CHAPTER 5 Geometry A/B Workbook
NAME:______
TEACHER:______
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Table of Contents
Date: Topic: Description: Page:
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Geometry Section 5.1 Notes: Bisectors of Triangles Date:
Question, Topics and Vocabulary Problems, Definitions and Work
Segment Bisector:
Perpendicular Bisector:
Perpendicular Bisector Theorem:
Converse of the Perpendicular Bisector
Theorem:
Example 1:
a) Find the length of BC.
What type of relationship should I use in order to find the length of BC?
b) Find the length of XY.
How is this diagram different to the previous diagram?
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c) Find the length of PQ.
Concurrent Lines:
Point of Concurrency:
Circumcenter
Theorem:
Example 2: A triangular-shaped garden is shown. Can a fountain be placed at the Where can the circumcenter and still be inside the garden? circumcenter be located
at?
Angle Bisector:
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Angle Bisector Theorem:
Converse of the Angle Bisector Theorem:
Example 3:
a) Find the length of DB.
What conclusion can be made if given a segment is an angle bisector?
b) Find mWYZ.
What conclusions can be made if given the two
segments from the angle bisector are equal and intersect at a right angle?
c) Find the length of QS.
Given the set up to the right, what can I do to my two expressions? And why?
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Incenter:
Incenter Theorem:
Example 4: Use the diagram to the right.
a) Find ST if S is the incenter of ΔMNP.
If S is the incenter, what do I know about my triangle?
b) Find mSPU if S is the incenter of ΔMNP.
Summary 1) Find the value of x. 2) Find the length of KL.
3) Find 4) Point A is the incenter of . Find the measure of .
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Geometry Name: ______Section 5.1 Worksheet
For numbers 1 – 6, find each measure.
1. TP 2. VU
3. KN 4. NJZ
5. QA 6. MFZ
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For numbers 7 & 8, point L is the circumcenter of ABC. List any segment(s) congruent to each segment.
7. BN
8. BL
For numbers 9 & 10, point A is the incenter of PQR. Find each measure.
9. YLA
10. YGA
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Geometry Section 5.2 Notes: Medians and Altitudes of Triangles Date:
Question, Topics and Problems, Definitions and Work Vocabulary
Median:
Centroid:
Centroid Theorem:
Example 1: In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Given the
information about triangle XYZ, how can I mark up the triangle if knowing P is the centroid?
Example 2: In ΔABC, CG = 4. Find GE.
Given the picture of the triangle, what
is G in my picture?
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Altitude:
Orthocenter:
Point of Concurrency :
Special Segments and Points in Triangles Special Property:
Perpendicular Bisector
Is there an acronym I could create to help me memorize these special segments/points in Point of Concurrency : triangles?
Special Property: Angle Bisector
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Median
Is there an acronym I could Special Property: create to help me memorize these Point of Concurrency : special segments/points in triangles?
Point of Concurrency :
Special Property: Altitude
Summary 5.2 Exit Slip
In , NQ = 6, RK = 3, and PK = 4. Find each measure using the figure below.
1) KM 2) KQ
3) LK 4) LR
5) NK 6) PM
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7. Draw an altitude from point B to ̅̅̅ ̅ in triangle ABC. Make all necessary markings.
For numbers 8 - 11, give the name of the point of concurrency for each of the following.
8. Angle Bisectors of a Triangle
9. Medians of a Triangle
10. Altitudes of a Triangle
11. Perpendicular Bisectors of a Triangle
For numbers 12 – 13, complete each of the following statements.
12. The incenter of a triangle is equidistant from the ______of the triangle.
13. The circumcenter of a triangle is equidistant from the ______of the triangle.
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Geometry Name: ______Section 5.2 Worksheet
For numbers 1 – 6, in ABC, CP = 30, EP = 18, and BF = 39. Find each measure.
1. PD 2. FP
3. BP 4. CD
5. PA 6. EA
For numbers 7 – 12, in MIV, Z is the centroid, MZ = 6, YI = 18, and NZ = 12. Find each measure.
7. ZR 8. YZ
9. MR 10. ZV
11. NV 12. IZ
13. DISTANCES For what kind of triangle is there a point where the distance to each side is half the distance to each vertex? Explain.
14. MEDIANS Look at the right triangle below. What do you notice about the orthocenter and the vertices of the triangle?
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Geometry Section 5.3 Notes: Inequalities in One Triangle Date:
Question, Topics and Vocabulary Problems, Definitions and Work
Definition of Inequality
Exterior Angle
Inequality
Example 1: Use the diagram to the right.
a) Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14.
b) Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5.
Longest Side Vs. Longest Angle
Smallest Side Vs. Shortest Side
Angle-Side Relationships in Triangles (5.9)
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Angle- Side Relationships in Triangles (5.10)
Example 2: List the angles of ΔABC in order from smallest to largest.
How do I know which side corresponds to which angle?
Example 3: List the sides of ΔABC in order from shortest to longest.
Example 4: HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?
How do I know which angle corresponds to which side?
Example 5: List the angles of each triangle in order from smallest to largest.
Example 6: List the angles of each triangle in order from smallest to largest.
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Example 7: List the angles of each triangle in order from smallest to largest.
∆XYZ, where XY = 25, YZ = 11, and XZ = 15
Example 8: List the sides of each triangle in order from shortest to longest.
Example 9: List the sides of each triangle in order from shortest to longest.
5.1-5.3 Vocab Skills Match each word to the sentence in which it best fills in the blank. Write your answer on the line to the left. Check _____1) The point of intersection of the angle bisectors in a triangle is called the_?_.
Answer Choices 12 _____2) The _?_ divides the medians in a triangle into the and parts. A. circumcenter 33 B. incenter C. sides _____3) The centroid is the intersection of the_?_in a triangle. D. vertices E. medians F. orthocenter _____4) The circumcenter is equidistant from the _?_in a triangle. G.centroid
_____5) The incenter is equidistant from the_?_in a triangle
_____6) The point of intersection of the perpendicular bisectors in a triangle is called the_?_.
_____7) The altitudes in a triangle intersect at a point called the _?_.
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Summary 1. a) List the sides from biggest to smallest. Write your answer as an inequality. B 20
E
110
M
b) List the angles from largest to smallest. Write your answer as an inequality.
A
25 12
W 15 P
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Geometry Name: ______Section 5.3 Worksheet
For numbers 1 – 4, use the figure at the right to determine which angle has the greatest measure.
1. 1, 3, 4 2. 4, 8, 9
3. 2, 3, 7 4. 7, 8, 10
For numbers 5 – 8, use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.
5. measures are less than m1.
6. measures are less than m3.
7. measures are greater than m7.
8. measures are greater than m∠2.
For numbers 9 – 12, use the figure at the right to determine the relationship between the measures of the given angles.
9. mQRW, mRWQ 10. mRTW, mTWR
11. mRST, mTRS 12. mWQR, mQRW
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Use the figure at the right to determine the relationship between the lengths of the given sides.
13. DH, GH 14. DE, DG
15. EG, FG 16. DE, EG
17. SPORTS The figure shows the position of three trees on one part of a disc golf course. At which tree position is the angle between the trees the greatest
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Geometry Section 5.5 Notes: The Triangle Inequality Date:
Question, Topics and Vocabulary Problems, Definitions and Work
Triangle Inequality Theorem
Example 1: a) Is it possible to form a triangle with side lengths of 6.5, 6.5, and 14.5? If not, explain why Is this triangle not. possible?
b) Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not.
Example 2: In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR?
a) 7 b) 9
c) 11 d) 13
Example 3: TRAVEL The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury.
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Example 4:The lengths of two sides of a triangle are given. Find the Range of the lengths: range of possible lengths for the third side.
a. 4, 8
b. 13, 8
c. 10, 15
Example 5: Error Analysis A student draws a triangle with a perimeter of 12 in. The student says that the longest side measures 7 in. How do you know that the student is incorrect? Explain.
Summary 1. Can you make a triangle out of the following lengths: 1 cm, 3 cm, and 8 cm? Why or why not?
2. Can you make a triangle out of the following lengths: 14 cm, 6 cm, and 9 cm? Why or why not?
3. Can you make a triangle out of the following lengths: 14 cm, 30 cm, and 16 cm? Why or why not?
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4. Can you make a triangle out of the following lengths: 2.5 cm, 6 cm, and 3.49 cm? Why or why not?
5. Can you make a triangle out of the following lengths: 21 cm, 8 cm, and 14 cm? Why or why not?
6. Can you make a triangle out of the following lengths: 13 cm, 14 cm, and 27 cm? Why or why not?
7. Can you make a triangle out of the following lengths: 35 cm, 30 cm, and 36 cm? Why or why not?
8. Can you make a triangle out of the following lengths: 1.4 cm, .9 cm, and 2.1 cm? Why or why not?
9. Two sides of an isosceles triangle have lengths 2 and 12, respectively. Find the length of the third side.
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Geometry Name: ______Section 5.5 Worksheet
For numbers 1 – 8, is it possible to form a triangle with the given side lengths? If not explain why not.
1. 9, 12, 18 2. 8, 9, 17 3. 14, 14, 19
4. 23, 26, 50 5. 32, 41, 63 6. 2.7, 3.1, 4.3
7. 0.7, 1.4, 2.1 8. 12.3, 13.9, 25.2
For numbers 9 – 16, find the range for the measure of the third side of a triangle given the measures of two sides.
9. 6 ft and 19 ft 10. 7 km and 29 km 11. 13 in. and 27 in.
12. 18 ft and 23 ft 13. 25 yd and 38 yd 14. 31 cm and 39 cm
15. 42 m and 6 m 16. 54 in. and 7 in.
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Geometry Section 5.6 Notes: Inequalities in Two Triangles Date:
Question, Topics and Vocabulary Problems, Definitions and Work
Inequalities in Two
Triangles 5.13
Inequalities in Two
Triangles 5.14
Example 1: Hinge Theorem a) Compare the measures AD and BD.
Hinge Theorem b) Compare the measures of ABD and BDC Converse
Example 2: HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Nitan raise higher above the table?
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When the included angle of one triangle is greater than the included angle in a second triangle, the Converse of the Hinge Theorem is used.
Range of Possible Example 3: Find the range of possible values for a.
Values
Complete with <, >, or =. Explain 1. ST ______VW
When do I know when the two sides or equal?
2. DE _____ EF
3. JK ____ LM When do I know one side is GREATER THAN the other?
4. m 1 _____ m 2
When do I know one side is 5. m 1 _____ m 2 LESS THAN the other?
6. m 1 _____ m 2
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Summary Example 1: Use the Hinge Theorem or its converse and properties of triangles to write and solve an inequality to describe a restriction on the value of x.
Example 2: Use the Hinge Theorem or its converse and properties of triangles to write and solve an inequality to describe a restriction on the value of x.
Example 3: Error Analysis Explain why the student's reasoning is not correct.
By the Hinge Theorem, AB > DC.
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Geometry Name: ______Section 5.6 Worksheet
For numbers 1 – 4, compare the given measures.
1. AB and BK 2. ST and SR
3. mCDF and mEDF 4. mR and mT
5. MOUNTAIN PEAKS Emily lives the same distance from three mountain peaks: High Point, Topper, and Cloud Nine. For a photography class, Emily must take a photograph from her house that shows two of the mountain peaks. Which two peaks would she have the best chance of capturing in one image?
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