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Geometry
Unit 5
Relationships in Triangles
Name:________________________________________________
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Geometry
Chapter 5 – Relationships in Triangles
***In order to get full credit for your assignments they must me done on time
and you must SHOW ALL WORK. ***
1.____ (5-1) Bisectors, Medians, and Altitudes – Page 235 1-13 all
2. ____ (5-1) Bisectors, Medians, and Altitudes – Pages 243-244 11-22 all
3. ____ (5-1) Bisectors, Medians, and Altitudes – 5-1 Practice Worksheet
4. ____ (5-2) Inequalities and Triangles – Pages 252-253 17-25, 29-34, 37-43, 46, 47
5. ____ (5-2) Inequalities and Triangles – 5-2 Practice Worksheet
6.____ (5-4) The Triangle Inequality – Pages 264-266 14-36 even, 57, 58
7. _____ Chapter 5 Review WS
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Date: _____________________________
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Section 5 – 1: Bisectors, Medians, and Altitudes
Notes – Part A
Perpendicular Lines:
Bisect:
Perpendicular Bisector: a line, segment, or ray that
passes through the __________________ of a side of
a ________________ and is perpendicular to that side
Points on Perpendicular Bisectors
Theorem
5.1:
Any
point
on
the
perpendicular bisector of a segment is
_____________________ from the endpoints
of the _________________.
Example:
Concurrent Lines: _____________ or more lines that intersect at a common
_____________
Point of Concurrency: the point of ___________________ of concurrent lines
Circumcenter:
the point of concurrency of the _____________________
bisectors of a triangle
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Circumcenter
Theorem:
the
circumcenter of a triangle is equidistant
from the ________________ of the
triangle
Example:
Points on Angle Bisectors
Theorem 5.4:
Any point on the angle
bisector is ____________________ from
the sides of the angle.
Theorem 5.5: Any point equidistant from
the
sides
of
an
angle
lies
on
the
____________ bisector.
Incenter: the point of concurrency of the angle ________________ of a triangle
Incenter Theorem: the incenter of a triangle is _____________________ from
each side of the triangle
Example:
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Example #1: RI bisects SRA . Find the value of x and mIRA .
Example #2: QE is the perpendicular bisector of MU . Find the value of m and
the length of ME .
Example #3: EA bisects DEV . Find the value of x if mDEV = 52 and
mAEV = 6x – 10.
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Example #4: Find x and EF if BD is an angle bisector.
Example #5: In ∆DEF, GI is a perpendicular bisector.
a.) Find x if EH = 19 and FH = 6x – 5.
b.) Find y if EG = 3y – 2 and FG = 5y – 17.
c.) Find z if mEGH = 9z.
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CRITICAL THINKING
1.) Draw a triangle in which the circumcenter lies outside the triangle.
2.) For what kinds of triangle(s) can the perpendicular bisector of a side
also be an angle bisector of the angle opposite the side?
3.) For what kind of triangle do the perpendicular bisectors intersect in a
point outside the triangle?
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Date: _____________________________
Section 5 – 1: Bisectors, Medians, and Altitudes
Notes – Part B
Median: a segment whose endpoints are a ______________ of a triangle and the
___________________ of the side opposite the vertex
Centroid: the point of concurrency for the ________________ of a triangle
Centroid Theorem: The centroid of a triangle
is located _________ of the distance from a
____________ to the __________________ of
the side opposite the vertex on a median.
Example:
Example #1:
Points S, T, and U are the midpoints of DE, EF , and DF ,
respectively. Find x.
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Altitude:
a
segment
from
a
_______________ to the line containing
the
opposite
side
and
_______________________ to the line
containing that side
Orthocenter: the intersection point of the
____________________
Example #2: Find x and RT if SU is a median of ∆RST. Is SU also an altitude
of ∆RST? Explain.
Example #3: Find x and IJ if HK is an altitude of ∆HIJ.
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CRITICAL THINKING
1.) R(3, 3), S(-1, 6), and T(1, 8) are the vertices of RST , and RX is a median.
a.) What are the coordinates of X?


b.) Find RX.
c.) Determine the slope of RX .

d.) Is RX an altitude of RST ? Explain.


2.) Draw any XYZ with median XN and altitude XO. Recall that the area
of a triangle is one-half the product of the measures of the base and the
altitude. What conclusion can you make about the relationship between



the areas of XYN and XZN ?


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Date: _____________________________
Section 5 – 2: Inequalities and Triangles
Notes
Definition of Inequality:
For any real numbers a and b, ____________ if and only if there is a positive
number c such that _________________.
Example:
Exterior Angle Inequality Theorem: If an angle is an ________________ angle
of a triangle, then its measures is ________________ than the measure of either of
its ________________________ remote interior angles.
Example:
Example #1: Determine which angle has the greatest measure.
Example #2: Use the Exterior Angle Inequality Theorem
to list all of the angles that satisfy the stated condition.
a.) all angles whose measures are less than m8
b.) all angles whose measures are greater than m2
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Theorem 5.9: If one side of a triangle is ________________ than another side,
then the angle opposite the longer side has a _______________ measure than the
angle opposite the shorter side.
Example #3:
Determine the relationship between the measures of the given
angles.
a.) RSU , SUR
b.) TSV , STV
c.) RSV , RUV
Theorem 5.10: If one angle of a triangle has a ________________ measure than
another angle, then the side opposite the greater angle is ________________ than
the side opposite the lesser angle.
Example #4: Determine the relationship between the lengths of the given sides.
a.) AE, EB
b.) CE, CD
c.) BC, EC
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CRITICAL THINKING
1.) Find The Error: Hector and Grace each labeled QRS .

Who is correct? Explain.
2.) Write and solve an inequality for x.
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Name ________________________
Chapter 5 (5.4)
Period ____________
Use your paper strips to determine whether a triangle can be formed.
Complete the following chart using the correct values.
Orange = 2 inches
Yellow = 3 inches
Blue = 4 inches
Green = 5 inches
Side
measure
First
side
Second
side
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7
Third
side
Is it a
triangle?
What can you conclude from the data in the table above?
Complete the following sentence:
In order to have a triangle, the sum of two smallest sides must be
_____________________________________________________
_________________.
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Date: _____________________________
Section 5 – 4: The Triangle Inequality
Notes
Triangle Inequality Theorem: The sum of the lengths of any two sides of a
_________________ is _________________ than the length of the third side.
Example:
Example #1: Determine whether the given measures can be the lengths of the
sides of a triangle.
a.) 2, 4, 5
b.) 6, 8, 14
Example #2: Find the range for the measure of the third side of a triangle given
the measures of two sides.
a.) 7 and 9
b.) 32 and 61
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Theorem 5.12: The perpendicular segment from a ____________ to a line is the
_________________ segment from the point to the line.
Example:
Corollary 5.1:
The perpendicular segment from a point to a plane is the
________________ segment from the point to the plane.
Example:
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CRITICAL THINKING
1.) Find The Error: Jameson and Anoki drew EFG with FG = 13 and
EF = 5. They each chose a possible measure for GE.

Who is correct? Explain.
2.) Find three numbers that can be the lengths of the sides of a triangle
and three numbers that cannot be the lengths of the sides of a triangle.
Justify your reasoning, and include a picture.
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