InterMath | Workshop Support | Write Up Template
Title
Classifying Triangles
Problem Statement
Triangles can be classified by their side lengths (scalene, isosceles, and equilateral), and
by their angle measure (acute, right, and obtuse). However, not all combinations of these
classifications exist.
Do each of the following triangles exist? If yes, draw the triangle accurately, with
measurements. If no, explain why it cannot exist.









a right isosceles triangle
a right equilateral triangle
a right scalene triangle
an acute scalene triangle
an acute isosceles triangle
an acute equilateral triangle
an obtuse scalene triangle
an obtuse isosceles triangle
an obtuse equilateral triangle
Problem setup
I am trying to find out which combinations of classifications (side lengths and angle
measures) can exist in different types of triangles.

Plans to Solve/Investigate the Problem
Prediction: I predict that a right equilateral and an obtuse equilateral angle cannot
exist. My reasoning is due to the individual distinct properties that make up
equilateral triangles.
Investigation/Exploration of the Problem
1. Before exploring the problem, we must make sure we understand the properties
that make up the different classifications of triangles. Triangles can be classified
by their side lengths (scalene, isosceles, equilateral) and also by their angle
measure (acute, right, and obtuse).

Right triangles have one 90º angle.

Acute triangles have three acute angles (measures less than 90º).

Obtuse triangles have an obtuse angle (more than 90º).

Equilateral triangles have 3 congruent sides and each angle is equal to 60º.

Isosceles triangles have at least two sides that are congruent and two equal
angles.

Scalene triangles have no congruent sides.
2. A right isosceles triangle CAN EXIST. It has one right angle with two congruent
sides and two equal angles.
m CB = 4.76 cm
mBAC = 90.00
j = 3.37 cm
mABC = 45.00
j ' = 3.37 cm
mACB = 45.00
B j
A
j'
C
3. A right equilateral triangle CANNOT EXIST. Equilateral triangles must have
equal side lengths and each angle must measure 60º.
m FE = 4.10 cm
k = 4.10 cm
k' = 4.10 cm
mEDF = 60.00
mDFE = 60.00
mFED = 60.00
D
k'
F
This means that if the sides are equal in length (1.96cm), then one side will not
belong enough to complete the triangle. Also, one of the angles would measure 90º,
which would not meet the properties of an equilateral triangle.
K Il
mHGI = 90.00
l ' = 1.96 cm
l = 1.96 cm
m HK = 1.96 cm
H l'
G
4. A right scalene triangle CAN EXIST. Since no sides are equal in scalene
triangles, this type of triangle can exist. All sides of the triangle are different
lengths, while one angle forms a 90º angle.
L
N
M
m MN = 3.31 cm
mNLM = 90.00
m NL = 3.12 cm
mLMN = 70.69
m ML = 1.10 cm
mMNL = 19.31
5. An acute scalene triangle CAN EXIST. All sides can be different lengths and all
three angles are less than 90º but still add up to 180º.
P
O
Q
m OP = 1.15 cm
mOQP = 50.56
m QO = 1.48 cm
mOPQ = 88.10
m PQ = 0.98 cm
mPOQ = 41.34
mOQP+mOPQ+mPOQ = 180.00
6. An acute isosceles triangle CAN EXIST. In the figure below, there are two sides
that are equal in length and two equal angles, where all angles are less than 90º.
R
T m'
S
m ST = 1.00 cm
mTRS = 30.00
m = 1.93 cm
mRST = 75.00
m' = 1.93 cm
mST R = 75.00
mTRS+mRST+mST R = 180.00
7. An acute equilateral triangle CAN EXIST. This shows it can exist because in the
figure below all the sides are equal and each angle measures 60º, which means the
figure contains three acute angles (measures less than 90º).
n'
nU
w
V W
n = 1.37 cm
mVUW = 60.00
w = 1.37 cm
mWVU = 60.00
n' = 1.37 cm
mUWV = 60.00
mVUW+mWVU+mUWV = 180.00
8. An obtuse scalene triangle CAN EXIST. The figure below shows that this type of
triangle can exist because there is one angle that is obtuse (greater than 90º) and
none of the sides are equal in lengths.
Z
m XY = 4.41 cm
m ZX = 7.21 cm
m ZY = 3.32 cm
Y
mZYX = 137.25
mYZX = 24.55
mZXY = 18.20
mZYX+mYZX+mZXY = 180.00
X
9. An obtuse isosceles triangle CAN EXIST. The figure below shows that this type
of triangle does exist because there is two sides that are equal in length and also
two equal angles, which means the third angle has to be the obtuse angle in the
triangle.
A1
o = 3.27 cm
m C1B1 = 5.36 cm
o'
B1
C1
o' = 3.27 cm
mB1A1C1 = 110.00
mA1C1B1 = 35.00
mC1B1A1 = 35.00
mB1A1C1+mA1C1B1+mC1B1A1 = 180.00
10. An obtuse equilateral triangle CANNOT EXIST. Since all sides have to be equal
in lengths, if the triangle has an obtuse angle, then the triangle will not be closed
by the third side. Two of the sides in the triangle can be equal in length but not all
three sides. Also, a property of equilateral triangles is that each angle is 60º.
Therefore, this means that there is no obtuse angle in an equilateral triangle.
A F1
p
B
p' = 2.65 cm
mE1D1F1 = 115.00
p = 2.65 cm
mBF1D1 = 39.43
m BA = 2.65 cm
D1
p'
E1
11. In conclusion, all of the following triangles can exist EXCEPT for a right
equilateral triangle and an obtuse equilateral triangle.
Extensions of the Problem
How would you construct an equilateral triangle using circles?
1. First I would construct a circle.
2. Then, I will construct a new circle by center and point. The two circles
will have the same radius.
3. Where the two circles meet, I will create an intersection point. From
there, I will create segments connecting the three points.
4. This will allow me to make an equilateral triangle. But to make sure, I
will measure each side lengths and the angles.
A
m AB = 2.31 cm
m CA = 2.31 cm
m BC = 2.31 cm
B
C
mBAC = 60.00
mACB = 60.00
mCBA = 60.00
Author & Contact
Lauren Johnson, Middle Grades Cohort at Georgia College and State University
Lauren_johnson@ecats.gcsu.edu
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2