# Unit Plan Information

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Oct 10, 2013 (4 years and 7 months ago)

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Geometry

Mrs. Franks

10
th

Unit Plan Information

Properties of Triangles

Jessica Franks

10
th

Mathematics

Geometry

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 yo
u 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 thi
s 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

Geometry

Mrs. Franks

10
th

Standards:

NY State

Geometry Standards

G.G.28
Determine the c
ongruence of
two triangles by using one of the five
congruence techniques (SSS, SAS,
ASA, AAS, HL), given sufficient
angles of two congruent triangles

G.G.29
Identify corresponding parts of
congruent triangles

G.G.30
In
vestigate, 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
lities,
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

NY State

Technology Standards

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 produc
tivity,
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 strategie
s for solving
problems in the real world.

Geometry

Mrs. Franks

10
th

DATE

OBJECTIVE:

DATE

OBJECTIVE:

12/05

-

Understand the key properties of
triangles using geometer’s

(GS)

-

Classify triangles by their sides
and angles

12/05

-

Identify congruent figures and
corre
sponding 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

-

to the students

-

Allow students to get the used
to
the new program by letting them
explore

-

explore basic properties of triangles

-

Go to computer lab

-

Bell Ringer

-

Go over homework

-

Using
GS

have class take notes,
practice, explore
and di
scuss
congruency of
triangles

MATERIALS NEEDED:

MATERIALS NEEDED:

Computers,
,
Calculators
, Worksheet for GS from
http://sierra.nmsu.edu/morandi
/Cour

(In Notes)

Computers,
,
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

.

Geometry

Mrs. Franks

10
th

DATE

OBJECTIVE:

DATE

OBJECTIVE:

12/05

-

Prove that triangles are congruent
using the ASA Congruence
Postulate and the AAS Congruence
Theorem.

-

Use propertie
s 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
l
argest.

-

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:

ACTIVIT
IES:

-

Go to computer lab

-

Bell Ringer

-

Go over homework

-

the

class take notes, practice
,
explore

and discuss congruency,
medians and altitudes of triangles

-

Go to computer lab

-

Bell Ringer

-

Go over homework

-

have

the

class take notes, practice
and discuss
Mid
-
segment and
inequalities in one
triangle
.

MATERIALS NEEDED:

MATERIALS NEEDED:

Computers,
,

Calculator
,

Worksheet for GS

from

http://sierra.nmsu.edu/morandi/Cour

(in
Notes)

Computers,
,
Calculator,
Sketch from
Key
Curriculum Press

on web page

ASSESSMENT:

ASSESSME
NT:

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

Textbo
ok Problems

In class
-

see written examples from
notes

Homework

-

T
extbook problems.

Geometry

Mrs. Franks

10
th

DATE

OBJECTIVE:

DATE

OBJECTIVE:

12/05

-

Write the equation of a line given
a point on the line and the slope of
the line

-

Write the equation of a line given
two poi
nts on the line

12/05

-

Assess Knowledge of Students

CONTENT:

CONTENT:

-

5.3 & 5.5: Writing equations of
lines with two points and Point
-
Slope Form

-

Chapters 4 & 5

ACTIVITIES:

ACTIVITIES:

-

Bell Ringer

-

Go over homework

-

Using the

graphing calculator
have class take notes, practice and
discuss the equations of lines.

-

Complete Worksheet on the
writing equations of lines

-

Go to computer lab

-

Bell Ringer

-

Go over homework

-

assess triangles

MA
TERIALS NEEDED:

MATERIALS NEEDED:

Computers,
Calculators
, notes

Computers,
,

Calculators,
Teacher created e
xam

ASSESSMENT:

ASSESSMENT:

Student responses

verbal and
written. Class participation

Sketch. Homework assignment.

Student responses

verbal and
written. Class participation

Sketch.
Teacher created exam

PRACTICE:

PRACTICE:

In class
-

see written examples from
notes

Homework

T
extbook problems

In class
-

Teacher created exam

Individually
-

Teacher creat
ed
exam

Geometry

Mrs. Franks

10
th

Chapter
s

4 & 5

Course 2R

Mrs. Franks

Geometry

Mrs. Franks

10
th

Geometry

Mrs. Franks

10
th

Triangles

What do you remember about triangles?
Geometry

Mrs. Franks

10
th

In this assignment we will learn how to use the program Geometer's Sketchpad. This program i
s
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 yo
u 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
sele
ct 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 h
ighlight objects can affect the resulting construction.

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 s
egment 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.
T
hese, 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 mo
use, 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 poin
t, 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 t
wo 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 resi
ze. Alternatively, click
and drag one of the sides.

Geometry

Mrs. Franks

10
th

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. The
n 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.
No
tice 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 c
ircles
: 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 seg
ments, 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 pa
ssing 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 la
bel 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 m
y web page
http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660

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:

Geometry

Mrs. Franks

10
th

4.2 Congruence and Triangles

Two Geometric Figures are __________________________ if they have exactly the

same _____________
________ and ___________________________.

When two figures are ________________________, there is a correspondence between

their angles and sides such that, corresponding ____________________ are congruent

and corresponding ________________________ ar
e congruent.

For the triangles below you can write

Corresponding Angles

Corresponding Sides

: Create Two Congruent Triangles
. S
how that Corresponding
Angles are Congruent and Corr
esponding Sides are Congruent (Using the Measure Tool).

Example 1
: Congruent Figures

In the diagram

a.

Find the value of x

b.

Find the value of y

B

C

P

Q

R

A

(2x

3) m

(7y + 9)º

110º

87º

72º

8 m

10m

H

G

F

E

M

L

P

N

Geometry

Mrs. Franks

10
th

Theorem 4.3 Third Angles Theorem

If two angles of one triangle ar
e congruent to two angles of another triangle, then the third angles
are also congruent.

If

and

then

Use

: Create two triangles (NOT Congruent).
Meas
ure 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
:

B

D

A

(2x + 30)º

65º

55º

T

S

R

L

N

M

E

C

F

Geometry

Mrs. Franks

10
th

4.3

Proving Triangles are Congruent: SSS and SAS

:
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 th
e 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

If three sides of one triangle are congruent to three sides of

a second triangle, then the

two triangles are ___________________________.

If

Side

Side

Side

Then

Using a Compass
Construct a triangle that i
s congruent to the given triangle ABC.

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.

M

Q

N

R

P

S

Geometry

Mrs. Franks

10
th

Use Geomet
: 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

If two sides and the included angle of one triangle are congruent to two sides and the

includ
ed angle of a second triangle, then the two triangles are __________________.

If

Side

Angle

Side

Then

Example 3
: Use the SSS Congruence Postulate

to

Prove the two triangles congruent.

Homework
: Go to
http://illuminations.nctm.org/tools/tool_detail.aspx?id=4

and play around
with the applet. Answer the quest
ions at the bottom of the page and print out your explorations.

P

Q

S

W

X

Y

Geometry

Mrs. Franks

10
th

4.4 Proving Triangles are Congruent: ASA and AAS

: Construct two triangles (NOT Congruent). Measure the length of
two angles and the side between the two them in b
oth 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

A
ngle

S
ide

A
ngle

Then

: Construct two triangles (NOT Congruent). Measure the length of
two angles and a side NOT between the two angles in both tria
ngles. 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) C
ongruence 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

A
ngle

A
ngle

S
ide

Then

C

B

A

D

E

F

A

B

C

F

E

D

Geometry

Mrs. Franks

10
th

This is Wonderful that Geometer’s Sketchpad is working to show us these
postulates and
theorems are true, but c
an 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 cong
ruent? If so, state the postulate or
theorem you would use. Explain your reasoning.

Example 2
: You want to describe the boundary lines of a triangular piece of proper
ty 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?

e
xplain.

N

The southern border is a line running

cherry tree

east from the apple tree, and the

western border is the north

south

line running from the cherry tree to

250ft

the apple tree. The bearing from the

easternmost point to the northernmost

point is W 53.1º N. The distance

betw
een these points is 250 ft.

53.1º

Geometry

Mrs. Franks

10
th

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.

Use Geometer’s Sket
: 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 notic
e
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 Ske
: 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?

Geometry

Mrs. Franks

10
th

Theorem

Theorem 5.7 Concurrency of Media
ns of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each
vertex to the midpoint of the opposite side.

If P is the centroid of ΔABC, then

Exampl
e 1
: P is the centroid of ΔQRS shown below and PT = 5, find RT and RP.

Example 2
: Find the coordinates of the centroid of ΔJKL.

Geometry

Mrs. Franks

10
th

Altitude of a Triangle

the ______________________ segment from a vertex to the opposite
side

or to the line that contains the opposite side.

: 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 c
onstruct 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. Acu
te Triangle

b. Right Triangle

c. Obtuse Triangle

Theorem

Theorem 5.6 Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent.

If
,

a
nd

are the altitudes of

ΔABC, then the lines

a湤n
=
††=

=
=
=
Geometry

Mrs. Franks

10
th

5.4 Mid
-
segment Theorem

A Mid
-
segment of a triangle is a segment that

__________________________________ of two
sides of a triangle.

Example 4
:
Show that the mid
-
segment

is parallel to side

and is half as long.

Hint: How do we know lines are para
llel?

Draw in the missing pieces (segments and measurements) from Sketchpad.

Midsegment Theorem

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.

Geometry

Mrs. Franks

10
th

Example 5
:

and
are mid
-
segments of ΔRST. Find UW and RT.

If

Geometry

Mrs. Franks

10
th

5.5 Inequalities in One Triangle

Theorems

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.

Theorem 5.11

If one angle of a triangle is larger than

another angle, then the side opposite

t
he larger angle is longer than the side

opposite the smaller angle.

Largest Angle

Shortest

Side

Longest Side

Smallest Angle

Example 1
: Write the measurements of the triangles in order from lea
st to greatest.

a.

b.

3

B

A

40º

60º

5

C

D

E

F

Geometry

Mrs. Franks

10
th

:
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?

Theorem

Theorem 5.12 Exterior Angle Inequality

The measure of an exterior angle of a

triangle is greater than the measure of

t interior

angles.

>

and
>

:
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 triang
le?

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

Geometry

Mrs. Franks

10
th

Example 3
: A triangle has one side of 10 centime
ters and another of 14 centimeters.

Describe the possible lengths of the third side.

Geometry

Mrs. Franks

10
th

5.3 & 5.5 Writing equations of lines with two points and Point
-
Slope Form

What two pieces of informatio
n do you
need

to write the equation of a line?

The __________________________ and the _______________________________.

What is the
Slope

Intercept Form

of a line? ________________________________.

What is the
Slope Formula
? _______________________
___________.

Example 1
: Write the equation of the line that passes through the points (1, 6) and (3,
-
4).

What do we need to write the equation? _____________________ and ____________________

What can we find with two points? _______________________
_____________

Writing an equation of a line given two points

Step 1

Find the Slope
. Substitute the coordinates of the two given points into the formula for

slope,
.

Step 2

Find the y
-
intercept
. Substitute the slope m and the coordinates of one of the points

into the slope
-
intercept form,
y

=
mx

+
b
.

Step 3

Write an equation of the line
. Substitute the slope
m

and the y
-
intercept
b

into the

slope
-
intercept form,
y

=
mx

+
b
.

Geometry

Mrs. Franks

10
th

Another strategy for writing the equation of a line is __________________________________.

Point

Slope Form of the equation of a li
ne

The
point

slope form

of the equation of the nonvertical line that passes through

a given point

with a slope of
m

is

Example 2
: Write an equation of the line given the point
(2, 5) and a slope
m

of
.

Example 3
: Write an equation of a line given the points (
-
2, 3) and (
-
1, 1).

Geometry

Mrs. Franks

10
th

Another way to write the equation of line when given two points is to use your graphing
Calculator
. Let’s use the last example:

Write an equation of a line given the points (
-
2, 3) and (
-
1, 1).

1. First hit the STAT button and then Edit.

-
values into L
1

and your

y
-
values into L
2

3.
Now hit 2
nd
, Y = to get into STAT PLOT

and then ENTER

4. Turn on your STAT PLOT

5.

Then GRAPH it.

6. Now we want the equation of the line.

Go back to STAT, CALC

7. We want the equation of a line #4. Then
type

L
1
, L
2

, Y
1

(Under VARS).

8. Hit ENTER and your coefficients will
appear and you can look at the graph to see