Download Triangle Inequality (third side)

Algebra / Geometry II:
Unit 5- Triangles
SUCCESS CRITERIA:
1. Use the concepts of perpendicular, perpendicular bisector, & angle bisector to find measures of
segments & angles.
2. Use the concepts of triangle inequality to determine if three lengths can be a triangle and their
relationship to each other.
3. Use the concept of triangle inequality to determine the relationship of two angles or sides.
INSTRUCTOR: Craig Sherman
Hidden Lake High School
Westminster Public Schools
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EMPOWER Recorded TARGET
SCALE THEME
MA.10.G.01.04
Geometric Theorems
RELATED LT’s
MA.09.H43.02.04
MA.10.H41.02.04
SCALE THEME
Prove the slope criteria for parallel and perpendicular lines and
use them to solve geometric problems
Points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
D.O.K.
3
3
PROFICIENCY SCALE:
SCORE
REQUIREMENTS
4.0
In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go
BEYOND what was taught in class.
Score 4.0 does not equate to more work but rather a higher level of performance.
3.5
In addition to Score 3.0 performance, in-depth inferences and applications with partial success.
3.0
The learner exhibits no major errors or omissions regarding any of the information and
processes (simple or complex) that were explicitly taught.
o Use the concepts of perpendicular, perpendicular bisector, & angle bisector to find measures
of segments & angles AND
o Use the concepts of triangle inequality to determine if three lengths can be a triangle and
their relationship to each other, AND
o Use the concept of triangle inequality to determine the relationship of two angles or sides.
2.0
Can do one or more of the following skills / concepts:
There are no major errors or omissions regarding the simpler details and processes as the
learner…
o Identify and use perpendicular, segment bisector, perpendicular bisector, and angle bisector
to find segment length and angle measurements, OR
o Use the Triangle Sum theorem to find a missing angle measurement in a triangle, OR
o List sides and angles in order from greatest to smallest or vice versa, OR
o Determine if three segment lengths could form a triangle, OR
o Determine the possible lengths of a third side of a triangle given the other two sides, OR
o Determine the relationship of two angle or two sides based on the Hinge Theorem, OR
o Determine the relationship of two angle or two sides based on the Hinge Theorem.
1.0
Know and use the vocabulary
Identify the Basic Elements
With help, a partial understanding of some of the simpler details and process
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Perpendiculars and Bisectors
WORD or CONCEPT
DEFINITION or NOTES
EXAMPLE or GRAPHIC REPRESENTATION
perpendicular
segment bisector
perpendicular
bisector
angle bisector
INSTRUCTION 1: SEGMENT BISECTOR TUTORIAL
INSTRUCTION 3: ANGLE BISECTOR TUTORIAL
INSTRUCTION 2: PERPENDICULAR BISECTOR TUTORIAL
BISECTORS
ANGLE
DEFINITION
SEGMENT
POINT ON BISECTOR
BISECTOR
DEFINITION
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PERPENDICULAR BISECTOR
POINT ON
BISECTOR
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Class Work
True or False #1-5
1. A bisector of a segment is perpendicular to the segment it bisects.
2. The angle bisector creates two adjacent congruent angles.
3. A point can bisect a segment.
AC then AC = 2 × AB.
1
If P lies in the interior of <CDE, and P lies on the angle bisector of <CDE then m < CDP = m < CDE
2
4. If A, B, and C are collinear and B lies on the perpendicular bisector of
5.
Identify the red-dashed line as the bisector, perpendicular, perpendicular bisector, or angle bisector.
6.
7.
8.
9.
2
1
m<1=m<2
10. Is enough information given to
determine if P lies on the perpendicular
bisector of
10
A
AB .
Explain.
P
11. Is enough information given to
determine if P lies on the angle
bisector of <ABC. Explain.
10
B
6
P
6
A
C
B
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Homework
Fill in the blank. #12-15
XY then XM @ ________.uuur
If W is on the angle bisector of <PQR then W is equidistant from QP and ______.
12. If M is on the perpendicular bisector of
13.
14. To find the distance from a point to a line we measure the _____ distance from the point to the line.
15. A bisector of a segment intersects the segment at its ______.
Decide if enough information has been given to determine if F lies on the perpendicular bisector of
your answer.
16.
17.
CD .
Explain
28.
C
F
F
5
25°
25°
F
5
D
C
C
D
D
Determine if enough information has been given to determine if P lies on the angle bisector of <ABC. Explain your
answer. #21-23
19.
20.
21.
B
4
A
4
B
C
A
P
C
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P
A
8
P
8
B
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C
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Triangle Inequality (comparison)
NOTES:
1. Triangle Sum Theorem: The sum of the interior angles of a triangle = 1800
m∠𝐴 + m∠𝐵 + m∠𝐶 = 1800
INSTRUCTION 1: TRIANGLE SUM THEOREM
2. Order of size
a. The side opposite the Largest angle is the Largest side.
b. The side opposite the Smallest angle is the Smallest side
INSTRUCTION 2: COMPARING SIDES and ANGLES TUTORIAL
Classwork
For each triangle list the sides from greatest to smallest #22-24.
22.
23.
24.
B
H
D
30°
A
C
65°
F
24°
112°
E
G
76°
I
75°
For each triangle list the angles in order from greatest to smallest #25-27.
25.
26.
27.
L M
5.7
4
O
Q
6.75
2.5
K
J
62
43
60
N
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3.8
P
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5.5
R
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Homework
For each triangle list the sides from greatest to smallest #28-30.
28.
29.
30.
E
B
H
57°
41°
95°
C D
A
G
F
63°
80°
18°
I
For each triangle list the angles in order from greatest to smallest 31-33.
31.
32.
33.
M
K
H
18
N
10
5
7
G
8
L
11
I J
7
27
33
O
List the sides in order from shortest to longest #34-35.
34.
35.
H
T
75° 40°
80°
S
V
40° 80°
G
75°
70°
U
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110° 40°
F
60°
I
J
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Triangle Inequality (third side)
NOTES:
1. The sum of the two smallest sides of a triangle must be bigger than the largest side
EXEMPLAR: Given sides of 4, 5 , and 6
4+5>6
9>6
INSTRUCTION 1: TRIANGLE INEQUALITY TUTORIAL
2. Given two sides of a triangle
a. The third side must be less than the sum
b. The third side must be greater than the difference
EXEMPLAR: Given sides of 4 and 7
7 – 4 < third side < 7 + 4
3 < third side < 11
Class Work
Will the three lengths given make a triangle?
36. 2, 3, and 4
37. 1, 3, and 4
38. 5, 6, and 7
39. 16, 8, and 7
40. 20, 10, and 10
41. 8x, 7x, and 14x
Given the lengths of two sides of a triangle, what lengths could the third side, x, have?
42. 12 and 14
44. 22 and 22
46. 8y and 10y
43. 15 and 6
45. 9 and 12
Homework
Will the three lengths given make a triangle?
47. 21, 34, and 49
48. 11, 31, and 44
49. 8, 6, and 5
50. 12, 5, and 7
51. 20, 30, and 11
52. 9x, 17x, and 26x
Given the lengths of two sides of a triangle, what lengths could the third side, x, have?
53. 10 and 21
55. 30 and 30
57. 4y and 14y
54. 19 and 8
56. 5 and 15
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Hinge Theorem
NOTES:
INSTRUCTION 1
HINGE THEOREM TUTORIAL
Class Work
Decide whether the Hinge Theorem can be used to determine if x or y is larger. If the Hinge Theorem
is applicable, state whether x or y is larger.
58.
59.
60.
61.
62.
Homework
Decide whether the Hinge Theorem can be used to determine if x or y is larger. If the Hinge Theorem
is applicable, state whether x or y is larger.
63.
64.
65.
66.
67.
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68.
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TRIANGLES
UNIT REVIEW
Multiple Choice:
1. If LN= 12, PR= 20, and QN= 14, then the perimeter of ∆𝑃𝐿𝑁 =
a. 23
b. 36
c. 72
d. 92
2. Using the figure at right, list the segments from least to greatest.
̅̅̅̅ , 𝐴𝐶
̅̅̅̅ , 𝐵𝐶
̅̅̅̅ , 𝐴𝐷
̅̅̅̅ , 𝐷𝐶
̅̅̅̅
a. 𝐵𝐶
̅̅̅̅, 𝐵𝐶
̅̅̅̅ , 𝐴𝐶
̅̅̅̅ , 𝐴𝐷
̅̅̅̅ , 𝐷𝐶
̅̅̅̅
b. 𝐴𝐵
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
c. 𝐴𝐷 , 𝐷𝐶 , 𝐴𝐶 , 𝐴𝐵, ̅̅̅̅
𝐵𝐶
d. cannot be determined
3. Use the diagram to
find x.
a. 130
b. 125
c. 120
d. 110
4. Which of the following values cannot be the third side of a triangle if two of the sides are 14 and 20?
a. 18
c. 32.5
b. 20
d. 34
5. Can the Hinge Theorem be used to find whether x or y is larger? If yes, which is larger?
a. yes; x is larger
b. yes; y is larger
c. yes; x and y are equal
d. no, the hinge theorem does not apply
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Extended Response:
6. French doors that open in the same direction from the center.
a. If the left door is open 35° and the right door is open 38°, which door is closer to being shut?
b. What theorem supports your answer from part a?
7. Given the figure at the right:
a. If the 𝑚∠𝐻𝐽𝐺 = 20 and 𝑚∠𝐺𝐻𝐽 = 145, which
side of the triangle HGJ is the shortest?
b. Using the angle measures from part a,
find 𝑚∠𝐴𝐺𝐻.
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