1

Geometry

Unit 5 Relationships in Triangles

Name:________________________________________________

2

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 

3

Date: _____________________________

4

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

5

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:

6

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.

7

Example #4: FindxandEFif BD isananglebisector.

Example #5: In∆DEF,GI isaperpendicularbisector.

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 EGHm = 9z.

8

 

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? 

 

 

   

   

9

 

 

10

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.

11

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: FindxandRTif SU isamedianof∆RST.Is SU alsoanaltitudeof∆RST?Explain.

Example #3: FindxandIJifHK isanaltitudeof∆HIJ.

12

 

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 ? 

   

13

 

 

14

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 8m

b.) all angles whose measures are greater than 2m

15

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

 

 

16

CRITICAL THINKING   

 

 

1.)  Find The Error:  Hector and Grace each labeled QRS . 

 

 

 

 

 

Who is correct?  Explain. 

 

 

 

 

2.)  Write and solve an inequality for x.   

17

18

Name ________________________ Period ____________ Chapter 5 (5.4)

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

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7

First side

Second side

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 ______________________________________________________________________.

19

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

20

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:

21

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.