Download Properties of Triangles

Geometry
Mrs. Franks
10th Grade
Unit Plan Information
Properties of Triangles
Jessica Franks
10th Grade
80 Minute Block Schedule
Mathematics
Geometry
Text Book: Geometry by McDougal Littell
Learning Objectives: See daily lessons
Essential Questions: What are the properties of a triangle?
How can I identify congruent triangles?
What is an altitude?
What are the different types of triangles?
What are corresponding parts?
What happens when the medians of a triangle meet?
What happens when the altitudes of a triangle meet?
When you connect the midpoints of a triangle what do you get?
Enduring Understandings:
Students are able to geometric relationships are evident in real-life
situations.
Students will be able to recognize math processes in the future and be able
to locate appropriate resource materials to assist them.
Students will be able to recognize reasoning and proof as fundamental
aspects of mathematics.
Students will be able to see relationships that exist between the angles and
sides of geometric figures can be proven.
At the conclusion of this unit the students should be able to use properties, theorems and
postulates to prove the congruency of triangles to one another.
Instructional Procedures: See Daily Lesson Plans
Day 1: Discuss Triangles & Explore Geometer’s Sketchpad
Day 2: Use Geometer’s Sketchpad to prove triangles are congruent
by SAS and SSS
Day 3: Use Geometer’s Sketchpad to prove triangles are congruent
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Geometry
Mrs. Franks
10th Grade
by ASA and AAS and look at medians and altitudes
Day 4: Use Geometer’s Sketchpad to explore the mid-segment
Theorem and Inequalities in One Triangle
Day 5: Chapter test
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Geometry
10th Grade
Mrs. Franks
Standards:
NY State
Geometry Standards
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G.G.28 Determine the congruence of
two triangles by using one of the five
congruence techniques (SSS, SAS,
ASA, AAS, HL), given sufficient
information about the sides and/or
angles of two congruent triangles
G.G.29 Identify corresponding parts of
congruent triangles
G.G.30 Investigate, justify, and apply
theorems about the sum of the
measures of the angles of a triangle
G.G.31 Investigate, justify, and apply
the isosceles triangle theorem and its
converse
G.G.32 Investigate, justify, and apply
theorems about geometric inequalities,
using the exterior angle theorem
G.G.33 Investigate, justify, and apply
the triangle inequality theorem
G.G.34 Determine either the longest
side of a triangle given the three angle
measures or the largest angle given the
lengths of three sides of a triangle
G.G.43 Investigate, justify, and apply
theorems about the centroid of a
triangle, dividing each median into
segments whose lengths are in the ratio
2:1
G.G.44 Establish similarity of
triangles, using the following theorems:
ASA, SAS, and SSS
G.G.45 Investigate, justify, and apply
theorems about similar triangles
NY State
Technology Standards
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1.a Students demonstrate a sound
understanding of the nature and
operation of technology systems
2.b Students practice responsible use of
technology systems, information, and
software.
2.c Students develop positive attitudes
toward technology uses that support
lifelong learning, collaboration,
personal pursuits, and productivity.
3.a Students use technology tools to
enhance learning, increase productivity,
and promote creativity.
5.b Students use technology tools to
process data and report results.
6.a Students use technology resources
for solving problems and making
informed decisions.
6.b Students employ technology in the
development of strategies for solving
problems in the real world.
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Geometry
10th Grade
Mrs. Franks
DATE
OBJECTIVE:
DATE
OBJECTIVE:
12/05
- Understand the key properties of
triangles using geometer’s
sketchpad (GS)
- Classify triangles by their sides
and angles
12/05
- Identify congruent figures and
corresponding parts.
- Prove that two triangles are
congruent.
- Prove that triangles are congruent
using the SSS and SAS Congruence
Postulates.
CONTENT:
CONTENT:
- Triangles
- 4.1: Triangles and Angles
- 4.2: Congruence and Triangles
- 4.3: Proving Triangles are
Congruent: SSS and SAS
ACTIVITIES:
ACTIVITIES:
- Class discussion on Triangles
- What make a triangle a triangle?
- Go to the computer lab
- Introduce geometer’s sketchpad
to the students
- Allow students to get the used to
the new program by letting them
explore
- Use geometer’s sketchpad to
explore basic properties of triangles
using geometer’s sketchpad
- Go to computer lab
- Bell Ringer
- Go over homework
- Using GS have class take notes,
practice, explore and discuss
congruency of triangles
MATERIALS NEEDED:
MATERIALS NEEDED:
Computers, Geometer’s Sketchpad,
Calculators, Worksheet for GS from
http://sierra.nmsu.edu/morandi/Cour
seMaterials/IntroToSketchpad.html
(In Notes)
Computers, Geometer’s Sketchpad,
Calculator, Compass
ASSESSMENT:
ASSESSMENT:
Student responses – verbal and
written. Class participation –
Sketch. Homework assignment.
Student responses – verbal and
written. Class participation –
Sketch. Homework assignment.
PRACTICE:
PRACTICE:
In class - sketches
Homework – Triangle Worksheet
In class - see written examples from
notes
Homework – Explore applet at
http://illuminations.nctm.org/tools/t
ool_detail.aspx?id=4 .
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Geometry
10th Grade
Mrs. Franks
DATE
OBJECTIVE:
DATE
OBJECTIVE:
12/05
- Prove that triangles are congruent
using the ASA Congruence
Postulate and the AAS Congruence
Theorem.
- Use properties of medians of a
triangle.
- Use properties of altitudes of a
triangle.
12/05
- Identify the mid-segments of a
triangle.
- Use properties of mid-segments of a
triangle.
- Use triangle measurement to decide
which side is longest or which angle is
largest.
- Use the triangle Inequality.
CONTENT:
CONTENT:
- 4.4: Proving Triangles are
Congruent: ASA and AAS
- 5.3: Medians and Altitudes of a
Triangle
- 5.4: Mid-segment Theorem
- 5.5: Inequalities in One Triangle
ACTIVITIES:
ACTIVITIES:
- Go to computer lab
- Bell Ringer
- Go over homework
- Using Geometer’s sketchpad have
the class take notes, practice,
explore and discuss congruency,
medians and altitudes of triangles
- Go to computer lab
- Bell Ringer
- Go over homework
- Using Geometer’s sketchpad
have the class take notes, practice
and discuss Mid-segment and
inequalities in one triangle.
MATERIALS NEEDED:
MATERIALS NEEDED:
Computers, Geometer’s Sketchpad,
Calculator, Worksheet for GS from
http://sierra.nmsu.edu/morandi/Cour
seMaterials/sketchpadFiles.html (in
Notes)
Computers, Geometer’s Sketchpad,
Calculator, Sketch from Key
Curriculum Press on web page
ASSESSMENT:
ASSESSMENT:
Student responses – verbal and
written. Class participation –
Sketch. Homework assignment.
Student responses – verbal and
written. Class participation –
Sketch. Homework assignment.
PRACTICE:
PRACTICE:
In class - sketches
Homework – Textbook Problems
In class - see written examples from
notes
Homework - Textbook problems.
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Geometry
10th Grade
Mrs. Franks
DATE
OBJECTIVE:
DATE
12/05
- Assess Knowledge of Students
12/05
CONTENT:
OBJECTIVE:
CONTENT:
- Chapters 4 & 5
ACTIVITIES:
ACTIVITIES:
- Go to computer lab
- Bell Ringer
- Go over homework
- Using Geometer’s sketchpad
assess triangles
MATERIALS NEEDED:
MATERIALS NEEDED:
Computers, Geometer’s Sketchpad,
Calculators, Teacher created exam
ASSESSMENT:
ASSESSMENT:
Student responses – verbal and
written. Class participation –
Sketch. Teacher created exam
PRACTICE:
PRACTICE:
In class - Teacher created exam
Individually - Teacher created exam
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Geometry
Mrs. Franks
10th Grade
Chapters
4&5
Course 2R
Mrs. Franks
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Geometry
Mrs. Franks
10th Grade
Triangles
What do you remember about triangles?
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Geometry
Mrs. Franks
10th Grade
Introduction to Geometer's Sketchpad
In this assignment we will learn how to use the program Geometer's Sketchpad. This program is
very useful for learning about geometry. We will discover several geometric facts this semester
through its use.
Here are several tasks to perform in Geometer's Sketchpad. You should use the
program enough to be able to do these tasks with ease. When you open the
program, you will see six icons on the left side of the screen. They are, from top to
bottom, the arrow tool, the point tool, the compass (or circle) tool, the
straightedge tool, the text tool, and the custom tool. The arrow tool is used to
select objects. The next three are used to draw points, circles, and lines.
One important thing to know about is how to highlight objects. By clicking on an
object it will be highlighted, and then can be used in further constructions. The
order in which you highlight objects can affect the resulting construction.
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Draw a point: Click on the point tool, then click where you want a point.
Draw a line segment: Click on the line tool. The icon should show two points and a segment
connecting them. To draw a line segment click the mouse where you want the segment to
begin, and holding the mouse, drag it until you get to where you want the line to end, then
release the mouse.
Draw a ray and line: Click and hold the mouse on the line tool until you see three icons.
These, from left to right, are the line segment, ray, and line tools. Click on the appropriate
one, then click and hold the mouse somewhere on the screen, then drag to get the ray or
line.
Draw a circle: Click on the circle tool, then click and hold the mouse, move to size the
circle. Alternatively, if you want a circle centered at a given point, with the circle tool, place
the cursor over the point and then draw the circle. If you want the circle centered at a
certain point and passing through another point, click on the center and then click on the
second point. Finally, click on construct, then circle by center and point. See what happens if
you highlight the points in reverse order and construct the circle by center and point.
Circles are determined by two points, one being the center and the other being a point on
the circle.
Resize the circle: Click on the arrow tool, then on the point on the circle. Drag this point to
resize the circle. Alternatively, click and drag the center.
Move the circle: Click on the arrow tool, then on the circle away from the point on the
circle. Drag to move the circle.
Draw a triangle: using the line segment tool, draw a line segment. Then draw a second
segment starting where the first segment ended. Finally, draw a third segment starting
where the second segment ended and ending where the first segment started.
Resize the triangle: Click the mouse on the arrow tool. Then click on one of the vertices of
the triangle (i.e., one of the endpoints), then drag the mouse to resize. Alternatively, click
and drag one of the sides.
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Geometry
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Mrs. Franks
10th Grade
Move the triangle: Click the arrow tool. Then click on two of the sides (or the three
vertices). Then drag one of the sides.
Measure the angles of the triangle: Click the arrow tool. Then click three of the vertices
in order. Then go to Measure, Angle.
Select more than one object: Click on the arrow tool. Click on the objects you wish to
select. You should see which objects are selected.
Draw the interior of a triangle: Click on the arrow tool. Then click on all three vertices of
the triangle. You should see large dots over each of them. Click the mouse on the menu item
construct, then on polygon interior.
Draw a four-sided figure: Once you have drawn it, resize it by moving one of the vertices.
Notice that you can make many different shapes.
Draw the four-sided figure's interior.
Draw an angle bisector. Geometer's Sketchpad views an angle as three points selected in
order. The middle point is the vertex, or corner, of the angle. You can then draw the angle
by drawing rays from the vertex through the other two points. Once you have drawn and
selected three points, click on construct, and then angle bisector. This line should cut the
angle into two equal pieces. If it does not appear to do so, look carefully at the order in
which you selected your three points, since there are three different angles that can be
made from the three points (the three angles of the triangle formed by the three points).
Find the intersection of two lines, segments, or circles: Draw two line segments (or rays
or lines or circles) that cross. With the point tool, put the mouse over the intersection and
click. Move one of the line segments and watch what happens to the intersection point.
Alternatively, select both line segments, then click on construct, then on intersection.
Draw perpendicular and parallel lines: Draw a line. Select the line and a point on the line.
Then click construct, then perpendicular line. This constructs a line perpendicular to the
given line and passing through the given point. Next, plot a point off of a given line. Select
the line and the point. Click construct, then parallel line. This produces a line through the
given point and parallel to the given line.
Label points or sides: Click on the label tool (the one that looks like a hand), then click on
whatever you want to label. If you want to change the label, double click on the label (after
selecting either the label tool or the arrow tool).
Open documents: Open the file Square.gsp. It is on my web page
http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660 . Read the
instructions once you open it and play around with them accordingly.
Print documents: Click on file, then on print preview. Click on fit to page if it shows your
sketch printing on two pages . Finally, click print. If you click print directly, your document
may print on two pages.
Resource: http://sierra.nmsu.edu/morandi/CourseMaterials/IntroToSketchpad.html
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Geometry
10th Grade
Mrs. Franks
4.2 Congruence and Triangles
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Two Geometric Figures are __________________________ if they have exactly the
same _____________________ and ___________________________.
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When two figures are ________________________, there is a correspondence between
their angles and sides such that, corresponding ____________________ are congruent
and corresponding ________________________ are congruent.
For the triangles below you can write !ABC " !PQR
A
Corresponding Angles
Corresponding Sides
B
C
P
Q
R
Using Geometer’s Sketchpad: Create Two Congruent Triangles. Show that Corresponding
Angles are Congruent and Corresponding Sides are Congruent (Using the Measure Tool).
Example 1: Congruent Figures
In the diagram NPLM ! EFGH
P
a. Find the value of x
b. Find the value of y
8m
N
E
72º
L
F
(7y + 9)º
110º
10m
H
(2x – 3) m
G
87º
M
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Geometry
10th Grade
Mrs. Franks
Theorem 4.3 Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles
are also congruent.
E
If !A " !D and !B " !E then !C " !F
B
D
A
F
C
Use Geometer’s Sketchpad: Create two triangles (NOT Congruent). Measure all of the angles
in each triangle. Now move your points around so that you have two sets of angles congruent.
Is the third set of angles congruent?
Example 2: Find the value of x if !MNL " !TRS :
M
(2x + 30)º
T
N
55º
R
65º
L
S
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Geometry
10th Grade
Mrs. Franks
4.3 Proving Triangles are Congruent: SSS and SAS
Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
the sides of the two triangles. Now move the triangles such that the sides of the first triangle are
congruent to the sides of the second triangle. Now without moving the triangles measure all the
angles of both triangles. What do you notice?
Postulate 19 Side – Side – Side (SSS) Congruence Postulate
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If three sides of one triangle are congruent to three sides of a second triangle, then the
two triangles are ___________________________.
N
If
Side
Then
Side NP ! RS
Side PM ! SQ
!MNP " !QRS
MN ! QR
R
M
Q
P
S
Using a Compass Construct a triangle that is congruent to the given triangle ABC.
A
C
B
Now that we’ve used the compass try using Geometer’s Sketchpad to construct congruent
triangles. Remember, you must show your arcs to have a valid construction. Hint use construct
a circle.
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Geometry
10th Grade
Mrs. Franks
Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two sides and the angle between the two sides of the two triangles. Now move the triangles such
that these three measurements are congruent to each other. Now without moving the triangles
measure the rest of the sides and angles of both triangles. What do you notice?
Postulate 20 Side – Angle – Side (SAS) Congruence Postulate
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If two sides and the included angle of one triangle are congruent to two sides and the
included angle of a second triangle, then the two triangles are __________________.
If
Side PQ ! WX
Angle !Q " !X
Side
Then
X
Q
QS ! XY
!PQS " !WXY
P
W
S
Y
Example 3: Use the SSS Congruence Postulate to Prove the two triangles congruent.
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B: (-7.00 , 5.00 )
F : (6.00 , 5.00 )
A: (-4.00 , 5.00 )
B
A
F
4
2
D
E
E: (6.00 , 2.00 )
D : (1.00 , 2.00 )
-10
C
-5
5
10
C: (-7.01 , 0.00 )
-2
-4
-6
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Homework: Go to http://illuminations.nctm.org/tools/tool_detail.aspx?id=4 and play around
with the applet. Answer the questions at the bottom of the page and print out your explorations.
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Geometry
10th Grade
Mrs. Franks
4.4 Proving Triangles are Congruent: ASA and AAS
Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two angles and the side between the two them in both triangles. Now move the triangles such
that these three measurements are congruent to each other. Now without moving the triangles
measure the rest of the sides and angles of both triangles. What do you notice?
Postulate 21 Angle – Side – Angle (ASA) Congruence Postulate
If two ______________ and the included _______________ of one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles are _________________.
If
Angle
Side
Angle
Then
!A " !D
AC ! DF
!C " !F
!ABC " !DEF
B
E
C
A
F
D
Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two angles and a side NOT between the two angles in both triangles. Now move the triangles
such that these three measurements are congruent to each other. Now without moving the
triangles measure the rest of the sides and angles of both triangles. What do you notice?
Theorem 4.5 Angle – Angle – Side (AAS) Congruence Theorem
If two _________________ and a non-included ______________________ of one triangle are
congruent to two angles and the corresponding non-included side of a second triangle, then the
two triangles are ________________________.
If
Then
Angle
Angle
Side
A
!A " !D
!C " !F
BC ! EF
!ABC " !DEF
C
F
B
E
D
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Geometry
10th Grade
Mrs. Franks
This is Wonderful that Geometer’s Sketchpad is working to show us these postulates and
theorems are true, but can anyone tell us why, or show us another way using Geometer’s
Sketchpad to prove these postulates to us?
- With a partner try to find another way to use geometer’s sketchpad to prove these to the class.
Example 1: Is it possible to prove that the triangles are congruent? If so, state the postulate or
theorem you would use. Explain your reasoning.
M
D
F
A
N
G
2
3
C
1
4
E
B
I
!D " !B
H
P
O
MN PO and PM ON
Example 2: You want to describe the boundary lines of a triangular piece of property to a
friend. You fax the note and the sketch below to your friend. Have you provided enough
information to determine the boundary lines of the property? Use Geometer’s Sketchpad to
explain.
N
The southern border is a line running
east from the apple tree, and the
western border is the north – south
line running from the cherry tree to
the apple tree. The bearing from the
easternmost point to the northernmost
point is W 53.1º N. The distance
between these points is 250 ft.
cherry tree
250ft
53.1º
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Geometry
10th Grade
Mrs. Franks
5.3 Medians and Altitudes of a Triangle
Median of a Triangle – a segment whose endpoints are a ____________ of the triangle
and the ____________________ of the opposite side.
A
C
M
B
Use Geometer’s Sketchpad: Construct a Triangle. Find the midpoint of Each Side. Now
connect the vertex of each angle to the midpoint on the opposite side. What do you notice?
Drag one vertex of the triangle to see an acute, obtuse and right triangle. What do you notice
now?
The medians of a triangle are __________________________.
Concurrent Lines – Lines that intersect at ____________________________________.
The point of concurrency is called the _________________________ of the triangle.
Use Geometer’s Sketchpad: Construct a point at the centroid. Now use the Measure Tool to
measure the distance from each vertex to the centroid. Use the Calculate Tool to find the ratio of
each Median. What do you notice?
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Geometry
10th Grade
Mrs. Franks
Theorem
Theorem 5.7 Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two thirds of the distance from each
B
vertex to the midpoint of the opposite side.
D
If P is the centroid of ΔABC, then
2
2
2
AP = AD, BP = BF , andCP = CE.
3
3
3
P
E
C
F
A
Example 1: P is the centroid of ΔQRS shown below and PT = 5, find RT and RP.
Q
P
S
R
Example 2: Find the coordinates of the centroid of ΔJKL.
12
L
10
8
6
K
4
2
J
5
10
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Geometry
10th Grade
Mrs. Franks
Altitude of a Triangle – the ______________________ segment from a vertex to the opposite
side or to the line that contains the opposite side.
Use Geometer’s Sketchpad: Construct a triangle. Construct perpendicular segments from a
vertex to the opposite side of the triangle. Repeat for all three sides. Do these lines intersect? If
they do construct a point at the intersection. Drag one of the vertices of the triangle, What do
you notice about the point of intersection? Think about the following questions.
The altitude of a triangle can be where?
How many altitudes does a triangle have?
Are the lines concurrent?
The point where they intersect is called the _______________________________________.
Example 3: Where is the orthocenter located in each type of triangle? Use Geometer’s
Sketchpad to see the sketch. Try to draw it.
a. Acute Triangle
b. Right Triangle
c. Obtuse Triangle
Theorem
Theorem 5.6 Concurrency of Altitudes of a Triangle
F
A
B
The lines containing the altitudes of a triangle are concurrent.
H
If AE , BF and CD are the altitudes of
ΔABC, then the lines AE , BF and CD
intersect at some point H.
E
D
C
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Geometry
10th Grade
Mrs. Franks
5.4 Mid-segment Theorem
A Mid-segment of a triangle is a segment that __________________________________ of two
sides of a triangle.
Example 4: Using Geometer’s Sketchpad Show that the mid-segment MN is parallel to side
JK and is half as long. Hint: How do we know lines are parallel?
8
6
K
4
J
2
-5
5
L
-2
Draw in the missing pieces (segments and measurements) from Sketchpad.
Midsegment Theorem
C
Theorem 5.9 Mid-segment Theorem
The segment connecting the midpoints of
two sides of a triangle is parallel to the
third side and is half as long.
DE AB and DE =
1
AB
2
D
A
E
B
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Geometry
10th Grade
Mrs. Franks
Example 5:
UV and VW are mid-segments of ΔRST. Find UW and RT. If RS = 12 and VW = 8
R
U
V
T
W
S
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Geometry
10th Grade
Mrs. Franks
5.5 Inequalities in One Triangle
Theorems
B
Theorem 5.10
If one side of a triangle is longer than
another side, then the angle opposite
the longer side is larger than the angle
opposite the shorter side.
5
3
A
Theorem 5.11
If one angle of a triangle is larger than
another angle, then the side opposite
the larger angle is longer than the side
opposite the smaller angle.
D
C
60º
40º
E
F
Largest Angle
Shortest
Side
Longest Side
Smallest Angle
Example 1: Write the measurements of the triangles in order from least to greatest.
a.
R
b.
F
100
8
7
35
H
45
P
5
Q
G
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Geometry
10th Grade
Mrs. Franks
Use Geometer’s Sketchpad: Construct a ray. Construct a point above the ray and a point on
the ray. Construct a triangle using the endpoint of the ray and the two new points that you have
created. Measure the exterior and interior angles. Play around with the calculations. Do you
notice anything?
B
A
C
D
Theorem
Theorem 5.12 Exterior Angle Inequality
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
m!1 > m!A and m!1 > m!B
A
1
D
C
B
Use Geometer’s Sketchpad: Go to my webpage
http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660 and open
Inequalities in One Triangle. Keep clicking random break and see if you can make a triangle.
What do you notice about the lengths of the sides when you can and cannot make a triangle?
Theorem
Theorem 5.13 Triangle Inequality
The sum of the lengths of any two sides of a triangle
is greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
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Geometry
Mrs. Franks
10th Grade
Example 3: A triangle has one side of 10 centimeters and another of 14 centimeters.
Describe the possible lengths of the third side.
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