Download Unit 4: Relationships Within Triangles

Unit 4: Relationships Within Triangles
Scale for Unit 4
4
Through independent work beyond what was taught in class, students could
(one example)analyze Orlando’s city planning by investigating the logic in the placement of
the Amway Arena as it relates to surrounding highways.
3
The students will:
 construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
 prove theorems about lines and angles; use theorems about lines and angles to solve
problems.
 prove theorems about triangles; use theorems about triangles to solve problems.
 make formal geometric constructions with a variety of tools and methods
 use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
2
1
The students will:
 determine the meaning of symbols, key terms, and other geometry specific words
and phrases.
 recall how to find the midpoint of a segment.
 recall how to find the distance between two points.
 recall how to construct a circle, lines, angles, and segments.
 understand how to distinguish congruence and similarity criteria for triangles
The student will be able to correctly use the following vocabulary: midpoint, triangle,
segment, lines, points, distance, and proof.
Ranking:
Date
Level
Notes: (what you didn’t understand from the chapter and want to work on)
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Sample Problem Scale:
Level 4
Level 3
Level 2
Level 1
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BELLWORK:
Date: __________________________ Date: __________________________
Date: __________________________ Date: __________________________
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BELLWORK:
Date: __________________________ Date: __________________________
Date: __________________________ Date: __________________________
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Vocabulary Activity:
Vocabulary Word
Definition
Picture
Altitude of a Triangle
Centroid
Circumcenter
Concurrent
Equidistant
Incenter
Median
Midsegment of
a Triangle
Orthocenter
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Section 5-1 – Midsegments of Triangles
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EXAMPLES:
Find the value of x:
Find the value of x:
Find the value of x:
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Section 5-2 – Perpendicular and Angle Bisectors
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EXAMPLES:
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Section 5-3 – Bisectors in Triangles
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EXAMPLES:
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Section 5-4 – Medians and Altitudes
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EXAMPLES:
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MIDUNIT REVIEW
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Section 5-5 – Indirect Proof
The goal of this game is to fill in the empty squares with numbers. The numbers 1, 2, 3, and
4 must appear once in each row and once in each column. Complete both games.
GAME A
1
GAME B
2
4
3
1
4
1
2
2
1
4
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Section 5-6 – Inequalities in One Triangle
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EXAMPLES:
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Section 5-7 – Inequalities in Two Triangles
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UNIT 5 REVIEW:
Lesson 5-1
Find the value of x.
1.
2.
3.
In the figure at the right, points Q, R, and S are midpoints of LMN.
4. Identify all pairs of parallel segments.
5. If LR = 25, find QS
6. If MN = 40, find QR.
7. If mLNM = 53, find mSQM.
8. A sinkhole caused the sudden collapse of a large section of
highway. Highway safety investigators paced out the triangle
shown in the figure to help them estimate the distance across the
sinkhole. What is the distance across the sinkhole?
Lesson 5-2
Use the figure at the right for Exercises 9–12.
9. Find the value of x.
10. Find the length of AD.
11. Find the value of y.
12. Find the length of EG.
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13. Given: AP  AQ, BP  BQ, CP  CQ
14. Given: CX
bisects BCN.
BX bisects CBM.
Prove: A, B, and C are collinear.
Prove: X is on the bisector of A
15. Find an equation in slope-intercept form for the perpendicular bisector of the segment with endpoints H(–3, 2) and
K(7, –5).
Lesson 5-3
Find the center of the circle that you can circumscribe about ABC.
16. A(2, 8)
17. A(–3, 6)
18. A(4, 3)
B(0, 8)
B(–3, –2)
B(– 4, –3)
C(2, 2)
C(7, 6)
C(4, –3)
Use the figure at the right for Exercises 19 and 20.
19. What is the point of concurrency of the angle bisectors of QRS?
20. Find the value of x if QC = 2x + 3 and
CS = 5 – 4(x + 1).
21. Find the center of the circle that you can circumscribe about the triangle with vertices A(1, 3), B(5, 8), and C(6, 3).
22. Draw an acute triangle and construct its inscribed circle.
Lesson 5-4
In DEF, L is the centroid.
23. If HL = 30, find LF and HF.
24. If KE = 15, find KL and LE.
25. If DL = 24, find LJ and DJ.
Is AB a median, an altitude, a perpendicular bisector, or none of these? How do you know?
26.
27.
28.
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Find the coordinates of the orthocenter of ∆ABC.
29. A(1, 0)
30. A(2, 5)
B(0, 4)
B(6, 2)
C(3, 0)
C(2, 1)
31. A(1, 1)
B(4, 0)
C(2, 4)
32. Tell which line contains each point for ∆ABC.
a. the circumcenter
b. the orthocenter
c. the centroid
d. the incenter
Lesson 5-5
Write the first step of an indirect proof.
33. ∆ABC is a right triangle.
34. Points J, K, and L are collinear.
35. Lines
36.
and m are not parallel.
Identify the two statements that contradict each other.
37. I. The perimeter of ∆XYZ is 10 cm.
II. No side of ∆XYZ is longer than 5 cm.
III. No side of ∆XYZ is shorter than 4 cm.
38.
XYZV is a square.
I. ∆QRS is isoceles.
II. ∆QRS has at least one acute angle.
III. ∆QRS is equiangular.
39. Suppose you know that A is an obtuse angle in ∆ABC. You want to prove that B is an acute angle.
What assumption would you make to give an indirect proof?
Lesson 5-6
Can a triangle have sides with the given lengths? Explain.
40. 2 in., 3 in., 5 in.
41. 9 cm, 11 cm, 15 cm
42. 8 ft, 9 ft, 18 ft
List the sides of each triangle in order from shortest to longest.
43.
44.
45.
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46. In ∆PQR, mP = 55, mQ = 82, and mR = 43. List the sides of the triangle in order from shortest
to longest.
47. In ∆MNS, MN = 7, NS = 5, and MS = 9. List the angles of the triangle in order from smallest to
largest.
48. Two sides of a triangle have side lengths 8 units and 17 units. Describe the lengths x that are
possible for the third side.
Lesson 5-7
Write an inequality relating the given side lengths. If there is not enough information to reach a conclusion, write no
conclusion.
49. AC and XZ
50. DF and FG
Find the range of possible values for each variable.
51.
52.
Copy and complete with > or <. Explain your reasoning.
53. mAEB
54. AB
55. BC
mEBD
BC
ED
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