Warm Up/
Activator
Using scissors and a ruler, cut out a
triangle from a piece of colored paper.
Clean up your area afterwards. We will
be using the triangle during our lesson
today.
Constructing
Triangles
Common Core 7.G.2
Vocabulary
• Uniquely defined
• Ambiguously defined
• Nonexistent
Triangle Inequality
Theorem
Let’s review what we learned yesterday with this
video.
http://www.youtube.com/watch?v=OoLb_NnnKSQ
Triangle Inequality
Theorem
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
z
x
y
𝑥+𝑦 >𝑧
𝑥+𝑧 >𝑦
𝑦+𝑧 >𝑥
Practice
Can these measures be the sides of a
triangle?
1.
2.
3.
4.
7, 5, 4
2, 1, 5
9, 6, 3
7, 8, 4
Practice
Can these measures be the sides of a
triangle?
1.
2.
3.
4.
7, 5, 4
2, 1, 5
9, 6, 3
7, 8, 4
yes
no
no
yes
Example 1
Using the measurements 6 in and 8 in, what is
the smallest possible length of the third side?
What is the largest possible length of the third
side?
Example 1
If you assume that 6 and 8 are the shorter
sides, then their sum is greater than the third
side. Therefore, the third side has to be less
than 14.
6+8>𝑐
14 > 𝑐 𝑜𝑟 𝑐 < 14
Example 1
If you assume that the larger of these values, 8, is
the largest side of the triangle, then 6 plus the
missing value must be greater than 8. Therefore,
the third side has to be more than 2.
6+𝑐 >8
−6
−6
𝑐>2
Example 1
If you put these two inequalities together, then
you get the range of values that can be the
length of the third side:
2 < 𝑐 < 14
Therefore, any value between 2 and 14 (but
not equal to 2 or 14) can be the length of the
third side.
Practice
Solve for the range of values that could be the length
of the third side for triangles with these 2 sides:
1. 2 and 6
2. 9 and 11
3. 10 and 18
(Be sure to look for patterns!)
Practice
Solve for the range of values that could be the length
of the third side for triangles with these 2 sides:
1. 2 and 6
2. 9 and 11
3. 10 and 18
𝟒<𝒙<𝟖
𝟐 < 𝒙 < 𝟐𝟎
𝟖 < 𝒙 < 𝟐𝟖
What patterns do you see?
Types of Triangles
If three side lengths do not make a triangle, you would
say that the triangle is nonexistent.
If three side lengths do make a triangle, you would say
that the triangle is uniquely defined because it
creates one, specific triangle.
Angles of Triangles
Tear off the corners of the triangle that you created in
the warm up/activator. Lay them on your paper with
all of the vertices pointing inwards and the edges
touching. Like this:
What do they create?
Angles of Triangles
What do they create? A straight line, which is equal
to 180 degrees; therefore, the sum of the angles in a
triangle always equal 180 degrees. This is called the
Triangle Angle Sum Theorem. Glue your triangle
corners in your math notebook and explain this in your
own words.
Practice
Given the following angle measurements, determine
the third angle measurement.
1. 60°, 80°
2. 110°, 20°
Do these measurements create triangles?
3. 55°, 75°, 𝑎𝑛𝑑 50°
4. 80°, 90°, 𝑎𝑛𝑑 80°
Practice
Given the following angle measurements, determine
the third angle measurement.
1. 60°, 80° 𝟒𝟎°
2. 110°, 20° 𝟓𝟎°
Do these measurements create triangles?
3. 55°, 75°, 𝑎𝑛𝑑 50° yes
4. 80°, 90°, 𝑎𝑛𝑑 80° no
Constructing Triangles
from Angles
Look at these triangles. They have the same angle
measurements, which is why they are similar in shape.
However, do they have the same side lengths?
Constructing Triangles
from Angles
Look at these triangles. They have the same angle
measurements, which is why they are similar in shape.
However, do they have the same side lengths? No.
Since they aren’t the same size, will angle
measurements construct unique triangles?
Constructing Triangles
from Angles
Look at these triangles. They have the same angle
measurements, which is why they are similar in shape.
However, do they have the same side lengths? No.
Since they aren’t the same size, will angle
measurements construct unique triangles? No.
Constructing Triangles
from Angles
Conditions, such as angle measurements, that can
create more than one triangle are called
ambiguously defined.
Summary
Take turns with your partner explaining the Triangle
Angle Sum Theorem and the Triangle Inequality
Theorem in your own words.