Download BLoCK 2 ~ LInes And AngLes

BLoCK 2 ~ LInes And AngLes
triangLes and QuadriLateraLs
L esson 7
L esson 8
L esson 9
L esson 10
L esson 11
L esson 12
L esson 13
r eview
cLassiFYing T riangLes ----------------------------------------------Explore! Naming By Sides
a ngLe sUM oF a T riangLe -------------------------------------------Explore! Add Them Up
sPeciaL T riangLes ---------------------------------------------------Explore! What Makes Me Special?
congrUenT and siMiLar T riangLes ----------------------------------ParaLLeL L ines and siMiLar T riangLes -------------------------------Explore! Parallel Similarity
a ngLe sUM oF a QUadriLaTeraL -------------------------------------Explore! Four Corners
sPeciaL QUadriLaTeraLs ---------------------------------------------BLock 2 ~ T riangLes and QUadriLaTeraLs ----------------------------
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Block 2 ~ Lines And Angles ~ Triangles And Quadrilaterals
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l At
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triAng
tr i A
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BLoCK 2 ~ trIAngLes And
QuAdrILAterALs
tic - tac - tOe
high school choices
exterior A ngles
triAngle moBile
Investigate math courses
offered at the high school
you will attend.
Discover and apply
a unique property
in all triangles.
Create an original piece
of art showing the
properties of triangles.
See page  for details.
See page  for details.
See page  for details.
l And oF sPeciAl Figures
sum oF interior A ngles
constructing segments
Write a fiction story about
the Land of Special Figures.
Include triangles and
quadrilaterals in the story.
Discover the rule for the
sum of the interior angles
in any polygon.
Use a compass and
straightedge to duplicate
and combine segments.
See page  for details.
See page  for details.
See page  for details.
triAngle ineQuAlity
theoreM
clAssiFicAtion gAme
too tAll to K now
Determine the possible
measurements for the third
side of a given triangle.
Make a memory card
game where players
classify triangles by
sides and angles.
Find the height of objects
that are too tall to measure
with a measuring tape.
See page  for details.
See page  for details.
See page  for details.
Block 2 ~ Triangles And Quadrilaterals ~ Tic - Tac - Toe
41
cLassiFying triangLes
Lesson 7
A triangle is a polygon with three sides. A triangle can be classified by
its angles and by the lengths of its sides. In Block 1, you learned about
four types of angles: acute, obtuse, right and straight. When a triangle is
classified by its angles, these same words are used. Since a triangle cannot
have a straight angle, there are three classifications of triangles by angles.
Acute triangles have three
acute angles
right triangles have one
right angle
obtuse triangles have one
obtuse angle
In the last block, angles were marked as congruent using tick marks on the arcs.
In the same way, tick marks can show that the lengths of two line segments are equal.
Lesson 7 ~ Classifying Triangles
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expLOre!
naming By sides
equilateral triangles
scalene triangles
2.7 cm
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cm
3.5
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2.4 c
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cm
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3.5
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Isosceles triangles
m
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3.5 cm
cm
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2.9
7
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7
7
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13
4
3
12
8
8
3
5
2
2
2.4
5
step 1: Examine each group of triangles in the table above. List any similarities you see in each category.
step 2: Identify how each group is different from the other groups. Specifically, identify why triangles
from one group do not belong in the other group.
step 3: Angles can be classified by the lengths of their sides. Triangles can be equilateral, scalene or
isosceles. Based on your observations of the triangles in the table above, write a definition
for each type of triangle.
step 4: Sketch your own triangle for each group.
All sides of an equilateral triangle are the same length. An isosceles triangle has at least two sides with the
same measure. A scalene triangle has no sides with the same measure. If a triangle is a scalene triangle, each
side of the triangle will be a different length.
exampLe 1
Classify each triangle by its sides and angle measures.
a.
b.
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4.5 cm
5
|
30°
120°
c.
60°
12
30°
12
12
13
60°
60°
12
solutions
a. Two sides are the same length so it is an isosceles triangle. One angle in the triangle
is more than 90°, so it is an isosceles, obtuse triangle.
b. No sides are the same length. There is a right angle. The triangle is a scalene, right
triangle.
c. All angles in the triangle are less than 90° and all sides are the same length. It is an
acute, equilateral triangle.
Lesson 7 ~ Classifying Triangles
43
exampLe 2
sketch a diagram to represent an acute, isosceles triangle named ∆ABC.
solution
In order to be an acute triangle all angles must be less than 90°. Since the triangle is
isosceles, at least two sides must be the same length. Tick marks can be used to show
which sides are equal lengths in the sketch.
B
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A
C
exercises
1. What are the three triangle classifications based on side lengths?
2. What are the three triangle classifications based on angle measures?
Classify each triangle by its angle measures.
3.
50°
4.
60°
30°
75°
60°
6. |
7.
8.
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40°
5.
60°
42°
115°
18°
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60°
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23°
18°
Classify each triangle by its side lengths.
10.
7
7
44
Lesson 7 ~ Classifying Triangles
7.1 m
11.
6.4 m
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7
2m
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9.
13.
12.
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4 ft
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4 ft
14.
6 ft
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Classify each triangle by its sides and angle measures.
15.
2 cm
16.
38°
117°
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71°
17.
4 cm
71°
18.
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60°
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60°
3 cm
60°
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19. A sail on a sailboat has one 90° angle and the sides are three
different lengths. Classify this triangle by its sides and angle measures.
20. Another sail on the sailboat is an acute triangle with side lengths
of 10 feet, 6 feet and 10 feet. Classify this sail by its sides and
angle measures.
draw and label each triangle to match each description.
21. Acute ∆CUT
22. Isosceles ∆SAM
23. Right ∆RGT
24. Isosceles, right ∆POE
25. Obtuse, scalene ∆CUP
26. Acute, equilateral ∆EQU
The side lengths of a triangle are given. Classify the triangle by its sides.
27. 1, 1, 1
28. 8, 8, 10
29. 6, 8, 10
30. h, h, h
31. 5p, 3p, 5p
32. 3j, 4j, 5j
33. Can a right triangle have more than one right angle? Support your answer with a diagram.
34. In an obtuse triangle one angle is obtuse. What type of angle are the other two angles?
Lesson 7 ~ Classifying Triangles
45
review
solve for x. Check your solution.
35.
(20x + 11)°
36.
>>
111° (6x − 3)°
>>
(15x + 36)°
37.
38.
(3x + 60)°
125°
>>
>>
(9x + 1)°
5x°
t ic -t Ac -t oe ~ c l A s s i F ic At ion g A m e
Create a memory card game where players must match cards with triangles
with given angle measures, side lengths and/or diagrams to other cards that
classify the triangle. You can include some cards that classify only by sides
or angles and others that classify by both sides and angles. Make sure each
information card has a classification card to match it. Make a set of at least
twelve pairs of cards. Try playing the game with a classmate. If it does not
work, make the needed adjustments before turning in the game.
Example of a matching set:
10
8
6
46
Lesson 7 ~ Classifying Triangles
Scalene
Right
Triangle
angLe sum OF a triangLe
Lesson 8
T
he sum of the measures of the angles of every triangle is the same. In this lesson you will determine the
angle sum of a triangle. This property of triangles will be very useful as you apply it to triangles in many
situations.
expLOre!
add them up
step 1: Draw the three triangles listed below on a blank sheet of paper. Make the triangles large enough to
measure their angles with a protractor.
right triangle
Acute triangle
obtuse triangle
step 2: Use a protractor to measure each angle in all three triangles. Write the measure of each angle inside
the triangle.
step 3: Find the sum of the angles in each triangle.
step 4: Do you notice any similarities in the sums of the angles in each triangle? If possible, write a rule
for the sum of the measures of the angles of any triangle.
step 5: Compare your triangle sums and rule with a classmate. Did he/she get the same or similar results?
The sum of the angles of a triangle can also be shown using the method below. A triangle is drawn on a piece
of paper and cut out. The angles are torn apart and lined up. The three angles form a straight angle. The
measure of a straight angle is 180°.
3
1
2
1
3
2
1 3 2
180°
Lesson 8 ~ Angle Sum Of A Triangle
47
exampLe 1
set up an equation and solve for x.
80°
65°
solution
x°
80 + 65 + x = 180
145 + x = 180
−145
−145
x = 35
The sum of the angles in a triangle is 180°.
Combine like terms.
Subtract 145 from each side of the equation.
The measure of the missing angle is 35°.
exampLe 2
∆You has the angle measures listed below.
m∠Y = 70°
m∠o = (3x – 10)°
m∠u = 7x°
a. set up an equation. solve for x.
b. Find the degree measure of each angle.
solutions
a. The angles of a triangle sum to 180°.
Combine like terms.
Subtract 60 from each side of the equation.
Divide both sides of the equation by 10.
70 + (3x – 10) + 7x = 180
10x + 60 = 180
−60 −60
10x = ___
120
___
10
10
x = 12
b. Write the given expression
for each angle.
Substitute 12 for x.
Multiply.
Subtract.
☑ m∠Y + m∠O + m∠U = 180°
70° +
48
Lesson 8 ~ Angle Sum Of A Triangle
26° + 84° =? 180°
180° = 180°
m∠O = (3x − 10)°
= 3(12) − 10
= 36 − 10
= 26°
m∠O = 26°
m∠U = 7x
= 7(12)
= 84
m∠U = 84°
exercises
Find the degree of each missing angle.
1.
120°
25°
2.
3.
x°
51°
76°
x°
x°
39°
5.
6.
12°
45°
99°
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||
4.
45°
147°
x°
x°
x°
set up an equation and solve for x.
7.
8.
4x°
9.
x°
29°
(x + 5)°
72°
x°
x°
6x°
(3x + 3)°
5x°
10.
11.
x°
10x°
x°
12.
(3x +5)°
2x°
(2x + 1)°
A
2x°
13. Use ∆AMT at the right.
a. Set up an equation to find the value of x.
b. Solve for x.
c. Find the measure of each angle.
M
(x − 6)°
T
14. The m∠C = 60°, m∠U = (7 + 5x)° and m∠P = (1 + 3x)° in ∆CUP.
a. Set up an equation and solve for x.
b. Find the m∠C, m∠U and m∠P.
Lesson 8 ~ Angle Sum Of A Triangle
49
15. ∆PRT is an isosceles triangle. The measure of ∠P is (2x + 1)°. The other two angles each measure 42°.
a. Set up an equation and solve for x.
b. Find m∠P.
A
7x°
16. Jeff determined that the value of x in the triangle at the right is 11.
a. Find the value of each angle by substituting 11 for x.
b. Was Jeff ’s solution of x = 11 correct? How do you know?
S
B
(50 − 5x)°
(17 −
(5 −
L
(5x +
(4x +
2)°
17. Siena determined that x = −8 in the triangle at the left.
3x)°
a. Find the value of each angle by substituting −8 for x.
b. Was Siena’s solution of x = −8 correct? How do you know?
c. Find the correct value of x.
K
4x)°
review
Classify each triangle by its sides and angles.
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59°
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20.
49°
72°
Fill in each blank with the appropriate word or number.
21. Same-side interior angles add up to ______ degrees when between parallel lines.
22. Alternate interior angles are _________ to each other when between parallel lines.
23. A ___________ is the line that cuts through a set of parallel lines.
24. Complementary angles add up to ______ degrees.
25. An angle that equals 180° is called a ______________ angle.
50
Lesson 8 ~ Angle Sum Of A Triangle
107°
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19.
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18.
2)°
C
t ic -t Ac -t oe ~ e x t e r ior A ngl e s
The sum of the remote interior angles in any triangle is congruent to the
measure of the corresponding exterior angle. Below is a diagram showing
the remote interior angles and the corresponding exterior angle.
2
3
1
An algebraic proof of the exterior angle and remote interior angles relationship shows that the
sum of the remote interior angles equals the measure of the corresponding exterior angle.
b
a
c
d
statement
a + b + c = 180°
c + d = 180°
a+b+c=c+d
−c −c
a+b=d
reason
The sum of the angles of a triangle is 180°.
Angles c and d are supplementary.
Substitute c + d for 180°.
Subtract c from both sides.
solve for x.
1.
2.
42°
x°
3.
138°
50°
2x°
x°
61°
133°
111°
4. One of the remote interior angles is 66°. The exterior angle is a right angle. What is the degree
measure of the other remote interior angle?
5. The exterior angle measures 78°. Give a possible pair of degree measures that the remote interior
angles could be.
solve for x. Find the measure of each angle inside the triangle.
B
6.
S
7.
(1 − 2x)°
x°
R
A
(5x − 5)°
8.
_1 x°
2
3x°
N
(x − 62)°
T
67°
M
(12 − 4x)°
P
91°
C
Lesson 8 ~ Angle Sum Of A Triangle
51
speciaL triangLes
Lesson 9
Y
ou have classified triangles as equilateral, isosceles or scalene depending on the lengths of their sides.
When a triangle has two or more sides that are the same length, the angles in that triangle have unique
properties. Complete the Explore! below to discover these properties.
expLOre!
what makes me speciaL?
step 1: Two equilateral triangles are drawn below. Measure the angles inside each triangle and list them
on your own paper.
E
B
D
C
A
F
step 2: Based on Lesson 8, what should the sum of the angle measures of each triangle equal?
step 3: Is the sum of ∠A, ∠B and ∠C equal to 180°? If not, check your measurements.
Is the sum of ∠D, ∠E and ∠F equal to 180? If not, check your measurements.
step 4: Do you notice anything about the measure of each angle in an equilateral triangle?
If so, what is your discovery?
step 5: Use division to show how you could calculate the degree measure of an
angle in an equilateral triangle.
step 6: Use a ruler to draw two isosceles triangles. Remember that two sides
must be the same length in an isosceles triangle.
step 7: Measure the angles in your triangles. There should be two angles
in each triangle that are equal to each other. Where are those angles
in comparison to the two sides that are equal?
52
Lesson 9 ~ Special Triangles
60°
60°
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60°
Equilateral triangles are also equiangular. Equiangular means that all angles have the same measure.
exampLe 1
∆MnP is an equilateral triangle. The measure of ∠M is (2x + 6)°.
Find the value of x.
solution
Each angle in an equilateral triangle is equal to 60°.
Set the angle equal to 60°.
Subtract 6 from each side of the equation.
Divide both sides of the equation by 2.
The value of x is 27.
exampLe 2
2x + 6 = 60
−6 −6
54
2x = __
__
2
2
x = 27
Find the value of x in the diagram below.
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112°
x°
The triangle above is isosceles. The angles that are across from the congruence
marks must also be equal so:
112°
|
|
solution
x°
The sum of the angles in a triangle is 180°.
Combine like terms.
Subtract 112 from both sides of the equation.
Divide both sides of the equation by 2.
The missing angles in the diagram are 34°.
x°
x + x + 112 = 180
2x + 112 = 180
−112 −112
68
2x = __
__
2
2
x = 34
Lesson 9 ~ Special Triangles
53
exercises
1. What is the measure of each angle in an equilateral triangle?
2. Which two angles in an isosceles triangle are equal? Draw a diagram to illustrate your answer.
Find the value of x in each diagram.
4.
5.
|
(x +3)°
6.
2x°
7.
3x − 5
60°
|
60°
8.
5x°
10
60°
(x + 24)° |
34°
|
|
|
x°
5
|
x°
71°
|
3.
|
9. ∆YUM has two angles that measure 78°.
a. Sketch a diagram of ∆YUM.
b. Find the measure of the other angle.
c. Classify ∆YUM by its sides and angles.
10. All three sides in ∆BET are 2 inches in length.
a. Sketch a diagram of ∆BET.
b. What is the degree measure of the angles?
c. Classify ∆BET by its sides and angles.
11. ∆JAM is isosceles and has an angle measuring 35°. Sketch two possible diagrams of ∆JAM
with the angle measures labeled.
12. One angle in an equilateral triangle is (8x − 24)°. Solve for x.
13. Hayley’s house makes an isosceles triangle with her school and the mall
as shown in the diagram below.
Hayley’s House
School
2 miles
10 − 4x
40°
40°
5 miles
Mall
a. Find the value of x.
b. How far is it from Hayley’s house to the mall?
54
Lesson 9 ~ Special Triangles
5
14. Hans designed a photo frame shaped like an isosceles triangle. He wanted two of the sides of the frame
to each be three times the length of the shortest side.
a. Hans made the shortest side of the triangle 4 inches long. Sketch a diagram of his photo frame.
b. The angle across from the shortest side of the frame is 20°. Determine the measures of the other
two angles. Explain your reasoning.
review
solve for x in each triangle.
15.
16.
x°
17.
40°
23°
23°
(10x − 1)°
(2x − 20)°
49°
(x + 10)°
18. The complement of ∠B is 37°. Find m∠B.
19. The supplement of ∠L is 146°. Find m∠L.
20. Two angles in a triangle are 18° and 102°. Find the measure of the third angle in the triangle.
21. One angle in a linear pair is 85°. Find the measure of the other angle in the linear pair.
22. Use the diagram at the right.
a. Find m∠1.
b. Find m∠2.
c. Find m∠3.
d. Find m∠4.
<
<
3 2
4 1
98°
t ic -t Ac -t oe ~ t r i A ngl e m oB i l e
A mobile is a type of art, often referred to as kinetic art. It hangs in space
and uses balance and motion. A mobile may be more commonly recognized
hanging above a baby lying in their crib. Create a mobile that shows all of the
different things you learned about triangles in this block. Use definitions, rules
and properties about triangles as information on the pieces of your mobile.
Lesson 9 ~ Special Triangles
55
cOngruent and simiLar triangLes
Lesson 10
T
wo triangles that are the exact same shape and the exact same size are called congruent figures. Two
triangles that have the exact same shape, but not necessarily the exact same size are called similar figures. The
parts of the figures that correspond are called corresponding parts. Look at the two similar triangles below.
D
C
3
35°
6
35°
120°
A
4
12
6
25°
T
120°
O
8
Corresponding Angles
Corresponding sides
∠C and ∠D
CT and DG
∠A and ∠O
___
___
___
___
___
___
25°
G
TA and GO
∠T and ∠G
AC and OD
The corresponding angles in similar triangles are congruent. The corresponding sides are proportional.
You can show two triangles are similar using the similar symbol: ∆CAT ~ ∆DOG.
exampLe 1
∆MnP is similar to ∆JKL. Find the value of x, y and z.
N
K
12
M
solution
y
50°
8
P
J
z°
10
x°
Since the triangles are similar, the corresponding
angles are congruent.
m∠L = m∠P
m∠L = x = 50
The sides of congruent triangles are proportional.
MP
NP = ___
___
JL
KL
y
12 = __
__
10
8
Substitute the values of each known side into the
proportion.
56
L
Set the cross products equal to each other.
Divide both sides of the equation by 8.
8y = 120
y = 15
To find the value of z, set the sum of the three angles
in the triangle equal to 180°.
Subtract 140 from both sides of the equation.
z + 50 + 90 = 180
z + 140 = 180
−140 −140
z = 40
Lesson 10 ~ Congruent And Similar Triangles
If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
This rule is referred to as the Angle-Angle Similarity Rule.
I
N
131°
A
27°
P
27°
131°
G
T
The triangles shown above are similar because ∠A ≅ ∠P and ∠N ≅ ∠I. The Angle-Angle Similarity Rule can
be proven by showing that if two angles are congruent in the triangles, the third angle is also congruent.
∠A + ∠N + ∠T = 180°
27° + 131° + ∠T = 180°
∠T = 22°
∠P + ∠I + ∠G = 180°
27° + 131° + ∠G = 180°
∠G = 22°
Since all three angles in the triangles are congruent, the triangles are similar.
In previous books you learned about slope triangles. You
were able to choose any two points on a line and form a slope
triangle.




On the graph at the left, a right angle is formed in each slope
triangle.
The corresponding sides of the triangles are parallel because
they are either both vertical or both horizontal.
Since the lines are parallel, the blue line is the transversal.
As you learned in Block 1, this makes the top angles in the
triangles corresponding angles. Therefore, those angles are
congruent. This is also true of the bottom angles.
Since two angles in the triangles are congruent, the slope
triangles are similar.
Lesson 10 ~ Congruent And Similar Triangles
57
exercises
Find the corresponding sides and corresponding angles to the ones given for each pair of figures.
M
1.
12
60°
A
80°
40°
18
2.
T
14
N
____
T
3 50°
6 80° 7
40°
O 60°
9
a. MN
___ corresponds to ____
b. ____
AN corresponds to ____
c. MA corresponds to ____
M
P
9
5
A
4
40°
R
50°
15
O
12
___
a. TA
___ corresponds to ____
b. ___
AR corresponds to ____
c. TR corresponds to ____
∠M ≅ ∠___
∠N ≅ ∠___
∠A ≅ ∠___
40°
P
∠M ≅ ∠___
∠O ≅ ∠___
∠P ≅ ∠___
3. Sketch a similar triangle to the right triangle below. Include angle and side lengths in your drawing.
3
54°
5
36°
4
4. Explain the difference between congruent triangles and similar triangles.
Find the missing side length(s) for each set of similar figures.
5.
6.
x
10
x
35
20
30
14
24.5
7.
x
8.
20
9
12
22
x
4
8
15
25
20
10
9.
10.
50°
y
x
60
12
y
15.4
x
30
18
50°
10
58
Lesson 10 ~ Congruent And Similar Triangles
y
20
11. Which set of triangles in exercises 5 through 10 are congruent figures? Explain your reasoning.
12. Are the triangles below similar? Explain your answer.
30°
30°
85°
85°
13. Are the triangles below similar? Explain your answer.
10
8
15
6
12
8
14. Which triangles below are similar to ∆RAP? Explain your reasoning for each.
80°
60°
6
R
80°
N
E
A
8
40°
T
50°
60°
10
20
G
4
6
5
70°
16
R
P
P
A
T
C
L
A
M
12
15. Explain how two slope triangles formed on the same line are similar.
Draw a diagram to support your answer.
16. Sketch a pair of triangles that are congruent. Label all the angles and sides.
Q
17. Draw two triangles of different sizes that are similar to ∆MQT.
5
Label all angle measures and side lengths.
M
98°
8
52°
10
T
Lesson 10 ~ Congruent And Similar Triangles
59
review
name the special angle relationship between the two angles. Find x.
18.
98°
19.
(x + 9)°
20.
(2x − 20)°
(x + 10)°
75°
(4x − 2)°
21. Two angles are supplementary. They measure (5x – 4)° and (9x + 16)°.
a. Write an equation and solve for x.
b. Find the measure of each angle.
t ic -t Ac -t oe ~ t oo t A l l
to
K now
Very tall objects are difficult to measure. This activity requires that you
go outside to find the height of objects that cannot be measured using
traditional means. It is important that all measurements are made within
about one hour total time. It must be sunny to complete this activity.
step 1: Begin by measuring your height in feet. Write the inches part of the
measurement as a fraction over 12.
step 2: Go outside and measure the length of your shadow in feet.
step 3: At the same time as you measure your own shadow, measure the shadow of at least 5 objects
such as a flagpole, the goal posts on a football field, a tree or a building. Measure in feet with
remaining inches written as a fraction of a foot.
step 4: Draw yourself as a stick-figure with your shadow on the ground. In the drawing, connect the
end of the shadow to the top of the stick figure’s head. This should make a right triangle.
Label all known lengths.
step 5: Draw triangles for each object you measured just as you did in step 4. These triangles are
similar to your stick figure triangle. Why is this true?
step 6: Find the height of each object using the similar triangles.
step 7: Explain why it was important to measure your shadow and the other object’s shadows at the
same time of day.
60
Lesson 10 ~ Congruent And Similar Triangles
paraLLeL Lines and simiLar triangLes
Lesson 11
In the first block of this book you learned about parallel lines, transversals and special angle pairs. In this
lesson, you will use some of these special angle pairs. Similar triangles can be formed by sets of parallel lines
and two transversals that intersect one another.
expLOre!
paraLLeL simiLarity
step 1: Copy or trace the diagram at the right. Label the points
and angles as shown.
C
D
>>
step 2: Angles 1 and 3 make what special angle pair?
step 3: What type of special angle pair are ∠4 and ∠2?
>>
A
step 4: What is true about the pairs of angles in steps 2 and 3?
3
E
1
2
B
T
step 5: What can you conclude about ∆ABC and ∆DEC? Why?
step 6: Copy or trace the diagram to the right. Explain how it is the
same type of problem as the diagram shown above.
4
S
step 7: What angle is shared by ∆STU and ∆RTV?
>>
55°
6
step 8: Find the measure of ∠STU.
step 9: What is the measure of ∠TVR?
4
50°
R
>>
x
U
5
V
step 10: Because the angles in ∆STU and ∆RTV are congruent, the
triangles are similar. Find the value of x using a proportion.
As you have seen in the Explore!, special angle pairs can be useful when trying to
determine if two triangles are similar. Once you have shown that two angles in a
triangle are equal to two angles in another triangle, then you know the triangles
are similar based on the Angle-Angle Similarity Rule.
Lesson 11 ~ Parallel Lines And Similar Triangles
61
exampLe 1
show that ∆JKL ~ ∆MnL.
J
K
>>
L
N
∠LMN ≅ ∠LJK because they are alternate interior angles.
J
<<
K
||
∠LNM ≅ ∠LKJ because they are alternate interior angles.
|
solution
>>
M
L
N
|
exampLe 2
<<
||
∆JKL ~ ∆MNL because two angles in each triangle
are congruent to one another. This is based on the
Angle-Angle Similarity Rule.
M
Find the missing measures in the two similar triangles.
a. m∠hAe
H
b. m∠Y
77°
c. x
6
A
x
>>
E
8
T
solutions
58°
21
>>
Y
a. Corresponding angles are congruent.
Substitute 58° for ∠T.
m∠T = m∠HAE
58° = m∠HAE
b. The sum of three angles in a triangle is 180°.
Substitute the angle values for ∠T and ∠H.
Combine like terms.
Subtract 135 from both sides of the equation.
m∠T + m∠H + m∠Y = 180°
58° + 77º + m∠Y = 180°
135º + m∠Y = 180°
m∠Y = 45°
c. Write a proportion with corresponding sides.
AE = ___
HE
___
TY HY
Fill in the known lengths.
x = __
6
__
21 14
Set the cross products equal to each other.
126 = 14x
Divide by 14 on both sides of the equation.
126 = ___
14x
___
14
14
9=x
62
Lesson 11 ~ Parallel Lines And Similar Triangles
exercises
1. Use the diagram at the right.
J
a. What special angle pair do ∠JLK and ∠LMN represent?
Are they congruent angles?
b. Choose the correct word to complete the statement:
∠JKL and ∠KNM are congruent or supplementary.
c. Complete the statement: ∆LJK ~ ∆_____
1
2
O
4
3
>
E
a. Name the two pairs of angles that are congruent inside
the triangles based on alternate interior angles.
b. Complete the statement: ∆POW ~ ∆_____
S
x
A
61°
>>
E
48°
Y
4. Use the diagram at the left.
>> D
8
N
56°
>
C
9
>>
2. Use the diagram at the left.
a. What angle in ∆AND measures 56°?
b. What is the m∠AND?
c. What is the m∠AYS?
d. How do you know ∆AND ~ ∆AYS?
A
N
M
3. Use the diagram at the right.
6
B 37°
>>
>
W
R
K
D
>
P
L
a. What angle besides ∠CBD measures 37°?
b. What is the measure of ∠CDB?
c. What is the measure of ∠C?
d. Find the value of x.
5. Sketch two similar triangles using parallel lines and transversals. Use congruence marks to show
the angles that are congruent.
Find the values of x and y in each pair of similar triangles.
7.
40°
>>
75°
x°
8.
y
9.
37°
52°
>
21
>
>
y°
40°
}
>>
x°
9
x°
27
>
y°
>
6.
39°
12
>
y°
70°
x°
71°
8
4
Lesson 11 ~ Parallel Lines And Similar Triangles
63
Find the measure of a, b, c, d and e.
4
d°
10.
a
e°
>>
11.
>
3
24
a°
6.25
b
c°
37°
5
e
>
c°
39°
14
21
76°
>>
42
b°
d
>>
12. Sketch similar triangles formed by parallel lines, transversals and alternate interior angles.
a. Label the vertex, or corner, of each triangle with a letter.
b. Identify the corresponding sides.
c. Identify the corresponding angles.
review
13. Use the triangle at the right.
a. Write an equation and solve for x.
b. Find the measure of ∠H.
c. Find the measure of ∠G.
F
G
H
(x + 12)°
2x°
|
14. The beams forming the roof of a house form an
isosceles triangle. Copy the diagram at right and
fill in the missing angles.
|
35°
15. An angle in an equilateral triangle measures (3x − 60)°. Find the value of x.
t ic -t Ac -t oe ~ h igh s c hool c hoic e s
Investigate the course offerings in mathematics at the high school you will
be (or are) attending by looking at their course catalog or on their website.
What courses are offered? How many of the courses are you required to take
to graduate? Is there a sequence of courses that students must follow? What
level of math must you reach to be eligible for entry into a four-year university?
Create a brochure illustrating your findings. Survey your friends to see if they
have any questions about high school mathematics. If the information is not already covered in your
brochure, try to locate the answer. Include the answers to their questions in a “Frequently Asked
Questions” section on the back of your brochure.
64
Lesson 11 ~ Parallel Lines And Similar Triangles
t ic -t Ac -t oe ~ c on s t ruc t i ng s e gm e n t s
Use a compass and straightedge to duplicate segments and make
combinations of segments. Use the segments below.
A
C
B
D
___
G
1. Duplicate AB.
J
K
H
E
F
___
a. Use a straightedge
to draw a segment longer than AB.
___
b. Measure AB by placing the stylus on A and opening the compass so the pencil is on B.
c. Using this setting, place the stylus on one endpoint of your segment then make a mark
that crosses your segment.
___
d. Label your congruent segment XY.
___
2. Duplicate GH.
___
___
3. Construct EF + GH.
___
___
a. Use a straightedge
to draw a segment longer than the combined length of EF and GH.
___
b. Measure EF by placing the stylus on E and opening the compass so the pencil is on F.
c. Using this setting, place the stylus on one endpoint of your segment then make a mark
that crosses
___your segment.
d. Measure GH by placing the stylus on G and opening the compass so the pencil is on H.
e. Using this setting, place the stylus on your intersection from part c, then make a mark
that crosses your segment.
This mark will be further down the___
segment
making it longer.
___
___
f. Label the segment EH that is equal to the combined length of EF and GH.
___
___
4. Construct AB + EF.
___
5. Construct 2 ∙ GH.
___
___
6. Construct JK – CD.
___
a. Use a straightedge
to draw a segment longer than JK.
___
b. Measure JK with a compass.
___
c. Using this___
setting, mark off the length of JK on your segment.
d. Measure CD with a compass.
e. Using this setting, place the stylus on the intersection
from part ___
c and turn the
___
compass backwards to mark off the length of CD. The length of JK should be
reduced or taken away
___
____ from.
___
f. Label the segment VW that represents the length of JK with CD removed from it.
___
___
7. Construct JK – GH.
___
___
8. Construct 2 ∙ EF – JK.
Lesson 11 ~ Parallel Lines And Similar Triangles
65
angLe sum OF a QuadriLateraL
Lesson 12
A
quadrilateral is a polygon with four sides. In previous
math classes, you have probably learned about some common
quadrilaterals such as squares, rectangles, parallelograms and
trapezoids. Try the Explore! to discover the sum of the angles in a
quadrilateral.
expLOre!
FOur cOrners
step 1: Trace the parallelogram at the right.
step 2: Connect two opposite corners in the quadrilateral with a line
segment. How many triangles are formed?
step 3: The sum of the angles in a triangle is 180°. Multiply the number of triangles formed in the
parallelogram in step 2 by 180°. What is the sum of the angles in the parallelogram?
step 4: Draw a square.
a. What is the degree measure of one angle in a square?
b. How many angles are in a square?
c. Find the sum of the angles in your square. Does it match your answer in step 3?
step 5: What is the degree measure of one angle in a rectangle? What is the sum of the four angles in a
rectangle?
step 6: Draw a large quadrilateral that is not a square, rectangle or parallelogram. Some examples are
shown below.
step 7: Use a protractor to measure the angles in your quadrilateral. Add the four angles together.
step 8: Write a rule about the angle sum of the four angles in any quadrilateral.
66
Lesson 12 ~ Angle Sum Of A Quadrilateral
exampLe 1
set up an equation and solve for x.
51°
33°
110°
solution
x°
The sum of the angles of a quadrilateral is 360°.
Combine like terms.
Subtract 194 from each side of the equation.
☑ 110 + 51 + 33 + 166 =? 360
110 + 51 + 33 + x = 360
194 + x = 360
−194
−194
x = 166
360 = 360
exampLe 2
In quadrilateral QuAd, m∠u = 2x°, m∠A = (x + 30)° , ∠u ≅ ∠d and
∠Q is a right angle.
a. draw a diagram and label it.
b. set up an equation and solve for x.
c. Find the degree measure for each angle.
solutions
a.
Q
D
2x°
2x°
U
(x + 30)°
A
b. The sum of the angles of a quadrilateral is 360°. 90 + 2x + (x + 30) + 2x = 360
Combine like terms.
120 + 5x = 360
Subtract 120 from each side of the equation.
−120
−120
5x
240
__
Divide both sides of the equation by 5.
= ___
5
5
x = 48
c. To find degree measures, substitute 48 for x.
∠Q is a right angle.
m∠U = m∠D = 2(48) = 96°
m∠A = (48 + 30) = 78°
m∠Q = 90°
Lesson 12 ~ Angle Sum Of A Quadrilateral
67
exercises
set up an equation and solve for x.
1.
x°
2.
109°
68°
148°
3.
100°
75°
x°
83°
89°
37°
x°
4.
142°
x°
5.
57°
6.
x°
98°
85°
x°
149°
116°
x°
set up an equation and solve for x. Find the degree measures of each unknown angle.
7.
B
(2x + 10)°
F
8.
C
104°
E
54°
L 5x°
(2x − 8)°
x°
165°
H
M
N
>
S
2x°
|
84°
R
12.
V
4x°
5x°
(9x + 6)°
W
3x°
||
Q
K
11.
(4x + 9)°
J
4x°
40°
P
U
>
||
10.
G
D
|
(3x − 30)°
A
I
9.
Y
Z (3x + 10)°
2x°
Y
13. A quadrilateral has angles that measure 146°, 75° and 84°. What is the measure of the missing angle?
68
Lesson 12 ~ Angle Sum Of A Quadrilateral
14. Patty owns a piece of farmland that is an irregular quadrilateral.
She wants to put a fence around her farm. She needs to know the
angles at each corner of her land. She measured three of the
angles and found they were 85°, 120° and 50°.
a. What is the angle measure of the fourth corner?
b. Sketch a diagram of Patty’s land.
15. In quadrilateral JUMP, ∠J and ∠U are both 65°. The measure
of ∠M is (3x + 1)° and the measure of ∠P is (2x – 6)°.
a. Draw a diagram and label it.
b. Set up an equation and solve for x.
c. Find the degree measure of ∠M and ∠P.
16. Each angle in quadrilateral WXYZ is (4x – 11)°, (7x + 2)°, (2x + 33)° and x°, respectively.
a. Draw a diagram and label it.
b. Set up an equation and solve for x.
c. Find the degree measure of each angle.
17. Quadrilateral GOAT has the following angle measures: m∠G = (x + 5)°, m∠O = (2x – 15)°, m∠A = 4x°
and ∠T is a right angle.
a. Draw a diagram and label it.
b. Set up an equation and solve for x.
c. Find the degree measure of each angle.
review
Find the value of x in each figure.
18. ∆RST ~ ∆MNP
S
4
6
5
R
T
19. ∆ABC ~ ∆UYZ
N
24
A
B
14
x
Z
P
20.
21.
x°
Y
x
C
M
8
U
7
8
73°
69°
>>
10
x
>
>
>>
Lesson 12 ~ Angle Sum Of A Quadrilateral
69
t ic -t Ac -t oe ~ s u m
oF
i n t e r ior A ngl e s
The sum of the three angles in a triangle is 180°. Discover how to
find the sum of the angles of any polygon based on its number of sides.
1. Copy and complete the chart.
number of triangles
sum of degree
measures in the
triangles
Conclusion: Angle
sum
Quadrilateral
4 sides
2
180° + 180°
or
2(180)°
360°
Pentagon
5 sides
3
Polygon name and
number of sides
diagram
hexagon
6 sides
heptagon
7 sides
octagon
8 sides
nonagon
9 sides
decagon
10 sides
2. Does a pattern form in the angle sums? How does the pattern relate to the number of sides
of the polygon?
3. Write a formula to calculate the degree measure of any polygon based on its number of sides, n.
4. Find the degree measure of these polygons:
a. Dodecagon (12 sides)
70
Lesson 12 ~ Angle Sum Of A Quadrilateral
b. 15-gon
c. 24-gon
d. 41-gon
speciaL QuadriLateraLs
Lesson 13
A
>
>
>
>
>
>
>
>>
>>
parallelogram is a quadrilateral with both pairs of opposite sides parallel. Rectangles and squares
are types of parallelograms.
>>
>>
>>
>
>>
>>
>>
In all parallelograms, the opposite sides are congruent and the opposite angles are congruent.
>> 70°
>
>
110°
70°
>>
110°
Two consecutive angles in a parallelogram are supplementary. Each pair of consecutive angles will always
add up to 180°. In the diagram above, two consecutive angles, 70° and 110°, sum to 180°.
Find the values of a, b and c.
13
>
b°
>>
54°
>
solution
7
>>
exampLe 1
c
a°
The variable a is opposite the angle which measures 54°, so a = 54.
The variable b is a consecutive angle with 54º.
This makes the angles supplementary.
Subtract 54 from both sides of the equation.
b + 54 = 180°
b = 126
The variable c is opposite the side which measures 13 units, so c = 13.
Lesson 13 ~ Special Quadrilaterals
71
A trapezoid is a quadrilateral with exactly one pair of parallel sides. A trapezoid has two bases and two legs.
If the legs are congruent, it is an isosceles trapezoid.
Base
>
>
>
Leg
Isosceles
trapezoid
|
trapezoid
>
Base
If a trapezoid is isosceles, then each pair of base angles is congruent.
|
114°
>
y°
|
|
z°
18
solution
>
114°
99°
x
66°
|
Find the values of x, y and z.
>
66°
A top base angle and a bottom base angle form a pair of same-side
interior angles. This means they are supplementary. In the
example at the right:
114° + 66° = 180°
exampLe 2
|
Leg
12
>
The variable x is congruent to the opposite leg, so x = 12.
The variable y forms a pair of base angles with 99°, so y = 99.
The angle represented by z is the supplement of 99° because
the angles form a pair of one top and one bottom base angle.
Subtract 99 from both sides of the equation.
72
Lesson 13 ~ Special Quadrilaterals
z + 99 = 180
−99 −99
z = 81
exercises
1. Draw a parallelogram. Write angle measures in the parallelogram so that opposite angles are equal
and consecutive angles are supplementary. The sum of the angles must be 360°.
2. Draw an isosceles trapezoid.
a. Place congruence marks on the appropriate sides.
b. Write angle measures in the trapezoid so that each pair of base angles is congruent and pairs
of top and bottom base angles are supplementary. The sum of the angles must be 360°.
Find the values of x and y in each figure.
>
82°
x°
4.
96°
x°
>>
>>
|
|
3.
>
y°
>
> y°
>
(x + 8)°
>
6.
(2x − 1)°
|
>>
>>
|
5.
>
(5y − 4)°
>
y°
7.
61°
115°
8.
3x + 5
>>
15
72°
|
>
>
2y
6
2y°
>
>
>>
17
|
11
10.
|
19
>
|
2x°
3 − 4y
(5x − 25)°
x°
>
>
(4x + 1)°
>>
9.
>>
10x
6
>
y+4
Lesson 13 ~ Special Quadrilaterals
73
11. Explain what special angle pair is used to determine that a top base angle and a bottom base angle in an
isosceles trapezoid are supplementary.
12. Lucy and Eddie each drew a parallelogram that measured 1 inch on every side. Lucy argues that Eddie’s
figure is a square, not a parallelogram. Do you agree or disagree with Lucy? Why?
Eddie’s
Drawing
Lucy’s
Drawing
13. A teacher asked her math class to draw an isosceles
trapezoid with one base that is 3 centimeters long. The
other base is 7 centimeters long. Their trapezoids should
have a pair of 60° angles and a pair of 120° angles. Will
everyone’s trapezoid look the same? Why or why not?
Support your answer with sketches.
14. Find the values of a, b, c, d and e in the figure below.
21
>>
a°
b°
9
>
>
5d − 1
>>
52°
(2c + 6)°
4e + 3
15. Find the values of a, b, c and d in the figure below.
>
a°
|
|
6d
67°
18
c°
74
Lesson 13 ~ Special Quadrilaterals
> (10b − 3)°
review
Find the values of x and y in each figure.
16.
17.
(−5 + 5y)° 120°
3x°
>
x°
113°
(2y − 7)° >
18.
19. ∆HJK ~ ∆PGR
>
3
y
H
2
4
3
x
P
>
12
13
y
4
J
t ic -t Ac -t oe ~ l A n d
oF
5
K
G
x
R
s P e c i A l F igu r e s
Write a fiction story about the Land of Special Figures. The story should
include the Special Triangles and Quadrilaterals from Lessons 9 and 13.
Also research and include at least one additional special figure and its
properties (i.e. rhombus, kite, pentagon, etc). Throughout the story, the
figures should reveal properties about their sides and angles. Title your
story and include illustrations.
Lesson 13 ~ Special Quadrilaterals
75
review
BLoCK 2
vocabulary
congruent figures
corresponding parts
equiangular
equilateral triangle
isosceles trapezoid
isosceles triangle
parallelogram
quadrilateral
scalene triangle
similar figures
trapezoid
Lesson 7 ~ Classifying Triangles
Classify each triangle by its sides and angle measures.
2.
4
|
5
3.
60°
108°
5
|
1.
7
60°
|
3
7
4.
5.
10
6.
5.7
4
6
||
||
100°
60°
70°
70°
4
sketch and label a triangle to match each description.
7. Scalene ∆GED
76
Block 2 ~ Review
8. Obtuse ∆PTA
9. Acute, isosceles ∆RAM
3
Lesson 8 ~ Angle Sum of a Triangle
solve for x in each triangle.
10.
30°
11.
122°
x°
(4x − 3)°
12.
2x°
2x°
30°
(2x − 3)°
A
3x°
13. Use ∆APE at the right.
a. Set up an equation to find the value of x.
b. Solve for x.
c. Find m∠A and m∠P.
(x + 2)°
P
E
14. In ∆CAR, m∠A = (15 – 3x)° and m∠R = (5 – 2x)°. The measure of ∠C is 80°.
a. Sketch and label a diagram of ∆CAR.
b. Write an equation and solve for x.
c. Find m∠A and m∠R.
R
2x °
15. Nancy determined that the value of x in the triangle at the right is 20.
a. Find the measure of each angle by substituting 20 for x.
b. Was Nancy’s solution of x = 20 correct? How do you know?
T
(3x + 6)°
(4x − 8)°
S
Lesson 9 ~ Special Triangles
Find the value of x in each diagram.
16.
x°
17.
||
18.
||
|
|
46°
(2x + 4)°
|
x°
|
25°
|
19. ∆TWL has two angles that measure 63°.
a. Sketch a diagram of ∆TWL.
b. Find the measure of the third angle.
c. What type of triangle is ∆TWL based on its side lengths and angle measures?
20. One angle in an equilateral triangle is (6x + 12)°. Solve for x.
Block 2 ~ Review
77
21. Francisco has a triangular window in his bedroom. It is the shape of an equilateral triangle.
a. What are the measures of each angle in the window?
b. He has equal-sized pieces of framing he wants to put around the window. It takes 6 pieces of
framing to cover one side of the window. How much will it take to go around the entire window?
22. Shannon’s home makes an isosceles triangle with her school and the park as shown in the diagram below.
Home
(3.5 − 2x) km
1.5 km
School
35°
2.5 km
35°
Park
a. Find the value of x.
b. How far is it from Shannon’s home to the park?
Lesson 10 ~ Congruent and Similar Triangles
23. ∆PLA ~ ∆NET. List the corresponding angles and sides.
L
P
E
T
A
N
solve for x in each set of similar triangles.
24.
x
x
5
2
26.
3
25.
5
20
4
27.
x
12
x
8
4
1.5
6
6
28. Are the triangles below similar? Explain your answer.
58°
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Block 2 ~ Review
58°
Lesson 11 ~ Parallel Lines and Similar Triangles
29. Use the diagram at the right.
a. What angle in ∆WSB measures 49°?
b. What is the m∠WAM?
c. What is the m∠BWS?
d. What property can you use to say ∆WAM ~ ∆WBS?
A
>
W
B
M
E
M
49°
51°
>
S
30. Use the diagram at the left.
58°
>
Y
a. What angle besides ∠YFG measures 42°?
b. What is the measure of ∠FYE?
c. What is the measure of ∠M?
d. Fill in the blank: ∆MYF ~ ∆____.
G
42°
>
F
Find the values of x and y in each pair of similar triangles.
31.
32.
50°
4
6
x
62°
>
71°
y°
6
6
8
>
42°
x
5
y°
>
>
10
33. Find the measure of a, b, c, d and e.
>
a°
4
52°
e°
5
b°
c°
>
d
78°
12.5
>
34. Sketch similar triangles formed by parallel lines, transversals and same-side interior angles.
Use congruence marks to show the congruent angles.
Block 2 ~ Review
79
Lesson 12 ~ Angle Sum of a Quadrilateral
35. What is the sum of the angles in a quadrilateral?
set up an equation and solve for x.
36.
102°
10x°
37.
x°
110°
38.
80°
(1 − 11x)°
3x°
64°
(1 − 6x)°
97°
57°
55°
39. Find the degree measure of the two unknown angles in ABCD.
B
(5x + 7)°
(2x + 1)°
C
38°
A
118°
D
40. A quadrilateral has one right angle, a 143° angle and a 56° angle. What is the measure of the fourth angle?
41. In quadrilateral TEND, ∠T and ∠E are both 60°. The measure of ∠N is (6x + 1)° and the measure of
∠D is (4x + 9)°.
a. Draw a diagram and label it.
b. Set up an equation and solve for x.
c. Find the degree measure of ∠N and ∠D.
Lesson 13 ~ Special Quadrilaterals
42. Draw an isosceles trapezoid.
a. Place congruence marks on the appropriate sides.
b. Write angle measures in the trapezoid so that each pair of base angles is congruent and pairs
of top and bottom base angles are supplementary. The sum of the angles must be 360°.
4x°
>>
Block 2 ~ Review
>>
>
46.
>
>>
70°
>
(3x + 14)°
|
(x + 20)°
80
10x°
>
45.
>
>
|
98°
44.
110°
>>
|
43.
|
Find the value of x in each figure.
>
x°
>
(5x − 2)°
47. A skate jump is the shape of an isosceles trapezoid. The bottom angles of
the trapezoidal jump are 40° each. What is the measure of one of the top
angles of the jump?
48. Explain what special angle pair is used to determine that two consecutive
angles in a parallelogram are supplementary.
t ic -t Ac -t oe ~ t r i A ngl e i n e QuA l i t y t h e or e m
The Triangle Inequality Theorem states that for any triangle, the length
of a given side must be less than the sum of the other two sides but
greater than the difference between the two sides.
For example, two sides of a triangle are 3 and 8.
The length of the third side, x, must be:
x < 11
x>5
◆ Less than the sum of the given two sides (3 and 8).
◆ Greater than the difference of the given two sides (3 and 8).
5 < x < 11
These findings can be written as a compound inequality.
Write a compound inequality showing the possible lengths of the third side of a triangle given the
other two lengths.
1. 5 and 9
2. 10 and 12
3. 6 and 7
Write the possible integer side lengths for the third side of a triangle given the other two lengths.
4. 4 and 7
5. 2 and 5
6. 1 and 9
determine if each triangle has measurements that can form a triangle. If not, re-draw the figure
with one side length changed. show all work.
3
7.
5
4
9
3
8.
10
12
9.
1
5
Block 2 ~ Review
81
A lden
A rchitect
PortlAnd, oregon
CAreer
FoCus
I am an architect. Architects are designers. Designs can be as small as a
piece of furniture or as large as a city. The primary focus for most architects
is buildings. Architects use a process to complete projects which includes
determining what a client wants, designing it and drawing up plans. Architects
also help oversee the bidding process and actual construction of a project.
Math is used everyday in architecture. Proportion, scale and dimension are important components
architects use to create a plan. Architects also have to estimate how much a project will cost. If a client
cannot afford to build what the architect has designed, then the design is of no use. Architects prepare
budgets and cost estimates throughout the entire architectural process to avoid this problem.
Architects have to do math calculations to account for every square inch of space. Calculations have
to be very precise to make sure the buildings designed will not fall over.
People need a Bachelor’s degree in Architecture from a college or university to become an architect.
This usually takes about five years. After getting a degree, people do on-the-job training. They must
pass tests to become a licensed architect. Architects often acquire a Master’s degree in Architecture.
The average starting salary for a college graduate architect is $35,000 - $45,000 per year depending on
where you live in the country. A mid-level architect makes between $55,000 - $75,000 per year. A senior
architect or a firm principal can make $80,000 - $100,000 or more.
Architecture offers something new every day. Whether it is putting together a design proposal,
preparing a set of construction drawings, overseeing construction in the field or picking materials for
a new project, architects are constantly being challenged with new tasks. Also, as a building goes up,
an architect gets to see their hard work paying off. It is truly a rewarding profession.
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