex
Interior
Exterior
Right Angle
Acute Angle
Obtuse Angle
Angle Bisector
Adjacent
Angles
Vertical Angles
Linear Pair
Complementary
Angles
Supplementary
Angles
Perpendicular
Transversal
Alternate
Interior
Angles
Alternate
Exterior Angles
Consecutive
interior Angles
Corresponding
Angles
Exterior Angle
Theorem
Parallelogram
Unit 7 Lesson 1 – Points, Lines, and Angles
Notes 7.1
Part 1
What is a point?
How should you sketch a point?
How do you label a
point?
What is a line?
How should you sketch a line?
How should you label a
line?
How should you sketch a plane?
How should you label a
plane?
Collinear Points –
What is a plane?
Examples of a plane:
What is a coplanar object? (Lines, points, etc)
Concept Check:
Name a point that is collinear with the other points.
1.
M and B
2.
C and E
3.
A and F
4.
E and J
5.
H and B
6.
K and G
7.
D and J
Notes on Intersection of lines, planes, etc:
A

B

C
M
H
D

E
G
K
J
F
Why can’t a line be “measured”?
Why can a line segment be measured?
What is a line segment? How do you sketch it? How is it labeled?
Example 1: (Finding Measurements)
̅̅̅̅
a) Find the measurement of AC
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A
B
13.6 cm
12.5 cm
̅̅̅̅
Find the measurement of QR
b)
M at h Com poser 1. 1. 5
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C
Q
4.0 ft
R
1.2 ft
S
In geometry you often times have to draw a picture or diagram to answer a question.
c)
Find the value of the variable and ST if S is between R and T
RS = 12, ST = 2x, RT = 34
d)
Find the value of the variable and LM if L is between N and M
NL = 5x, LM = 3x, NL = 15
Vocabulary
congruent –
Congruent segments –
Example 2 (Determine whether segments are congruent)
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M at h Com poser 1. 1. 5
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DC, CD
Determine whether each pair of segments is congruent
D
DE, CD
6 in
5 in
EC, ED
CE, DC
E
6 in
Part 2
Distance Formula –
Example 3:
a) Find the distance between X(7, 11)and Y(−1, 5)
b) Find the distance between D(2, 0)and E(8, 6)
C
c) Find the distance between the points (–2, –3) and (–4, 4).
Midpoint formula –
Example 4:
a) Find the coordinates of the midpoint of a segment having X(−4, 3)and Y(0, 1)as endpoints.
b) With endpoints A(8, 6)and B(10, 4)
c) With endpoints D(−9, 5)and E(−7, 4)
Example 5
̅̅ if Y is the midpoint of ̅XZ
̅̅̅?
What is the measure of ̅̅
YZ
M at h Com poser 1. 1. 5
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5x+4
X
3x-2
Y
Z
Unit 7 Lesson 1 – Points, Lines, and Distance
Ready, Set, Go! – Assignment 7.1
Name______________________________
Date_________ Hour_______
Ready
For 1 – 5, use the diagram to determine if each statement is TRUE or FALSE. JUSTIFY your answer.
1.
Point A lies on line m.
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2.
3.
B, C and D are collinear.
A
B
l
F
D
A, B and F are coplanar.
E
C
n
4.
5.
A, B, C, and D are collinear.
⃗⃗⃗⃗⃗
𝐶𝐷 𝑎𝑛𝑑 ⃗⃗⃗⃗⃗
𝐶𝐸 𝑎𝑟𝑒 𝑐𝑜𝑝𝑙𝑎𝑛𝑎𝑟.
For 6 – 9, name a point that is COLLINEAR with the given points.
6.
E and D
7.
C and A
8.
D and B
9.
B and G
m
Set Note: Figures may not be drawn to scale.
If AC=58, find the value of each of the following (10-12)
10.
x ________________
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3x-4
11.
3x-4
B
A
AB _______________
C
12. BC ________________
If GJ=32 find the value of each of the following. (13-15)
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13.
𝑥 __________________
14.
GH ________________
15.
HJ ________________
G
3x
x+16
J
H
In exercises 16-18, use the figure at the right to find PT.
Note: Recall slashes on a figure indicate which segments are congruent.
16. 𝑃𝑇 = 5𝑥 + 3 𝑎𝑛𝑑 𝑇𝑄 = 7𝑥 − 9
M at h Com poser 1. 1. 5
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P
17.
𝑃𝑇 = 4𝑥 − 6 𝑎𝑛𝑑 𝑇𝑄 = 3𝑥 + 4
18. 𝑃𝑇 = 7𝑥 − 24 𝑎𝑛𝑑 𝑇𝑄 = 6𝑥 − 2
T
Q
Go!
Find the coordinates of the midpoint of the segment with the given endpoints.
19.
𝑆(4, −1) 𝑎𝑛𝑑 𝑇(6, 0)
20.
𝐿(4, 2) 𝑎𝑛𝑑 𝑃(0, 2)
21.
𝐻(−5, 5)𝑎𝑛𝑑 𝐼(7, 3)
22.
𝐺(−2, 8) 𝑎𝑛𝑑 𝐻(−3, −12)
Use the given endpoint R and midpoint M of RS to find the coordinates of the other endpoints.
23.
𝑅(6, 0) , 𝑀(0,2)
24.
𝑅 (3, 4), 𝑀(3, −2)
25.
𝑅(−3, −2), 𝑀(−1, −8)
26.
𝑅(11, −5), 𝑀(−4, −4)
The endpoints of two segments are given. Find each segment length. Tell whether the segments are
congruent.
̅̅̅̅
27.
𝐴𝐵 : 𝐴(2, 6), 𝐵(0, 3)
̅̅̅̅: 𝐶(−1, 0), 𝐷(1, 3)
𝐶𝐷
28.
̅̅̅̅: 𝑅(5, 4), 𝑆(0,4)
𝑅𝑆
̅̅̅̅ : 𝑇(−4, 3), 𝑈(−1, 1)
𝑇𝑈
29.
̅̅̅̅: 𝐾(−4, 13), 𝐿(−1, −11)
𝐾𝐿
̅̅̅̅̅
𝑀𝑁: 𝑀 (−1, −2), 𝑁(−1, −11)
30.
̅̅̅̅
𝑂𝑃: 𝑂(6, −2), 𝑃(3, −2)
̅̅̅̅: 𝑄(5, 2), 𝑅(1, 5)
𝑄𝑅
Unit 7 Lesson 2 – Angles, and Other Vocabulary
Notes 7.2
Degree
Ray
Opposite Rays
Angle
How can you label/name an angle?
Acute Angles
Right Angles
Obtuse Angles
Straight Angles
Congruent Angles
Angle Bisector –
Example 1:
M at h Com poser 1. 1. 5
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AB bisects CAD the measure of CAB is 56 find the measure of BAD.
M at h Com poser 1. 1. 5
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C
A
B
D
Example 2: Using the same figure….
M at h Com poser 1. 1. 5
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AB bisects CAD the measure of CAD is 100 find the measure of BAD.
Complementary Angles
Supplementary Angles
Vertical Angles
Linear Pair
All angle relationships from both Lesson 1 and Lesson 2 will be present in these examples.
Example 3
Always ask….. What is the angle relationship in this picture?
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4x+25
6x-10
Example 4
What are the angle relationships in this picture?
Find “x” and “y”
M at h Com poser 1. 1. 5
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38
65
x
y
Example 5
What is the angle relationship in this diagram?
Find x, and then find the measures of BOTH angles
M at h Com poser 1. 1. 5
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4x+30 x+15
Find “x” and state each angle measure.
Example 6:
What angle relationship is present in this picture?
a) Find x
M at h Com poser 1. 1. 5
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3x+29
x+11
b) Find the measure of angle 2.
Example 7: Find…
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4 1
3 48º
M at h Com poser 1. 1. 5
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m 1=
m 3=
m 4=
Example 8: Two angles are complementary. One angle is 15 degrees less than the other. Find both
angles.
Example 9:
Find “x” and state the measure of BOTH angles.
M at h Com poser 1. 1. 5
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x
5x
Unit 7 Lesson 2 – Angles, and Vocabulary
Ready, Set, Go! – Assignment 7.2
Name______________________________
Date_________ Hour_______
Ready
Name each angle in 4 ways.
1.
2.
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D
M at h Com poser 1. 1. 5
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H
2
4
I
E
J
C
3.
4.
M at h Com poser 1. 1. 5
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M at h Com poser 1. 1. 5
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C
Q
B
4
1
R
A
C
Name all the angles that have V as a vertex.
5.
6.
M at h Com poser 1. 1. 5
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M at h Com poser 1. 1. 5
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H
V
E
I
J
F
1
V
G
7.
8.
M at h Com poser 1. 1. 5
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M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
Z
V
A
C
B
Y
V
X
D
Set
9.
10.
Angles A and B are complementary angles, A  48, Find mB.
Angles C and D are supplementary angles, mC  78, Find mD.
Use the following figure to answer questions # 11-14.
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1
2
3
5
4
11.
Angles 1 and 2 are ______________________ angles.
12.
Angles 3 and 5 are ______________________ angles.
13.
Which two angles are adjacent AND complementary?
14.
Which two pairs of angles are supplementary?
15.
Find “x”
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6x + 10
4x + 20
16.
Find “x”. Given that Angle 1 and Angle 2 are a linear pair.
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6x + 12
1
2
4x + 24
17.
Angles A and B are _____________________________ angles. A  53, find mB.
M at h Com poser 1. 1. 5
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A
B
For #18-20 draw the angle pairs that are described:
18. Draw two acute
vertical angles
19. Draw two adjacent
angles.
20. Draw two perpendicular
angles.
Determine whether each statement is sometimes, always, or never true:
21.
If two angles are supplementary and one is acute, the other MUST be obtuse.
22.
If two angles are complementary, they are both acute angles.
23.
If two angles are a linear pair, then they add up to 180 degrees.
24.
If two angles are a linear pair, then they are vertical angles.
25.
If two angles are equal, then they are adjacent angles.
Go!
26.
Two complementary angles must add to what degree measure?
27.
Two supplementary angles must add to what degree measure?
28.
What is the complement of a 30 degree angle?
29.
If two angles are supplementary and one of the angles is 54 degrees, then what is the other angle
measure?
30.
If mABD  90, find mDBC
If m1  30, find m2 .
31.
M at h Com poser 1. 1. 5
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M at h Com poser 1. 1. 5
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A
D
1
B
2
C
Use the following figure for # 32 – 34. 𝑚∠1 = 45°
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2
1
4
32.
5
𝐹𝑖𝑛𝑑 𝑚∠3
34.
True or False : 2 and 4 are vertical angles
35.
True or False : 2 and 3 are a linear pair.
3
33.
.
Find m2.
36.
Find both angles measures
(not just “x”)
37.
Find x and y
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M at h Com poser 1. 1. 5
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x
38.
2x
5x
mABD  87, find mDBC.
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A
D
B
C
39.
Find x and state the measure of BOTH angles.
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3x+7 4x-27
40.
If m1  x and m2  x  17 the measure of BOTH angles
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1 2
41.
If m2  67, find m1.
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1 2
42.
Find all of the angle measures if 1  2  3
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1
2 3
5 4
7y
40 º
Unit 7 Lesson 1&2
Summary Task/Assignment
Name______________________________
Date_________ Hour_______
Ready:
1. Using the following illustration:
a) Identify two acute vertical angles.
A
C
B
b) Identify two obtuse adjacent angles.
D
F
E
Set:
2. Is the following statement always, sometimes or never true? Justify.
If two angles are supplementary and one of them is acute, then the other one is obtuse.
3. Is the following statement always, sometimes or never true? Justify.
If X is supplementary to Y and Y is supplementary to Z , then X is supplementary to Z .
4. Is the following statement always, sometimes or never true? Justify.
If AB  BC , then ABC is acute.
5. The measures of two complimentary angles are 16y + 13 and 4y – 3. Find the measure of the two
angles.
6. The measure of an angles supplement is 32 less than the measure of the angle. Find the measure of
the angle and its supplement.
7. In the following picture, what value of x guarantees that ABD and DBC form a linear pair?
D
(2x - 5)
A
(8x + 15)
C
B
11. Given that m1  32 , and mR  90 find the measure of the other seven numbered angles and be
sure to label each one in your answer.
R
2
3
1
4
5
6
8 7
12. Two angles are supplementary. One angle measures 32 more than the other. What are the two
angles?
13. Two angles form a linear pair. The measure of one of the angles is twenty less than three times the
angle. What are the two angles?
Measuring and Drawing Angles Review
Summary Task/Assignment
Name______________________________
Date_________ Hour_______
1.
Estimate the size of each angle, then measure it with a protractor.
a.
M at h Com poser 1. 1. 5
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Estimate_______________
Actual Measure______________
b.
M at h Com poser 1. 1. 5
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Estimate_______________
Actual Measure______________
c.
Estimate_______________
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
Actual Measure______________
d.
Estimate_______________
M at h Com poser 1. 1. 5
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Actual Measure______________
e.
M at h Com poser 1. 1. 5
ht t p: / / www. m at hcom poser . com
Estimate_______________
Actual Measure______________
2. Draw angles with the following sizes. The entire page below is blank for you to use. Label each of
your answers. If you need more room, use another piece of paper.
a.
b.
50°
e.
c.
70°
f.
100°
d.
82°
g.
175°
42°
h.
160°
143°
Use your protractor and extend the lines to measure each angle in degrees. Be as exact as possible.
3.
4.
5.
6.
7.
8.
8.
9.
10.
Using the diagram, tell whether the angles are vertical angles, a linear pair, or neither.
a.
∠1 and ∠2
13.
∠1 and ∠3
b.
∠1 and ∠4
15.
∠1 and ∠5
c.
∠1 and ∠6
17.
∠1 and ∠7
d.
∠1 and ∠8
19.
∠2 and ∠4
11. ∠ 1 and ∠ 2 are complementary angles. Given the measure of ∠1, find m∠2.
a. m∠1=52 , m∠2 =
b. m∠1=76 , m∠2 =
c. m∠1=19 , m∠2 =
12. ∠ 1 and ∠ 2 are supplementary angles. Given the measure of ∠1, find m∠2.
a. m∠1=52 , m∠2 =
b. m∠1=76 , m∠2 =
c. m∠1=19 , m∠2 =
.
Unit 7 Lesson 3 – Angles and Transversals Part 1
Notes 7.3
Examples
Give two examples of each angle pair in the picture provided to the right.
1. Same-side interior angles (consecutive interior)
2. Alternate exterior angles
3. Corresponding angles
4. Alternate interior angles
5. Linear pair
Identify the transversal and classify each angle pair.
5. ∠ 2 and ∠ 3
6. ∠ 4 and ∠ 5
7. ∠ 2 and ∠ 4
8. ∠ 1 and ∠ 2
9.
Name a pair of alternate interior angles with transversal n.
10.
Name a pair of corresponding angles with transversal m.
11.
Identify the transversal for the angle pair ∠3 & ∠7.
12.
Identify and classify the angle pair for ∠1 & ∠6.
Unit 7 Lesson 3 – Angles and Transversals Part 1
Ready, Set, Go! - Assignment 7.3
Name______________________________
Date_________ Hour_______
http://goo.gl/dvL3C
Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, corresponding or
vertical.
Ready
1.  3 and  7
1
2.  1 and  5
2
3 4
3.  2 and  7
5
6
7
4.  2 and  3
8
5.  1 and  8
6.  3 and  6
7.  4 and  6
8.  4 and  5
Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, corresponding,
vertical, or none.
9.
∠1 𝑎𝑛𝑑 ∠5
10. ∠5 𝑎𝑛𝑑 ∠15
11. ∠1 𝑎𝑛𝑑 ∠4
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7 8
6 1
9 10
2 11
12. ∠5 𝑎𝑛𝑑 ∠6
13. ∠15 𝑎𝑛𝑑 ∠1
14. ∠12 𝑎𝑛𝑑 ∠14
5 4
16 15
3 12
14 13
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1. ∠2 𝑎𝑛𝑑 ∠10
7 8
6 1
16. ∠11 𝑎𝑛𝑑 ∠3
9 10
2 11
17. ∠16 𝑎𝑛𝑑 ∠3
18.
5 4
16 15
∠16 𝑎𝑛𝑑 ∠8
3 12
14 13
19. ∠10 𝑎𝑛𝑑 ∠14
20. ∠8 𝑎𝑛𝑑 ∠4
Set
Using the figures at the right answer questions 21-31. Classify each pair of angles as one of the
following:
a)
b)
c)
d)
e)
f)
g)
h)
alternate interior angles
consecutive interior angles
corresponding angles
alternate exterior angles
vertical angles
consecutive exterior
linear pair
none
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2
1 3
4
5 6
87
Use for problems # 21-26
9 10
13 14
11 12
15 16
Use for problems # 27-32
_______21. ∠1 & ∠3
_______22. ∠3 & ∠6
_______23. ∠3 & ∠5
_______24. ∠2 & ∠8
_______25. ∠4 & ∠7
_______26. ∠7 & ∠3
_______27. ∠9 & ∠12
_______28. ∠9 & ∠11
_______29. ∠15 & ∠11
_______30. ∠9 & ∠15
_______31. ∠14 & ∠15
_______32. ∠13 & ∠14
Classify each pair of angles as alternate interior, same-side interior, or corresponding angles.
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F
E
A
G
B
D
C
L
H
K
J
33.
∠EBA and ∠FCB _______________________________________
34.
∠DCH and ∠CBJ ________________________________________
35.
∠ FCB and ∠CBL ________________________________________
36.
∠FCL and ∠BLC ________________________________________
37.
∠HCB and ∠CBJ ________________________________________
38.
∠GCH and ∠GLJ _________________________________________
Unit 7 Lesson 4 – Angles and Transversals Part 2
Task 7.4
Name________________________________________
Date____________________ Hour_________
You will need a protractor and a ruler from your teacher to complete the following task.
Directions: Measure and label the degree of each angle in the following figures.
Figure 1:
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c
a
b
Figure 2:
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c
a
b
Questions:
1.
What is the transversal line in Figure 1? ____________ Figure 2?______________.
2.
What similarities and differences do you notice between Figure 1 and Figure 2. List a few.



3.
What do you notice about measurments the alternate interior angles in Figure 2?
3b.
Does this relationship hold true for the alterenate interior angles in Figure 1?
4.
What do you notice about the measurements of the alternate exterior angles in Figure 2?
4b.
Does this relationship hold true for the alternate exterior angles in Figure 1?
5.
What do you notice about the measurements of the pairs of vertical angles in Figure 2?
5b.
Does the relationship you found for vertical angles in Figure 2 hold true for Figure 1?
6.
What do you notice about the measurements of the corresponding angles in Figure 2?
6b.
Does that relationship hold true for the corresponding angles in Figure 1?
7.
What do you notice about the measurements of the consecutive interior angles in Figure 2?
7b.
Does that relationship hold true for the consecutive interior angles in Figure 1?
Summarize your findings from questions #1-7.
Unit 7 Lesson 4 – Angles and Transversals Part 2
Ready, Set, Go! - Assignment 7.4
Name______________________________
Date_________ Hour_______
http://goo.gl/u3yf2
Ready
State the transversal that forms each pair of angles. Then, identify the special name for the angle pair.
1. ∠10 & ∠13 transversal ___________ type
of
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n
m
angles_________________________
2. ∠6 & ∠15 transversal ___________ type of
angles_________________________
3. ∠4 & ∠10 transversal ___________ type of
angles________________________
1 2
3 4
9 10
11 12
56
7 8
13 14
15 16
4. ∠5 & ∠16 transversal ___________ type of
angles________________________
5. ∠2 & ∠6 transversal ___________ type of
angles________________________
6. ∠10 & ∠15 transversal ___________ type
of
angles________________________
7. Add the following pairs of angles and state whether they are complementary, supplementary, or
neither.
a. 20o, 70o
b. 30o, 150o
o
p
c. 110o, 140o
d. 47o, 43o
8. Find the measurement of the angle complementary to:
a. 30o
b. 5o
9. Find the measurement of the angle supplementary to:
a. 100o
b. 90o
10. Classify the following angle pairs as complementary, supplementary or neither:
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a. ∠BOC & ∠COD
C
B
b. ∠AOC & ∠COE
D
c. ∠COD & ∠DOE
A
d. ∠AOB & ∠BOE
O
E
Set
11. For the following picture with lines a and b being cut by a transversal, find all eight angles for the
given figure if 𝑎||𝑏.
a
1 2
4 3
6
5
8
7
b
a) Given 𝑚∠2 = 113𝑜 , find the measure
of all 8 angles.
b) Given 𝑚∠7 = 50𝑜 , find the measure
of all 8 angles.
𝑚∠1=
𝑚∠1=
𝑚∠2=
𝑚∠2=
𝑚∠3=
𝑚∠3=
𝑚∠4=
𝑚∠4=
𝑚∠5=
𝑚∠5=
𝑚∠6=
𝑚∠6=
𝑚∠7=
𝑚∠7=
𝑚∠8=
𝑚∠8=
Go!
12. Which of the following statements guarantee that lines m and n are parallel?
m
1 2
4 3
6
5
8
n
7
a) m1  42 and m5  42
b) m4  64 and m5  64
c) m3  118 and m6  62
d) m2  57 and m8  57
e) m1  42 and m7  138
f) m2  (3x  7) and m6  (3x  7)
g) m3  y and m7  (180  y)
h) m4  3 x and m5  (180  3x)
City Project Task
Name_________________________________
Date_________ Hour_______
Overview: City planners and designers must be able to accurately draw parallel and
perpendicular lines to create a city map.
Objective: Draw a city map using only a compass and straightedge that meets conditions
below.
Materials: Poster or blank paper, colored pencils, eraser, compass and straightedge.
Directions: Assume no two buildings can occupy the same space. Make your constructions
lines light so that they can be easily be erased. Draw a city with the following
conditions:
1. Use a straight edge to draw and label a street across your paper.
2. Draw and label a street that intersects the previous street drawn.
3. Construct and label three streets that are parallel to one of the streets you just drew.
4. Construct at least two transversal streets that are perpendicular to the parallel streets.
5. Sketch a house and a school on a pair of consecutive interior angles.
6. Sketch a bank and a post office on a pair of corresponding angles.
7. Sketch a grocery store and an electronic store on a pair of alternate interior angles.
8. Sketch a movie theater and a pet store on a pair of alternate exterior angles.
9. Sketch a water tower halfway between the bank and the post office.
10. Sketch a park exactly halfway between the grocery store and the school.
11. Sketch traffic lights on at least four intersections.
Unit 7 Lesson 5 – Introduction to Proof
Task 7.5
Name____________________________________
Date___________ Hour________
Directions: Use the diagram below to answer questions 1-3.
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l
m
A B
E F
C D
GH
J K
Q R
N P
T W
e
f
1.
Name all of the angles that would be congruent to ∠ 𝐶 if and only if lines e and f are parallel.
2.
Assume lines l and m are parallel, and lines e and f are parallel. List all of the angles that would
be congruent to ∠ 𝐾.
3.
Assume we do not know if any of the lines in the figure above are parallel. Determine which of
the following angle relationships would prove lines e and f are parallel. For each of the
problems, explain why or why.
a)
∠𝐴 ≅ ∠ 𝐽
b)
∠𝐴 ≅ ∠𝐶
c)
∠𝐷 ≅ ∠𝑇
d)
∠𝐺 ≅ ∠𝑃
e)
∠𝑁 ≅ ∠𝑊
f)
∠𝐴 + ∠𝑄 = 180°
e)
∠𝐾 + ∠𝑁 = 180°
Unit 7 Lesson 5 – Introduction to Proof
Notes 7.5
This lesson will focus on different kinds of proofs.
When we defined vertical angles, we mentioned that vertical angles are always congruent. We are now
going to prove this fact. Below is a diagram containing 4 angles, explain in words how you know that
∠1 ≅ ∠2. (without measuring the angles)
We will now organize the information above into what is called a two column proof. In this type of
proof, the first column consists of a series of mathematical statements and the second column
consists of the explanation as to why the each statement is true.
Statements
Explanation/Reasons
Given Information: ∠2 is a right angle and all segments that look straight are straight
Write a paragrah proving that ∠1 is an acute angle
B
C
A
1
2
D
F
E
Prove that ∠1 is acute in a two column proof.
Statements
Explanation/Reasons
Given Information: ∠2 is a right angle
B
m∠CFD=15°
C
m∠AFE=105°
∠AFD= is a straight angle
Prove that ∠BFE is a straight angle
Statements
Explanation/Reasons
Prove that ∠AFE and ∠1 are a linear pair
Statements
Explanation/Reasons
A
1
2
D
F
E
B
Given Information: FB and FE are opposite rays.
C
FA and FD are opposite rays.
m∠CFD = 10°
A
∠2 is obtuse
Prove that ∠DFE is acute
Statements
2
D
F
E
Explanation/Reasons
Prove that ∠AFE is obtuse
Statements
1
Explanation/Reasons
Suppose A and B are two distinct points in the plane and line L is the perpendicular bisector of segment
AB as pictured below.
a.
If C is a point on L, show that C is equidistant from A and B, that is show that AC and BC are
congruent.
b.
Conversely, show that if P is a point which is equidistant from A and B, then P is on L.
c.
Conclude and prove that the perpendicular bisector of AB is exactly the set of points which are
equidistant from A and B.
Unit 7 Lesson 5 – Introduction to Proof
Ready, Set, Go! - Assignment 7.5
Name______________________________
Date_________ Hour_______
http://goo.gl/Z1yu2
Ready
1.
Linear pairs could be defined as being supplementary angles because they always
add up to 180º. Are all supplementary angles linear pairs? Explain your answer.
2.
a.
Find the supplement of the given angle. Then draw the two angles as linear pairs.
Label each angle with its measure.
m/ ABC = 72º
B will be the vertex.
b.
m/ GHK = 113º H will be the vertex.
c.
m/ XYZ = 24º
Y will be the vertex
d.
m/ JMS = 168º
M will be the vertex
___
___
3. Throughout the study of mathematics, you have encountered many symbols that help you write
mathematical sentences and phrases without using words. Below is a set of common mathematical
symbols. Your job is to match them to their definitions.
Symbol
Matching
definition
1. =
2. 𝑚∠𝐶
3. GH
4. ∆𝐴𝐵C
5. ⊥
6. ∠ABC
7.
GH
8. ≅
9. ∼ 
10. GH

11. GH
12. ∥
13.
| ±
14.
Type xequation here.
Definitions
A.) Absolute value – it is always equal to the positive value of the
number inside the lines. It represents distance from zero.
B.) Congruent – Figures that are the same size and shape are said
to be congruent.
C.) Parallel – used between segments, lines, rays, or planes
D.) Line segment with endpoints G and H. Line segments can be
congruent to each other. You would not say they were equal.
E.) Ray GH – The letter on the left indicates the endpoint of the
ray.
F.) Used when comparing numbers of equal value.
G.) Plus or minus – indicates 2 values, the positive value and the
negative value
H. Triangle ABC
J.) Indicates the measure of an angle. It would be set equal to a
number.
K.) Perpendicular --- Lines, rays, segments, and planes can all be
perpendicular
L.) Angle ABC – The middle letter is always the vertex of the
angle.
M. Similar – Figures that have been dilated are similar.
N.) The length of GH. It would equal a number.
P.) Refers to the infinite line GH. Lines are not equal or congruent
to other lines.
Set
4. Prove the Alternate Interior Angles Theorem. (if two parallel lines are cut by a transversal, then each
pair of alternate interior angles is congruent)
Given: 𝑎 ∥ 𝑏; 𝑝 𝑖𝑠 𝑎 𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙 𝑜𝑓 𝑎 𝑎𝑛𝑑 𝑏.
Informal Paragraph Proof:
Prove: ∠2 ≅ ∠7, ∠3 ≅ ∠6
5.
Given: 𝑝 ∥ 𝑞, 𝑡 ⊥ 𝑝
Prove: 𝑡 ⊥ 𝑞
(you cannot state the perpendicular transversal
theorem, that is what you are proving)
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t
1
2
p
q
Two-Column formal proof.
Statements
Reasons
6. If two lines are both perpendicular to the transversal, are they parallel to each other? Why or why
not?
7. Two lines are cut by a transversal, and the resulting alternate interior angles are supplementary. Is
this always, sometimes or never true? Explain.
8. Two lines form alternate exterior angles that are complementary, and the two lines are parallel. Is
this always, sometimes or never true? Justify.
9. Use the following picture to answer the following questions.
(8x + 4)
a
(14y + 12)
b
(5y + 16)
(9x - 11)
What value of x will guarantee that lines a and b are parallel?
10. Using the same picture as #9, but not the same results (x and y may not be simultaneous), what
value of y will guarantee that lines a and b are parallel?
11. Identify which, if any, of the lines in the picture are parallel with the following information.
a
1 2
4 3
5 6
7 8
b
9 10
12 11
13 14
16 15
a) 1  11
m
b) 9  15
c) 6  12
n
d) m2  m9  180
e) 5  14
f) 7  13
g) m3  m6  180
Go!
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1 5
2 6
3 7
4 8
Use the figure above to answer the following questions.
Identify the types of angles AND tell if they are congruent or supplementary.
12. 1 and 8
13. 2 and 3
14. 3and 8
15. 3 and 4
16. 5 and 7
17. 2 and 7
18. 1 and 7
19. 2 and 6
20. If m1  30 find m3
21. If m2  150 find m7
22. If m4  145 find m8
23. If m4  160 find m5
24. If m4  2x  7 and m8  3x 12 find x, m4, and m8
25. If m8  14x  56 and m6  6x find x, m8, and m6
26. If m1  5x  8 and m2  12x  2 find x, m1, and m2
27. If m6  5x  25 and m7  3x  5 find x, m6, and m7
Unit 7 Enrichment Lesson
Using Geometric Relationships
Notes
a) Set up an equation to solve for the indicated variable using geometric relationships.
(Even if you can figure it out simply – this is practice for more complex problems)
b) Solve that equation. Circle your final answer.
a.
1.
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147º
bº
b.
2.
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a.
b.
cº
62º
cº
3.
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a.
dº
30º
b.
4.
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a.
yº yº
52º
50º
b.
Given that line l is parallel to line m, name the relationship, solve for x in each of the problems, and
find the measures of the specified angles.
5.
Angle Relationship:
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l
Equation:
m
(9x-4)º
1
140º
𝑥 = ________________
𝑚∠1 =______________
6.
Angle Relationship:
Equation:
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3
(2x+12)º
(5x-15)º
𝑥 = ________________
2
𝑚∠2 =______________
𝑚 ∠3 =_____________
7.
Angle Relationship:
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Equation:
(4x-6)º
4
(3x+8)º
l
5
m
𝑥 = ________________
𝑚∠4 =______________
𝑚 ∠5 =_____________
Unit 7 Enrichment Lesson
Solving for a Variable When Using Geometric Relationships
Assignment
Name______________________________
Date_________ Hour_______
Solve for the indicated variable.
1.
2.
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fº
fº
48º hº
74º
hº
38º
3.
4.
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48º
tº
tº tº
xº
xº
xº
5.
6.
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M at h Com poser 1. 1. 5
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6º
12º
pº
pº
.
eº
33º
7.
8.
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M at h Com poser 1. 1. 5
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gº
32º
gº
aº
25º
Find x and y in each figure.
9.
10.
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M at h Com poser 1. 1. 5
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(4x)º
(8y+2)º
56º
(25y-20)º
10xº
(3y-11)º
Solve for x.
11.
M at h Com poser 1. 1. 5
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(9x-5)º
(7x+3)º
Assume lines that appear to be parallel are parallel.
For each of the following: a) State the angle relationship used to solve for x. b) Set up the
corresponding equation and solve. c) circle your final answer.
12.
a)
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b)
(5x+90)º
(14x+9)º
c)
13.
a)
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b)
(8x+4)º
(9x-11)º
c)
14.
a)
(7x-1)º
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b)
90º
c)
15.
a)
(4-5x)º
(7x+100)º
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b)
c)
16.
a)
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b)
(4x+20)º
6xº
c)
Unit 7 Lesson 6 –Classifying Triangles
Task 7.6
Name__________________________________
Date_________ Hour_________
Part 1
Measure and label the measurements of each INTERIOR angle of the following triangle.
(use a protractor)
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B
C
A
What do you notice about the measurements? The sum of the measurements?
Draw and measure your own triangle different from the one given on the front of this page. Label your
measurements. What do you notice?
Draw a triangle that has 4 inch sides. Then, measure the sides. What do you notice about these
measurements?
Part 2
1. Take a piece of graph paper. Draw any type of triangle on it. Make sure it looks at least a little
different from the example below:
2. Cut or tear each angle off of your triangle.
3. Take the three angles that you cut off of your triangle and line them up so that the vertices are
touching each other. Write a paragraph explaining what you notice.
CONCLUSION:
4. Let’s see if you can prove generally the statement you just wrote on #8. Below, are two parallel lines
(XY and BC) with a couple of transversals running through them (AB and AC). Label some of the
alternate interior angles. Individually, or as a class, try and prove that the measurements of the interior
angles of a triangle sum to 180 degrees.
Unit 7 Lesson 6 –Classifying Triangles
Notes 7.6
A triangle is a 3-sided polygon.
How to Label Triangles:
CLASSIFYING TRAINGLES BY ANGLES
Acute Triangle
In an acute triangle, all of the
angles are acute.
Obtuse Triangle
In an obtuse triangle, one angle
is obtuse.
Right Triangle
In a right triangle, one angle
measures 90 degrees.
An acute triangle with all angles congruent is a equiangular triangle.
CLASSIFYING TRIANGLES BY SIDES
Scalene Triangle
No two sides of a scalene
triangle are congruent.
Isosceles Triangle
At least two sides of an
isosceles triangle are congruent.
Equilateral triangle
All of the sides of an equilateral
triangle are congruent.
An equilateral triangle is a special kind of isosceles triangle.
Examples
1. Find the missing angle:
2. Find the missing angle:
3.
Find the measures of the sides of ∆𝐴𝐵𝐶. Classify the triangle by its sides.
(Use the distance formula)
𝐴(5, 4), 𝐵(3, −1), 𝐶(, 7, −1)
4.
∆𝐹𝐺𝐻 is equilateral with 𝐹𝐺 = 𝑥 + 5, 𝐺𝐻 = 3𝑥 − 9, 𝐹𝐻 = 2𝑥 − 2.
Unit 7 Lesson 6 – Classifying Triangles
Ready, Set, Go! - Assignment 7.6
Name______________________________
Date_________ Hour_______
http://goo.gl/wrJ3O
Ready
1. Is the following really a triangle? Explain why or why not.
2. Find the measure of the missing angle.
3. Solve for x:
4.
5.
Find the measure of angle A:
6.
7.
In exercises 5 and 6, find the measure of the missing angle:
8.
9.
Set
Match each triangle with its description.
_______ 10. Side lengths: 2cm, 3cm, 4cm
A. Equilateral
_______ 11. Side lengths: 3cm, 2cm, 3cm
B. Scalene
_______ 12. Side lengths 1cm, 4cm, 5cm
C. Obtuse
_______ 13. Side lengths 4cm, 4cm, 4cm
D. Not a triangle
_______ 14. Angle measures 60°, 60°, 60°
E. Equiangular
________ 15. Angle measures 30°, 60°, 90°
F. Isosceles
_______ 16. Angle measures: 20°, 145°, 15°
G. Right
17.
What is the side length of an equilateral triangle with a perimeter of 36
2
3
in?
18.
An isosceles triangle has a perimeter of 34 cm. The congruent sides measure (4x-1)cm. The
length of the third side is x cm. Find x.
17.
Draw a triangle with 2 obtuse angles.
18.
Draw an equilateral triangle with side lengths 2cm, 3cm, 2cm.
19.
∆𝐿𝑀𝑁 is isosceles, ∠𝐿 is the vertex angle, 𝐿𝑀 = 3𝑥 − 2, 𝐿𝑁 = 2𝑥 + 1, 𝑎𝑛𝑑 𝑀𝑁 = 5𝑥 − 2.
Find the measures of the sides of ∆𝐴𝐵𝐶 and classify each triangle by its sides.
20.
𝐴(−3, −1), 𝐵(2, 1), 𝐶(2, −3)
21.
𝐴(−4, 1), 𝐵(5, 6), 𝐶(−3, −7)
22.
𝐴(0, 5), 𝐵(5√3, 2), 𝐶(0, −1)
Unit 7 Lesson 7 – Isosceles Triangles
Task 7.7
Name___________________________________
Date___________ Hour____________
1. On a piece of blank paper, with the help of a ruler, draw a large isosceles triangle. Cut the triangle
out and fold it as shown.
2. After you fold the triangle in half, what do you notice about the two halves?
Do you think this will always be the case – no matter the triangle you draw?
3. Compare your folded triangle with another individual/group’s triangle. Talk about any similarities
and differences you see.



4. Below, draw two more isosceles triangles. (two congruent sides)
Triangle 1
Triangle 2
Measure the angles directly across/opposite the congruent sides you drew
with a protractor. We call these base angles. Label those angles with their
corresponding measures.
5. Compare your results from #4 with another individual/group. Discuss any similarities and/or
differences that you notice.


Summarize:
Review of SSS congruency
6. Try to prove in a two column proof what you wrote in the summary generally. (for any isosceles
triangle). Below is a generic isosceles triangle. Point x is the midpoint of line segment AC. In groups,
or as an entire class, try to prove that the measure of angle A will always equal the measure of angle C.
(Hint: Remember SSS)
Thoughts/Insights:
Unit 7 Lesson 7 – Isosceles Triangles
Ready, Set, Go! - Assignment 7.7
Name______________________________
Date_________ Hour_______
http://goo.gl/lYqWN
Ready
Refer to the Figure for the following questions:
1. If AB  AD , name two congruent angles.
2. If GA  HA , name two congruent angles.
3. If AF  AI , name two congruent angles.
4. If ACD  ADC , name two congruent sides.
5. If AHI  AIH , name two congruent sides.
6. If DBA  BDA , name two congruent sides.
7.
Explain how many angles in an isosceles triangle must be given to find the measures of the other
angles.
8.
̅̅̅̅̅ .
Name the congruent sides and angles of an isosceles ∆𝑊𝑋𝑍 𝑤𝑖𝑡ℎ 𝑏𝑎𝑠𝑒 𝑊𝑍
̅̅̅̅.
In the figure, ̅̅̅̅
𝑱𝑴 ≅ ̅̅̅̅̅
𝑷𝑴 𝒂𝒏𝒅 ̅̅̅̅̅
𝑴𝑳 ≅ 𝑷𝑳
9. If 𝑚∠𝑃𝐿𝐽 = 34, 𝑓𝑖𝑛𝑑 𝑚∠𝐽𝑃𝑀.
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J
P
L
M
10. 𝐼𝑓 𝑚∠𝑃𝐿𝐽 = 58, 𝑓𝑖𝑛𝑑 𝑚∠𝑃𝐽𝐿.
̅̅̅̅̅ ≅ 𝑮𝑯
̅̅̅̅̅ 𝒂𝒏𝒅 𝑯𝑲
̅̅̅̅̅ ≅ 𝑲𝑱
̅̅̅̅.
In the figure, 𝑮𝑲
11. If 𝑚∠𝐻𝐺𝐾 = 28, 𝑓𝑖𝑛𝑑 𝑚∠𝐻𝐽𝐾.
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12. If 𝑚∠𝐻𝐺𝐾 = 42, 𝑓𝑖𝑛𝑑 𝑚∠𝐻𝐽𝐾.
G
H
K
J
13. Find x.
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(3x+8)º
(2x+20)º
14.
Triangle LMN is equilateral, and MP bisects LN. Find the measure of each side of LMN.
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M
4x-2
3x+1
L
5yº
P
N
Go!
Find the missing value.
15.
16.
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3x-6
(4x-20)º
3xº
x+10
𝑥 = _________
𝑥 = _________
Carfeful, should your answer have a degree
symbol?
18.
19.
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5xº
28º
2xº
xº
𝑥 = _________
𝑥 = _________
Determine whether each satement is always, sometimes, or never true.
20.
A right triangle is an isosceles triangle.
21.
A right triangle is an acute triangle.
22.
An isosceles triangle is an equilateral triangle.
23.
A scalene triangle is an obtuse triangle.
Unit 7 Lesson 8 – Exterior Angle Theorem
Task 7.8
Name___________________________________
Date_____________ Hour_________
Exterior Angles of a Triangle Task (adapted from Math Visions Projec Text)
When a side of a triangle is extended, as in the diagram below, the angle formed on the exterior of
the triangle is called an exterior angle. The two angles of the triangle that are not adjacent to the
exterior angle are referred to as the remote interior angles. In the diagram, 4 is an exterior
angle, and 1 and 2 are the two remote interior angles for this exterior angle.
Use the figure to answer the following questions.
1. Which of the angles in the figure are not exterior angles of the triangle?
1
b) altogether?
3
4
2. How many exterior angles does a triangle have…
a) at one vertex?
2
6 5
9
8
7
a) Is it possible that all of the triangle’s exterior angles of a triangle are equal? Explain.
b) Is it possible that all of the triangle’s exterior angles of a triangle are unequal? Explain.
3. Recall: The sum of the interior angles of any triangle is equal to ________ degrees.
U
4. Given  CUP, identify:
3
a) an exterior angle pictured
b) an interior angle adjacent to the exterior angle
C
2
1
4
P
c) 2 remote (meaning the two angles not adjacent to the exterior) interior angles.
5. With a protractor measure the angles on triangle CUP and make a conjecture about the relationship
between an exterior angle and its 2 remote interior angles.
U
3
C
2
1
4
P
Exterior Angle Theorem:
Let’s Practice using this new Exterior Angle Theorem along with previously learned theorems to solve
for the variable(s) in the following problems.
6.
7.
x°
y°
61°
39°
x°
48°
8.
57
(y + 10)
110
9.
10.
33°
(7x)
90°
87°
(5x + 50)
(3x)
x°
54°
11.
12.
x°
x°
95°
148°
91°
151°
We are going to prove the Exterior Angle Sum Theorem using a flow-chart proof.
Given: ∆𝑨𝑩𝑪
Prove: 𝒎∠𝑪𝑩𝑫 = 𝒎∠𝑨 + 𝒎∠𝑪
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D
C
B
A
Unit 7 Lesson 8 – Exterior Angles Theorem
Ready, Set, Go! - Assignment 7.8
Name______________________________
Date_________ Hour_______
http://goo.gl/kfUPu
Ready
Use the figure at the right for problems 1-3.
1. Find m3 if m5 = 130 and m4 = 70.
5
2
3
4
1
2. Find m1 if m5 = 142 and m4 = 65.
3. Find m2 if m3 = 125 and m4 = 23.
Use the figure at the right for problems 4-7.
4. m6 + m7 + m8 = _______.
5. If m6 = x, m7 = x – 20, and m11 = 80,
11
then x = _____.
6.
If m8 = 4x, m7 = 30, and m9 = 6x -20,
then x = _____.
7. m9 + m10 + m11 = _______.
9
6
8
7
10
Set
For 8 – 11, solve for x.
140°
x°
x°
8.
9.
35°
120
(5x)°
10.
(2x)º
11.
(4x)°
(3x + 54)º
x°
12.
x°
(x - 20)°
Find m1, m2, m3, m4, and m5. Be sure to label your answers to be easily graded.
65°
46°
82°
1
2
5
4
3
142°
Find the value of the 2 remote interior angles in the figures below. Be sure to label which angle is
which in your answer.
13.
14.
Go! (These problems are originally found in the Math Visions Project Secondary 2 Book)
Label each picture as showing parallel lines with a transversal, vertical angles, or an exterior
angle of a triangle. Highlight the geometric feature you identified. Can you find all 3 features
in 1 picture? Where?
15.
Unit 7 Lesson 9 – Mid-Segment of a Triangle
Task 7.9
Name___________________________________
Date_________ Hour________
Part 1 - Directions:
1. Use a straight-edge or ruler to draw a triangle in the space below. Label your triangle ABC.
2.
3.
4.
Using a compass, or ruler, bisect side AB of the triangle and side BC.
Locate the midpoint on each side and label them E and F respectively.
Connect the midpoints to form the mid-segment.
5.
Compare the length of the mid-segment to the large triangle’s third side. How do they relate?
6. Using your triangle drawn in step 1, connect the midpoints of the thriee sides to form three midsegments. Your picture should look similar to the one below.
7. Using your ruler to measure the sides (in cm) and record your data in the table provided.
DE =
BC =
EF =
AC =
FD =
AB =
Make a conjecture reguarding the relationship between the length of the mid-segment and the length of
the third side of the triangle.
Our Group’s Conjecture:
Part 2:
1. The three midsegments of a triangle divide it into four congruent triangles. Refer to your picture in
Part 1 and measure the angles to convince yourself this is true for your given example.
2. Draw another triagnle in the space below. Mark all three mid-segments of this triangle.
3. Mark all the congruent angles in your triangle. Be sure to check your answer with another group or
the teacher to be sure you are accurate.
4. Focus on one of the mid-segments and the third side of the triangle (the side the midsegment
doesn’t intersect). Look at the pairs of alternate interior angles and corresponding angles assosiated
with these segments. What conclusion can you make?
5. Look at the angles associated with each side of the toher mid-segments and corresponding third side.
What conclusion can you make?
Triangle Mid-Segment Conjecture:
Summary
Unit 7 Lesson 9 – Mid-Segment of a Triangle
Notes 7.9
Ex 1:
Plot the following points on the given coordinate plane.
𝐴(−2, −4), 𝐵(4, 8), 𝐶(6, −2), 𝐷(1, 2), 𝐸(2, −3).
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8
y
̅̅̅̅ ∥ ̅̅̅̅
Show that 𝐷𝐸
𝐵𝐶 .
7
6
̅̅̅̅ = 1 𝐵𝐶
Show that 𝐷𝐸
5
4
2
3
2
x
1
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
1
2
3
4
5
6
7
8
Ex 2:
Given: 𝐴(4,4), 𝐵(5, 9), 𝐶(1, 3) as the midpoints of the sides of triangle DEF. Find the coordiantes of
the vertecies of triangles DEF.
Ex 3: Solve for x. Note XY is a mid-segment.
Unit 7 Lesson 9 – Mid-Segments of a Triangle
Ready, Set, Go! - Assignment 7.9
Name______________________________
Date_________ Hour_______
http://goo.gl/xb8OU
Ready
1. DE is a mid-segment of triangle ABC. Find
DE.
C
2. D, and E are midpoints. Find AB.
C
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32
D
E
E
D
A
A
B
B
32
3. Given: 𝐴𝐶 = 42, 𝐶𝐵 = 46, 𝐴𝐵 = 48 𝑎𝑛𝑑 𝐷, 𝐸, 𝐹 𝑎𝑟𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡𝑠.
Find the perimeter of triangle DEF.
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C
32
D
A
E
B
F
4. D and E are midpoints. Find the measure of
angle A.
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C
D
A
46º
5. Given DE, DF, and FE are the lengths of
mid-segments. Find the perimeter of triangle
ABC.
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E
C
8
D
B
7
A
E
6
F
B
6. Show that the midsegment MN is parallel to side JK and is half as long.
𝐽(−2, 3), 𝐾(4, 5), 𝐿(6, −1), 𝑀(2, 1), 𝑁(5, 2).
7.
In this problem, GH, HJ, and JG are mid-segments of ∆𝐷𝐸𝐹. Each statement has a “?” decide
what the quesiton mark will equal/be and write it on the line.
̅̅ ∥ ? __________
a. ̅̅
𝐽𝐻
b. ̅̅̅̅
𝐸𝐹 = ? _________
c. ? ∥ ̅̅̅̅
𝐷𝐸 _________
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24 in
J
D
E
10.6
8
G
H
d. ̅̅̅̅
𝐺𝐻 = ? _________
e. ̅̅̅̅
𝐷𝐹 = ? __________
̅̅ = ? __________
f. ̅̅
𝐽𝐻
g. Find the perimeter of triagnle GHJ.
F
Set
8.
Given: 𝐴(4,4), 𝐵(5, 9), 𝐶(1, 3) as the midpoints of the sides of triangle DEF. Find the
coordiantes of the vertecies of triangles DEF.
9.
Prove that the perimteer of a tirangle formed by the three mid-segments of a triangle is half the
permineter of the original triagnle. Use a two-column proof.
10. For each triangle below, explain why line segment DE would NOT be the midsegment of the
triangle ABC.
11.
Given that line segment XY is a mid-segment of triangle PQR, solve for x, y, and z.
12. Given that line segment AB is a mid-segment of the triangle below, finda ll of the missing
interior angles.
13 .
Given that line segment XY is a midpoint of triangle PQR, solve for x, y, and z.
Go!
14. In Exercise 14, use ∆𝐴𝐵𝐶 where L, M, and B are midpoints of the sides.
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B
N
L
A
M
C
a. ̅̅̅̅
𝐿𝑀 ∥ ? __________
b. ̅̅̅̅
𝐴𝐵 ∥ ? ___________
c. 𝐼𝑓 𝐴𝐶 = 20, 𝑡ℎ𝑒𝑛 𝐿𝑁 = ? __________
d. 𝐼𝑓 𝑀𝑁 = 7, 𝑡ℎ𝑒𝑛 𝐴𝐵 = ? __________
e. 𝐼𝑓 𝑁𝐶 = 9, 𝑡ℎ𝑒𝑛 𝐿𝑀 = ? __________
f. 𝐼𝑓 𝐿𝑀 = 3𝑥 + 7 𝑎𝑛𝑑 𝐵𝐶 = 7𝑥 + 6, 𝑡ℎ𝑒𝑛 𝑤ℎ𝑎𝑡 𝑑𝑜𝑒𝑠 𝐿𝑀 𝑒𝑞𝑢𝑎𝑙?
g. 𝐼𝑓 𝑀𝑁 = 𝑥 − 1 𝑎𝑛𝑑 𝐴𝐵 = 6𝑥 − 18, 𝑡ℎ𝑒𝑛 𝑤ℎ𝑎𝑡 𝑑𝑜𝑒𝑠 𝐴𝐵 𝑒𝑞𝑢𝑎𝑙?
h. Which angles in the diagram are congruent? Explain your reasoning.
Unit 7 Lesson 9b – Medians of a Triangle
Task 7.9b
Name_______________________________________
Date_________ Hour________
Draw a triangle ABC below, construct the three medians of the triangle. (A median is a line segment that
goes from the midpoint of one side of the triangle to the opposite vertex of the triangle). You may want
to turn your paper sidewise which is fine.
If you constructed the medians correctly, they should all meet at a single point. The point where
the three medians meet is called the centroid of a triangle. Call the centroid of your triangle above G.
Additional Notes:
Unit 7 Lesson 10 – Parallelograms
Task 7.10
Name_______________________________________
Date_________ Hour________
This task is derived from the Math Visions Project Secondary Math 2 Book.
1. Explain how you would locate the center of rotation for the following parallelogram.
What convinces you that the point you have located is the center of rotation?
2. If you haven’t already, draw one or both of the diagonals in the above parallelogram.
Use this diagram to prove this statement: opposite sides of a parallelogram are congruent
3. Use this diagram to prove this statement: opposite angles of a parallelogram are congruent
4. Use this diagram to prove this statement: the diagonals of a parallelogram bisect each other
The statements we have proved above extend our knowledge of properties of all
parallelograms: not only are the opposites sides parallel, they are also congruent; opposite angles
are congruent; and the diagonals of a parallelogram bisect each other. A parallelogram has 180°
rotational symmetry around the point of intersection of the diagonals—the center of rotation for
the parallelogram.
If we have a quadrilateral that has some of these properties, can we convince ourselves that the
quadrilateral is a parallelogram?
How many of these properties do we need to know before we can conclude that a quadrilateral is a
parallelogram?
Consider the following statements. If you think the statement is true, create a diagram and
write a convincing argument to prove the statement.
a. If opposite sides and angles of a quadrilateral are congruent, the quadrilateral is a
parallelogram.
b. If opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
c.
If opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
d. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
e. If one pair of opposite sides of a quadrilater is both parallel and congruent, then the
quadrilateral is a parallelogram.
The following theorems all concern parallelograms:
 Opposite sides of a parallelogram are congruent.
 Opposite angles of a parallelogram are congruent.
 Consecutive angles of a parallelogram are supplementary.
 The diagonals of a parallelogram bisect each other.
Concept Summary:
“Tests” for Parallelograms





Additional Notes
Unit 7 Lesson 10 – Parallelograms
Ready, Set, Go! - Assignment 7.10
Name______________________________
Date_________ Hour_______
http://goo.gl/oxPhb
Parts of this assignment are taken from the Math Visions Project Secondary 2 Book.
Ready
For Problems #1-9. Rewrite the phrases below using correct mathematical symbols.
Example: Eleven plus eight is nineteen.
11 + 8 = 19
1. Triangle ABC is congruent to triangle GHJ.
2. Segment BV is congruent to segment PR.
3. Three feet are equal to one yard.
4. Line TR is parallel to line segment WQ.
5. Ray VP is perpendicular to segment GH.
6. Angle 3 is congruent to angle 5.
7. The distance between W and X is 7 feet.
8. The length of segment AB is equal to the length of TR.
9. The measure of angle SRT is equal to the measure of angle CDE.
Set
10. Use what you know about triangles to write a paragraph proof that proves that the sum of
the angles of a quadrilateral is 360 degrees.
11.
Find the measure of x in the quadrilateral ABGC.
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C
A (4x+23)º
(16x+15)º
(6x-5)º
G
(7x-3)º
B
12.
Match the equation with the correct line in the graph of lines p, q, r, and s.
3
𝒂. 𝑦 = 𝑥 + 2,
4
13.
3
𝒃. 𝑦 = − 𝑥 + 2,
4
3
𝒄. 𝑦 = 𝑥 + 4,
4
3
𝒅. 𝑦 = − 𝑥 + 4
4
a. Desribe the shape made by the intersection of the 4 lines in #12.
b. Whare the slopes of the four lines?
14.
List as many observations as you can about the shape and its features in #12.
For problems 15-16.
Sketch the quadrilateral by connecting the points in alphabetical order. Close the figure
15.
In both figures, the lines are perpendicular bisectors of each other.
A. Are the quadrilaterals you sketched congruent?
B.
16.
What additional requirement(s) is/are needed to make the figures congruent?
In both figures one set of opposite sides are parallel and congruent.
A. Are the quadrilaterals ou sketched congruent?
B. What additional requirement(s) is/are needed to make the figures congruent?
Go!
17. Give a reason why it is NOT possible for each figure below to be a parallelogram. List ALL that
apply.
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21 cm
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128º
53º
52º
127º
14 cm
13 cm
22 cm
Each quadrilateral below is a parallelogram. Find the values of x, y, and z.
19.
18.
𝑥=
𝑥=
𝑦=
𝑦=
𝑧=
𝑧=
20.
F
(7x+11)
6x
G
𝑥=
𝑦=
(10z+8)
(5y-12)
E
21.
Given 𝑚∠𝑊 = 63°, find the measures of the other three angles.
X
W
Y
Z
22.
𝑧=
H
Solve for a and b.
3b-17
A
B
3a+11
a+15
D
4a+2
C
23. The scissor lift uses parallelograms to guarantee a straight lift every time. Identify all of the angles
that are congruent to angle C. Then identify all of the angles congruent to angle EKF.
A
B
J
D
C
K
E
F
L
G
H
24. Use the following sketch of parallelogram ABCD to find the measures of :
a) mA 
b) mB 
c) AB 
d) AD 
B
A
4.2
123
D
25.
6
C
Determine whether a figure with the given vertices is a parallelogram by using the slope formula.
𝐵(−6, −3), 𝐶(2, −3), 𝐸(4, 4, ), 𝐺(−4, 4)
26.
Determine whether a figure with the given vertices is a parallelogram by using the distance
formula. 𝑄(−3, −6) , 𝑅(2, 2), 𝑆(−1, 6), 𝑇(−5, 2)
Unit 7 Lesson 11 –Rectangles, Rhombi, and Squares
Notes 7.11
 Quadrilateral just means “four sides”. (Quad mean four, lateral means side).
But, the sides have to be straight, and it has to be a closed 2-D shape.
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A rhombus is a parallelogram
with four congruent sides.
If the diagnals of a
parallelogram are
perpendicular, then the
parallelogram is a rhombus.
A rectangle is a parallelogram
with four right angles.
A squre is a parallelogram with
four congruent sides and four
If a parellelogram is a rectangle, right angles.
then the diagnals are congruent.
If a quadrilateral is both a
rhombus and a rectangle, then it
is a square.
Each diagnal of a rhombus
bisects a pair of opposite
angles.
Decide whether the statement is sometimes, always, or never true. Explain your answers.
1.
A rectangle is a parallelogram.
2.
A rectangle is a rhombus.
3.
A parallelogram is a rhombus.
4.
A square is a rectangle.
5
All square are rectangules.
6.
A rhombus is always a square.
7.
A parallelogram can have 3 obtuse angles.
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The flowchart on the right
has the most general 4-sided
polygon at the top and the
most specific at the bottom.
Around each box, write in
the details that make the
specific quadrilateral
unique. Then, explain why
the arrows points up instead
of down.
Quadrilaterals
Parallelagrams
Rectangles
Rhombi
Squares
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Quadralaterals
Parallelograms
Rhombi
Rectangles
Squares
The above Venn-Diagram also helps illistrate this relationship.
Unit 7 Lesson 11 –Rectangles, Rhombi, and Squares
Task 7.11
Name________________________________________
Date__________ Hour____________
Kyle and Marie are playing a guessing game in which one person describes some of the features of a
quadrilateral they have drawn and the other person has to name the type of quadrilateral.
Here are some of the clues they gave each other. Decide what type of quadrilateral they are
describing and explain how you know. Draw a picture if you need to.
1.
The diagonals of this quadrilateral are perpendicular to each other.
2.
The diagonals of this quadrilateral are congruent.
3.
When rotated 90 degrees, each diagonal of this quadrilateral gets superimposed on top of the
other.
4.
Consecutive angles of this quadrilateral are congruent.
5.
Consecutive angles of this quadrilateral are supplementary.
6.
The diagonals of this quadrilateral are congruent and perpendicular to each other.
Determine whether each quadrilateral is a parallelogram. Write YES if it is. If it is NOT a parallelogram,
state why not.
7.
One pair of opposite sides is parallel and it has two consecutive right angles.
8.
The quadrilateral has 4 right angles.
9.
One pair of opposite sides is parallel and congruent.
10.
One pair of opposite sides is parallel. The other pair of opposite sides is congruent.
11.
Consecutive angles are supplementary.
12.
The diagonals of the quadrilateral are perpendicular.
Additional Examples/Notes:
Unit 7 Lesson 11 –Rectangles, Rhombi, and Squares
Ready, Set, Go! - Assignment 7.11
Name______________________________
Date_________ Hour_______
http://goo.gl/o5dYP
Ready
Decide which quadrailteral is being described in the “Who Am I?” questions below.
1.
All four of my sides are the same length but I don’t have any right angles. What am I?
2.
My vertecies are all right angles and a side length of 4 is adjacent to a side length of 7 What am I?
3.
All four of my sides are the same length, but my vertecies don’t all have the same measure. What
am I?
4.
I have two pairs of sides that are parallel. I don’t have any right angles. A side length of 8 is
adjacent to a side length of 4. What am I?
5.
I was a rectangle until a strong wind came along and tipped me over 90 degrees. What am I now?
6.
I have four right angles and four sides that measure 2 cm. What am I?
7.
I have four equal sides and four angles that are 90 degrees each, What am I?
8.
I am almost a square since my angles are all 90 degrees, but I am not as tall as I am wide. What
am I?
Saure
Rhombus
#9
Rectangle
Parallelogram
In the chart below, place a X if you think the quadrilateral listed along the top row has peroperties listed
in the left column.
Opposite sides ≅
Opposite angles ≅
Consecutive angles
sum to 180°
Diagnals ≅
Diagnals ⊥
Diagnals bisect each
other
Set
10.
Given: 𝑊𝑋𝑌𝑍 is a rectangle with diagnals ̅̅̅̅̅
𝑊𝑌 𝑎𝑛𝑑 ̅̅̅̅
𝑋𝑍.
̅̅̅̅̅ ≅ 𝑋𝑍
̅̅̅̅.
Prove: 𝑊𝑌
11.
Draw two congruent right triangles with a common hypotenuse. Do the legs form a rectange?
12. Mrs. Peterson has a rectangular picture that is 12 inches by 48 inches. Because it is not a
standard size, a special frame must be built for her. What can the framer do to guarentee that the frame
is a rectangle? Justify your reasoning.
Quadrilateral JKMN is a rectangle. Use the figure below for questions 13-18.
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K
J
Q
M
N
13.
If 𝑁𝑄 = 5𝑥 − 3 𝑎𝑛𝑑 𝑄𝑀 = 4𝑥 + 6, 𝑓𝑖𝑛𝑑 𝑁𝐾.
14.
If 𝑁𝑄 = 2𝑥 + 3 𝑎𝑛𝑑 𝑄𝐾 = 5𝑥 − 9, 𝑓𝑖𝑛𝑑 𝐽𝑄.
15.
If 𝑁𝑀 = 8𝑥 − 14 𝑎𝑛𝑑 𝐽𝐾 = 𝑥 2 + 1, 𝑓𝑖𝑛𝑑 𝐽𝐾.
16.
If 𝑚∠𝑁𝐽𝑀 = 2𝑥 − 3 𝑎𝑛𝑑 𝑚∠𝐾𝐽𝑀 = 𝑥 + 5, 𝑓𝑖𝑛𝑑 𝑥.
17.
If 𝑚∠𝑁𝐾𝑀 = 𝑥 2 + 4 𝑎𝑛𝑑 𝑚∠𝐾𝑁𝑀 = 𝑥 + 30, 𝑓𝑖𝑛𝑑 𝑚∠𝐽𝐾𝑁.
18.
If 𝑚∠𝐽𝐾𝑁 = 2𝑥 2 + 2 𝑎𝑛𝑑 𝑚∠𝑁𝐾𝑀 = 14𝑥, 𝑓𝑖𝑛𝑑 𝑥.
19. WXYZ is a rectangle. Find each measure if 𝑚∠1 = 30.
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W
7
X
8
1 2
9
11
10
𝑚 ∠2 =
𝑚 ∠7 =
𝑚 ∠3 =
𝑚 ∠8 =
𝑚 ∠4 =
𝑚 ∠9 =
𝑚 ∠5 =
𝑚 ∠10 =
𝑚 ∠6 =
𝑚 ∠11 =
12
6
Y
5
4
3
Z
Go!
Use rectangle STUV for questions 20-23.
T
S
20. If m1 = 30, m2 = _______
7
6
2
K
21. If m6 = 57, m4 = _______
22. If m8 = 133, m2 = _______
1
8
4
5
V
3
U
23 . If m5 = 16, m3 = _______
Use rhombus ABCD for problems 24-29
B
A
24. If mBAF = 28, mACD = ______.
F
25. If mAFB = 16x + 6, x = _______.
26. If mACD = 34, mABC = _______.
D
C
27. If mBFC = 120 – 4x, x = ______.
28. If mBAC = 4x + 6 and mACD = 12x – 18, x = ______.
29. If mDCB = x2 – 6 and mDAC = 5x + 9, x = ______
30. ABCD is a square. AB = 5x + 2y, AD = 3x – y, and BC = 11. Find x and y.
A
B
D
C
Unit 6 Pretzel Sticks Activity
Name______________________________
Date_________ Hour_______
Directions: You have nine pretzel sticks. Make the triangle shape below using your pretzel sticks.
Follow the directions for each question below and draw a diagram for each one.
(If your teacher did not give you pretzels, you can do the same task with pencil and paper)
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1.
Remove two pretzels to get three triangles.
2.
Remove two pretzels leaving two triangles.
3.
Remove three pretzels leaving one triangle.
4.
Remove three pretzels leaving two triangles.
5.
Remove six pretzels leaving one triangle.
Answer Key:
(x +10)
o
(2x)
(2x +1)
(101)
o
o
Corresponding
Angles
Complementary
Angles
o
x +1+ 80 = 180
x = 55
2x +1 = 101
x = 10
x +10 = 2x
x = 30
2x = 110
x = 15
5x + 40 = 90
x = 50
3x = −x + 80
x = 99
2x +1 = x +16
x = 10
x + 5x = 180
x = 20
Vertical Angles
(x) o
(5x)
o
Vertical Angles
(5x) o (40) o
Alternate
Interior Angles
110 o
(2x) o
(x +16) o
(2x +1)
Supplementary
Angles
o
Alternate
Exterior Angles
(3x) o
(−x + 80)
(80) o
(x +1) o
o
Supplementary
Angles
Instructions:
Cut out each card and arrange them into rows so that each figure is grouped with its angle relationship
name, equation that models the relationship, and solution to the equation.
(x +10) o
(2x) o
(2x +1) o
(101)
o
(x) o
(5x)
110 o
o
(x +16) o
(2x +1)
x +1+ 80 = 180
Complementary
Angles
2x +1 = 101
x = 10
Vertical Angles
x +10 = 2x
x = 30
Vertical Angles
2x = 110
x = 15
Alternate
Interior Angles
5x + 40 = 90
x = 50
Supplementary
Angles
3x = −x + 80
x = 99
Alternate
Exterior Angles
2x +1 = x +16
x = 10
Supplementary
Angles
x + 5x = 180
x = 20
o
(5x) o (40) o
(2x)
Corresponding
Angles
o
(3x) o
(−x + 80) o
x = 55
(80) o
(x +1) o