Constructing
Triangles
Common Core 7.G.2
Vocabulary
• Uniquely defined
• Ambiguously defined
• Nonexistent
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?
How many triangles can
be constructed?
Remember our “I can” statement:
“I can determine if 1, more than 1, or no
triangles can be constructed given 3 side or 3
angle measures.”
• The organizer below should be filled out and glued in your
notebook.
The Triangle is…
Ambiguous
Unique
Definition: Able to draw more than Definition: Only able to draw 1
1 triangle
triangle
Non-Existent
Definition: Not possible to draw a
triangle
Use when given 3 angles that equal
180˚.
Use when the two theorems WILL
NOT work.
Use when given 3 side lengths the
satisfy the Triangle Inequality
Theorem.
Triangles are:
Nonexistent - If three side lengths or angle measures do not make
a triangle, you would say that the triangle is nonexistent because
a triangle cannot be formed.
Unique - If three side lengths do make a triangle, you would say
that the triangle is unique because it creates one, specific
triangle.
Ambiguous – If three angle measures do make a triangle, you
would say that the triangle is ambiguous because it creates more
than 1 triangle.
Ambiguous Triangles
• Used when 3 angle measures add up to equal 180˚.
• 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.
• This proves why more than 1 triangle can be drawn.
Unique Triangles
• Used when 3 side lengths are given and satisfy the
Triangle Inequality Theorem.
• Look at the triangle below. It has 3 side lengths that
will make a triangle. You can flip it, rotate it, or
translate it, but there is still ONLY ONE triangle that
can be made.
Non-existent Triangles
• When angle measures or side lengths DO NOT
satisfy our 2 theorems, no triangles can be created.
• If the sum of the 2 smaller sides is NOT greater than
the longest side, it WILL NOT make a triangle.
• If the sum of the angle measures DO NOT equal
180˚, it WILL NOT make a triangle.
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.