Unit 3.1
Lesson 3
TRIangles
TABLE OF CONTENTS
Essential Questions…………………………………………………………………..
Key Definitions and Skills………………………………………………………….
Welcome to the Triangle World …………………………………………………..
Exploring triangle properties…………………………….………………………
Conditions 1
Conditions 2
To be or Not to be a triangle that is the question………………………………
Putting It All together……………………… ……………………………………
Triangle Post Assessment……………………………..…………………………
Brain warm up section………………………………………………………….
TOTAL LESSONS:
5
Vocabulary Knowledge Test: January 24, 2013
POST ASSESSMENT: January 24, 2013
UNIT TEST 3.1 :
January 28, 2013
Essential Questions
1.
2.
3.
4.
What are the different triangles?
What makes up a triangle?
How are triangles defined?
How do we use information such as sides and angles to construct triangles?
COMMON CORE OBJECTIVES
Cluster:
Draw construct, and describe geometrical figures and describe the relationships
between them.
Standard:
Draw (freehand, with a ruler and protractor, and with technology) geometric shapes
with given conditions. Focus on constructing triangles from three measures of
angles or sides, noticing when the conditions determine a unique triangle, more
than one triangle, or no triangle.
Learning Targets:
I can draw a geometric shape with specific conditions
I can construct a triangle when given three measurements: 3 side lengths, 3 angle
measurements, or a combination of side and angle measurements.
I can determine when three specific measurements will result in one unique
triangle, more than one possible triangle, or no possible triangle.
KEY DEFINITIONS AND SKILLS
Acute Triangle: Triangles that have all interior acute angles.
Right Triangle: Triangles that have one right angle.
Obtuse Triangle: Triangles that have one obtuse angle.
Equilateral Triangle: Triangles that have congruent sides and congruent angles
Isosceles Triangle: Triangles that have two sides congruent.
Scalene Triangle: Triangles that have neither sides nor angles are congruent
Ambiguously defined: Classification used when three side lengths can create
several triangles
Uniquely Defined: Classification used when only one triangle can be made from a
set of side lengths
Nonexistent: When no triangles can be formed by a set of side lengths
Triangle Angle Sum Theorem: States the sum of the angles in any triangle is
180⁰.
SKILLS NEEDED
To complete this lesson book, students will need to know how to do the following.
-
Add and subtract two to three digit numbers
Solve two-step equations
Combine like terms
Quickly identify the missing angle in a triangle
Identify basic inequality signs such as less than and greater than.
Using a ruler and protractor
Welcome to the Triangle World!
Hello Everyone! Today we are going to explore angles in a different part of the angle
spectrum. This time we are going to find out what happens when angles come
together. In many cases, some might say we are show offs. However, I think we are
just talented. Moreover, we angles come together we can form many things such as
polygons and triangles. Today, we will start our journey off with learning about
triangles, and let me tell you they are interesting characters.
ACUTE TRIANGLES
When we have an acute triangle, we shall notice that all angles are acute. Well is
that not convenient.
RIGHT ANGLES
Those picky right angles strike once again. When we have a right triangle, it will
have one right angle. There are only two right triangles: 30-60-90 and 45-45-90.
OBTUSE ANGLES
Triangles that have one obtuse angle
EQUALATERAL TRIANGLE
Triangles that have equal sides and equal angles
ISOSOCLESES TRIANGLE
Triangles that have two sides congruent, the other cool thing about these triangles
is that the bottom two angles are congruent.
M
4
40
U
4
40
D
SCALENE TRIANGLE
Triangles that have either sides or angles equal.
Triangle Angle Sum Theorem
This theorem states that the sum of the angles in any triangle is 180⁰.
This means that before any triangle can be form, we have to make sure their angles
add up to be 180⁰.
_________________________INDEPENDENPENT PRACTICE___________________
1. Tyronica drew ∆NET, which is
shown below.
B. 100⁰,50⁰,30⁰
90⁰,20⁰
D. 80⁰,
3. Derek used drew the following
triangle:
What is the name of this triangle?
Based on the information in the
triangle, what type is it?
A. Acute Scalene triangle
A. Right
C. Scalene
B. Acute
D. Obtuse
C. Acute Isosceles triangle
4. Ciara constructs a triangle with
angles measuring 65⁰ and 38⁰. What
must be true of the measure of the
third angle in her construction?
D. Right Triangle
A. It must measure exactly 77⁰
2. Which of the following set of angle
measures would make a triangle?
B. It must measure exactly 87⁰
B. Obtuse Isosceles triangle
A. 10⁰, 50⁰, 70⁰
100⁰,60⁰,30⁰
C.
C. It can have measure less than 103⁰
D. It can have any measure greater than
27⁰
5.. Driving on a road trip, D’Asia and
Ne’Asya notice a yield sign:
7. When drawing the triangle PIN,
Ericka wants to find the measure of
the missing angle.
If all the sides are congruent, what
type triangle is it?
A. 40⁰
A. Equilateral
B. Scalene
C. Obtuse
D. Isosceles
6. One day, Mrs. Carter, Coach
Lombard, Mr. Hill, and Mrs.
Blackledge went for trail walk for
some exercise. The path they walked
is shown below.
B. 60⁰
C. 30⁰
D. 50⁰
8. What statement is true about
Isosceles triangles?
A. They have two congruent sides
only
B. They have no sides congruent but
all angles congruent
C. They have congruent sides and
congruent angles
D. They have two congruent sides and
two congruent angles
Their route resembles a
A. Right Triangle
B. Obtuse Triangle
C. Straight Triangle
D. Acute Triangle
DRAWING TRIANGLES WITH GIVEN CONDITIONS 1
Sometimes in the triangle world we must figure them, because everyone knows that
sometimes triangles get complicated. When constructing triangles we know that we
have side measures and angle measures.
One thing that we have to use is the Triangle Inequality Theorem.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
x+y>z
x+z>y
y+z>x
COACHED EXAMPLE
Is it possible to construct a triangle with sides measuring 8 feet, 9 feet, and 12 feet?
The triangle inequality theorem states that sum of the lengths of any tw sides of a
triangle must be________________________ than the length of the third side.
Use the theorem to determine if this triangle is possible or not.
Is 8 + 9 > 12?______________________________
Is 8 + 12 > 9?______________________________
Is 9 + 12 > 8?______________________________
The inequalities above are all true, so it____________possible to draw a triangle
with side lengths of 8 feet, 9 feet, and 12 feet.
Lets explore these for ourselves by completing some practice.
For questions 1-6, tell whether a triangle can be constructed using the give measure
1. 2 in, 2 in, 3 in
4. 8 in, 7 in, 15 in
2. 3 ft, 15 ft, 9ft
5. Lincoln constructs a triangle with
one side 5 inches long and another
side 7 inches long. Which is not a
possible length for the third side?
A. 3 inches
B. 6 inches
C. 11 inches
D. 12 inches
3. 4 yd, 6 yd, 5 yd
6. Maggie constructs a triangle with
one side 7 centimeters long and
another side 10 centimeters long.
Which is not a possible length for the
third side?
A. 2 centimeters
B. 4 centimeters
C. 7 centimeters
D. 16 centimeters
DRAWING TRIANGLES WITH GIVEN CONDITIONS 1(PT 2)
Beside using the Triangle Inequality Theorem, we might have to draw the triangle
given the three sides. First learn about some terms that while we are drawing
triangles.
When trying to construct a triangle with given side lengths or angle measures,
there are several possibilities:
-The triangle may be uniquely defined. You are only able to draw one triangle.
-The triangle may be ambiguously defined. That just means you may be able to
draw more than one triangle.
-The triangle may be nonexistent. It may not be possible to draw a triangle with
those measures.
Let us take a look at an example:
Using a ruler, construct a triangle with side lengths of 3 centimeters, 4 centimeters,
and 5 centimeters. What kind of triangle is it? Is it possible to draw another kind of
triangle?
STEP 1: Try drawing a triangle with one obtuse angle—an obtuse triangle.
3 cm
5 cm
4 cm
STEP 2: Try drawing a triangle with only acute angles—an acute triangle
5 cm
3 cm
4 cm
STEP 3: Try drawing a triangle with one right angle – a right triangle.
3 cm
5 cm
4 cm
You can draw one unique triangle with those side lengths, and it is a right triangle.
EXAMPLE 2
Two sides of a triangle are 6 centimeters and 4 centimeters long. The angle between
those two sides measures 50⁰. Use a ruler and protractor to draw this triangle.
DRAWING TRIANGLES WITH GIVEN CONDITIONS 2
Just like in the previous, we can draw triangles when given certain conditions.
Unlike the other section where we focused on sides, we will focus on angles. Who
knows, the problem may only provide us with angle measures. Let us explore some
examples using the condition of angles.
Remember that the sum of the angles in a triangle must add up to the sum of 180⁰.
EXAMPLE 1
Is it possible to construct a triangle with angles measuring 61⁰, 33⁰, and 86⁰. If so,
can you only draw one unique triangle, or can you draw many different triangles?
STEP 1: First let us see if the angles add up to be 180 degrees. If not, there is no
reason to continue.
61 + 33 + 86 = 180
180 = 180
So, a triangle with these angle measures is possible.
STEP 2: Use a protractor to draw one or more triangles with those angle measures.
The two triangles above have the correct angle measures, but they have different
lengths.
That is because triangles with the same angle measures are similar to one another.
So, it is possible to draw many different triangles with those angle measures.
A triangle with angles measuring 61, 33, and 86 is ambiguously defined because no
side lengths are mentioned. It is possible to draw many different similar triangles
with those angle measures.
EXAMPLE 2
A triangle has angles that measure 40⁰ and 75⁰. The length of the side between
these angles is 5 centimeters. Which measures are closest to the lengths of the other
two sides of this triangle?
A. 3 cm and 4 cm
B. 3.5 cm and 5.5 cm
C. 4 cm and 6 cm
D. 4.5 cm and 6.5 cm
BRAIN WARM UP SECTION
TUESDAY
Using your ruler, what is the length of the line below?
Using a scale factor of ½ inch = 3 feet, what is the measurement of the line in feet?
____________feet
THURSDAY
Dameka drew the diagram below.
T
S
A
5X - 10
W
60
M
What is the value of x?
x =__________
FRIDAY
Use the diagram below to determine the answer.
2
3
4
1
5
In the figure above, the measure of ∠1 = 30. What is the measure of ∠2?
m∠2 = ____________________