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Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Do Now 32
Name:
Date:
Do Now 32 – Slope from Graphs
Directions: draw a slope triangle and use it to find the slope of each line shown.
1.
2.
8
8
6
6
4
4
2
2
–5
–5
5
5
–2
–2
m=
m=
3.
4.
8
6
6
4
4
2
2
–5
–5
5
5
–2
–2
–4
m=
m=
5. All slope triangle are (circle one) right / acute / obtuse triangles.
Geometry Week 12 Packet Page 1
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Intro to Triangles Notes
Name:
Date:
Introduction to Triangles – Guided Notes
Classifying Triangles By Sides
Classification
Description
Figure
A triangle with three
congruent sides.
Scalene Triangle
Geometry Week 12 Packet Page 2
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Intro to Triangles Notes
Name:
Date:
Classifying Triangles by Angles
Classification
Description
Figure
Right Triangle
A triangle with all acute
angles.
127°
Geometry Week 12 Packet Page 3
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Intro to Triangles Notes
Name:
Date:
Triangle Sum Investigation:
You probably know what the sum of the degrees in a triangle is, but you probably haven’t seen it
proved. The exercise that we are about to do shows one way to understand this theorem.
Materials:
•
•
•
•
A sheet of colored paper.
A straight edge
Scissors
Scotch tape
Step 1: Draw a triangle on your sheet of paper. If your birthday is in January, February, March,
or April, draw an acute triangle. If your birthday is in May, June, July or August, draw a right
triangle. If your birthday is in September, October, November or December, draw an obtuse
triangle.
Step 2: Write the letters a, b, and c in the interiors of the three angles of one of the triangles, and
carefully cut out the triangle.
Step 3: Tear off the three angles. Arrange them so that their vertices meet at a point. (See
diagram below.) How does this arrangement show the sum of the angle measures?
__________________________________________________________________
__________________________________________________________________
Step 4: Tape your corners onto this paper on the lower right-hand corner.
Triangle Sum Theorem:
The sum of the angles in any triangle is _____________.
Geometry Week 12 Packet Page 4
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Intro to Triangles Notes
Name:
Date:
Example 1: Classify each of the triangles by sides and angles.
a.
b.
c.
d.
Example 2: Use the triangle sum theorem to write an equation and solve for x in each triangle.
Then find the measure of each unknown angle in the triangle.
a.
b.
Geometry Week 12 Packet Page 5
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Intro to Triangles Notes
Name:
Date:
Triangle Sum White Board Practice
Directions: Use the triangle sum or another theorem to write a true equation for each figure.
Solve for x and hold up your board silently to allow the teacher to check your answer.
Geometry Week 12 Packet Page 6
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Homework 26
Name:
Date:
Homework 26 – Sides and Angles in Triangles
Directions: Classify each triangle by BOTH its sides and its angles.
1.
2.
3.
Directions: Find the measure of angle 1. Show your work.
4.
5.
6.
7.
8.
9.
Directions: For question 10, show or explain how you got your answer to each part of the
question.
10. Consider triangle ABC with ∠𝐴 measures 40 degrees.
a. Give possible measures of ∠𝐵 and ∠𝐶 that would make triangle ABC an acute triangle.
b. Give possible measures of ∠𝐵 and ∠𝐶 that would make triangle ABC an obtuse
triangle.
c. Is it possible to give measures of ∠𝐵 and ∠𝐶 that would make triangle ABC an
equilateral triangle? Explain why or why not.
Geometry Week 12 Packet Page 7
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Do Now 33
Name:
Date:
Do Now 33 – Classify Triangles
Directions: Classify each of the triangles by their sides and angles based on the markings.
Figure
Classify By Sides
Classify by Angles
1.
R
T
S
2.
V
U
W
3.
E
102°
7 cm
9 cm
F
13 cm
D
4.
O
Q
114°
P
Geometry Week 12 Packet Page 8
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Exterior Angles
Name:
Date:
Triangle Exterior Angle Conjecture – Guided Notes
We often discover theorems _________________________, by using examples and
generalizing.
Today, we will derive our theorem ________________________, by using previously agreed
upon facts to reach a new conclusion.
Our goal is to discover a relationship between the exterior angle of a triangle and the remote
interior angles of the triangle.
Exterior Angle of a Triangle
D
________________________________________
________________________________________
C
________________________________________
________________________________________
Remote Interior Angles of a Triangle
A
________________________________________
B
________________________________________
________________________________________
F
________________________________________
Name each Exterior Angle in the diagram above and name each of the remote interior angles to
that angle.
Exterior Angle
Remote Interior Angle
Remote Interior Angle
Geometry Week 12 Packet Page 9
E
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Exterior Angles
Name:
Date:
B
Exterior Angle _____________
b
Adjacent Interior Angle ____________
Remote Interior Angles _________ & ___________
a
A
y
x
C
D
Step 1: Write an equation that relates the exterior angle and the adjacent interior angle. What
conjecture supports this statement?
Step 2: Write an equation that relates the three interior angles of the triangle. What conjecture
supports this statement?
Step 3: Use substitution to combine the two equations.
Step 4: Use the subtraction property of equality to eliminate the adjacent interior angle.
Step 5: State the Triangle Exterior Angle Conjecture:
Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal to __________________________
_______________________________________________________________________.
Geometry Week 12 Packet Page 10
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Exterior Angles
Name:
Date:
Examples: Using the Triangle Exterior Angle Conjecture
Directions: Use the triangle exterior angle conjecture to find the measure of the angle marked
with a question mark.
Directions: Use the triangle exterior angle conjecture to solve for x.
3)
4)
Geometry Week 12 Packet Page 11
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Exterior Angles
Name:
Date:
Triangle Exterior Angle Conjecture Independent Practice
Directions: Use the triangle exterior angle conjecture to find the measure of the angle marked
with a question mark. Use an answer card from a teacher to check your work at the end of each
section.
Section A
1)
2)
3)
4)
6)
5)
Section B
Directions: Use the triangle exterior angle conjecture to solve for x.
Geometry Week 12 Packet Page 12
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Exterior Angles
Name:
Date:
7)
8)
9)
Section C
Directions: Find the measure of the angle indicated.
10)
11)
12)
13)
Geometry Week 12 Packet Page 13
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Homework 27
Name:
Date:
Homework 27 – Triangle Exterior Angles
Directions: Find the measure of angle 3 using the triangle exterior angle theorem.
1.
2.
3.
Directions: Write a true equation and solve it to find the value of the variable. You must show the
equation and a value for the variable to earn full credit.
4.
5.
6.
7.
8.
9.
10. Fernando drew triangle ABC and an exterior ∠CBD.
a. Draw and correctly label Fernando’s triangle.
b. Name the alternate interior angles to ∠CBD.
c. Fernando measured ∠ABC and found that it was 52° . What must the measure of ∠CBD be?
Show or explain how you got your answer.
d. Fernando measured ∠C and found it to be 30° . Is this possible? Explain why or why not.
e. If Fernando’s measurement of ∠𝐵 is possible, use it to find ∠A. If not, suggest an alternate
measurement for ∠𝐵 and use it to find ∠A. Show or explain how you got your answer.
Geometry Week 12 Packet Page 14
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Do Now 34
Name:
Date:
Do Now 34 – Triangle Exterior Angle Theorem
Directions: Use the triangle exterior angle theorem to find the measure of the angle marked
with a question mark.
1.
2.
3.
4.
5.
6.
7.
8.
Geometry Week 12 Packet Page 15
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Isosceles and Equilateral Tri.
Name:
Date:
Isosceles and Equilateral Triangles – Investigation and Practice
Parts of an Isosceles Triangle:
B
Vertex
angle
Legs
Base angles
A
C
Base
Vertex Angle of an Isosceles Triangle ________________________________________
__________________________________________________________________
Legs of an Isosceles Triangle ______________________________________________
__________________________________________________________________
Base Angles of an Isosceles Triangle _________________________________________
__________________________________________________________________
Base of an Isosceles Triangle ______________________________________________
__________________________________________________________________
Geometry Week 12 Packet Page 16
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Isosceles and Equilateral Tri.
Name:
Date:
Investigation: Base Angles in an Isosceles Triangle
Group Roles:
Facilitator ____________________
Materials Manager _____________________
Reader ______________________
Technician __________________________
Materials:
● patty paper
● a straightedge
● a protractor
Let’s examine the angles of an isosceles triangle. Each person in your group should draw a
different angle for this investigation. Your group should have at least one acute angle and one
obtuse angle.
Step 1: Draw an angle on patty paper. Label it ∠C. This angle will be the vertex angle of your
isosceles triangle.
Step 2: Place a point A on one ray. Fold your patty paper so that the two rays match up. Trace
point A onto the other ray.
Step 3: Label the point on the other ray point B. Draw 𝐴𝐵. You have constructed an isosceles
triangle.
a. How do your know that it is isosceles?
b. Name the base and the base angles of the triangle you created.
Base _________
Base angles (there are 2) ______________
Step 4: Use your protractor to compare the measures of the base angles. What relationship do
you notice? How can you fold the paper to confirm your conclusion?
Step 5: Compare results in your group. Was the relationship you noticed the same for each
isosceles triangle? State your observations as your next theorem.
Geometry Week 12 Packet Page 17
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Isosceles and Equilateral Tri.
Name:
Date:
The Isosceles Triangle Theorem
Verbal
If:
Then:
Pictorial
H
H
I
I
G
G
Symbolic If:
Then:
The Converse of the Isosceles Triangle Theorem
Verbal
If:
Then:
Pictorial
H
H
I
G
Symbolic If:
I
G
Then:
Geometry Week 12 Packet Page 18
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Isosceles and Equilateral Tri.
Name:
Date:
Equilateral Equiangular Triangle Theorem
A triangle is equilateral if and only if it is equiangular.
Stated as a conditional,
Verbal
If:
Then:
Pictorial
E
D
E
F
Symbolic If:
D
F
Then:
And the converse of the conditional.
Verbal
If:
Then:
Pictorial
E
D
Symbolic If:
E
F
D
F
Then:
Geometry Week 12 Packet Page 19
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Isosceles and Equilateral Tri.
Name:
Date:
Isosceles and Equilateral Independent Practice
Directions: Find the measure of each angle marked with an x.
1.
2.
43°
x°
x°
41°
3.
4.
99°
x°
66°
5.
x°
6.
x°
40°
7.
x°
8.
x°
x°
54°
46°
Geometry Week 12 Packet Page 20
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Homework 28
Name:
Date:
Homework 28 – Properties of Isosceles and Equilateral Triangles
16. Shanice drew triangle PQR on the coordinate plane with coordinates P (5, -2), Q (5, 2), and R
(1, 2).
a. Plot and label Shanice’s triangle ON GRAPH PAPER.
b. Classify Shanice’s triangle by its sides. Show or explain how you know your answer is
correct.
c. Classify Shanice’s triangle by its angles. Show or explain how you know your answer is
correct.
Geometry Week 12 Packet Page 21
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Do Now 35
Name:
Date:
Do Now 35 – Angles and Theorems
Use your theorems to find each of the lettered angles. Name the theorem that you use for each
angle.
Theorem Bank
Linear Pair Theorem
Vertical Angles Theorem
Triangle Sum Theorem
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Isosceles Triangle Theorem
Equilateral/Equiangular
Triangle Theorem
•
•
•
•
•
•
•
•
Angle
Measure
Theorem Name
a
b
c
d
e
f
g
h
k
n
p
Geometry Week 12 Packet Page 22
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Side Investigation
Name:
Date:
Triangle Side Length Investigation
Group Roles
Facilitator _________________ in charge of making sure the group completes the
task in a timely manner and that all group members are involved and understand
Reader/Recorder _______________ responsible for reading each direction out loud.
Also responsible for recording the results of each trial and making sure that all other
group members also record the data
Materials Manager _______________ collects all material (see list below) from the
table and returns it at the end of the investigation
Technician ________________ responsible for directing and/or performing the
triangle trials
Materials
•
•
•
•
•
3 1-inch straws
2 2-inch straws
2 3-inch straws
2 4-inch straws
1 5-inch straw
Investigation
For each set of three lengths, use your straws to determine whether it is possible to make
a triangle with those three lengths. Record whether or not it is possible and draw an
example or counterexample (a picture that shows why the triangle is or is not
possible.)
Side
Lengths
1,1, and 2
Possible (P) or
Impossible (I)
Example or
Counterexample
1, 2, and 3
Geometry Week 12 Packet Page 23
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Side Investigation
Name:
Date:
2, 2, and 3
2, 3, and 4
3, 3 and 4
1, 3 and 4
3, 3 and 4
3, 4, and 5
Geometry Week 12 Packet Page 24
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Side Investigation
Name:
Date:
2, 2, and 4
1, 4, and 5
Use what you have determined in the investigation so far to create your own, different set of side
lengths. Create 2 sets that you think will make a triangle and 2 sets that will not. Then use your
straws to test your sets. Record the results below.
Side Lengths
Possible (P) or
Impossible (I)
Example or
Counterexample
Geometry Week 12 Packet Page 25
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Triangle Side Investigation
Name:
Date:
Triangle Inequality Theorem:
The sum of the lengths of any two sides of a triangle is _________________________ the
length of the third side.
Extra Credit Challenge: Given the two side lengths and the possibility or impossibility, find a
possible third length and draw an example or counterexample.
Side
Lengths
1.5, 2.7, and
Possible (P) or
Impossible (I)
Possible
Example or
Counterexample
______
Impossible
3 13 , 4 34 , and
______
Examples: Determine whether each of the following sets of lengths could be the side lengths of
triangles. Write yes or no. If the triangle is possible, classify it as scalene, isosceles, or
equilateral.
1) 3 cm, 4 cm, 4 cm
2) 5 miles, 5 miles, 3
3) 5 in, 5 in, 5 in
4) 4.5 yds, 5 yds, 9 yds
miles
Possible:
Possible:
Possible:
Possible:
Classify (if possible):
Classify (if possible):
Classify (if possible):
Classify (if possible):
Geometry Week 12 Packet Page 26
Geometry Level 2
Ms. Sheppard-Brick
617-596-4133
Homework 29
Name:
Date:
Homework 29 – Triangle Side Lengths
You may complete this assignment on this sheet of paper.
13. Desirae has a triangle which has one side that is 5 m and one side that is 9 m. She wonders
what the possible lengths are for the third side.
a) Which of the following side lengths are possible third sides for Desirae’s triangle?
Circle the ones that are.
3m
4m
5m
13 m
15 m
16 m
b) What is the smallest integer value that the third side could have? Explain how you
know your answer is correct.
c) What is the largest integer value that the third side could have? Explain how you know
your answer is correct
d) Write an inequality that shows all the possible values, x, that the third side could
have.
Geometry Week 12 Packet Page 27