4.1 Notes
Classifying Triangles
You can classify triangles by their angle measures or by their sides
Angle Measures
(You use angle measures to classify
Triangles)
Equiangular Triangle:
A triangle with three congruent angles.
Side Measures
(You use side measures to classify Triangles)
Equilateral Triangle:
A triangle with three congruent sides.
X
Y
Z
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅
𝑋𝑌
𝑋𝑍 ≅ 𝑌𝑍
∆𝑋𝑌𝑍 𝑖𝑠 𝐸𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙
Acute Triangle
All Angles in the triangle are acute angles.
All angles are between 0˚ & 90˚. (An
Equiangular Triangle is also an Acute
Triangle).
B
Isosceles Triangle
At least two sides of the triangle are
congruent. (An equilateral triangle is also
an isosceles triangle).
X
30˚
Y
70˚
80˚
A
T
∆𝑨𝑩𝑻 𝒊𝒔 𝒂𝒏 𝑨𝒄𝒖𝒕𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆
Z
̅̅̅̅ ≅ ̅̅̅̅
𝑋𝑌
𝑋𝑍
∆𝒀𝑿𝒁 𝒊𝒔 𝒂𝒏 𝑰𝒔𝒐𝒔𝒄𝒆𝒍𝒆𝒔 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆
Classifying Triangles: (continued)
You can classify triangles by their angle measures or by their sides
Obtuse Triangle
There is one obtuse angle in the triangle.
Largest side is across from largest angle.
Smallest side is across from the smallest
angle.
G
Scalene Triangle
No sides are congruent. No angles are
congruent.
Smallest angle is across from the smallest
side and the Largest angle is across from
the largest side.
D
R
K
M
∆𝑲𝑮𝑴 𝒊𝒔 𝒂𝒏 𝑶𝒃𝒕𝒖𝒔𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆
Right Triangle
There is one right angle in the triangle.
Pythagorean Theorem, special right
formulas and SOHCAHTOA is used with
Right Triangles. You must remember that
the hypotenuse is ALWAYS the largest side
and always across from the right angle.
The legs always make the right angle.
W
Leg
Hypotenuse
K
Leg
Z
∆𝑲𝑾𝒁 𝒊𝒔 𝒂 𝑹𝒊𝒈𝒉𝒕 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆
Q
∆𝑫𝑹𝑸 𝒊𝒔 𝒂 𝑺𝒄𝒂𝒍𝒆𝒏𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆
Knowing how to classify a triangle helps us to find angle measures or side measures
when an algebraic expression has been attached to the angle or the side, or finding
angles or sides to classifying triangles.
a. You can use angle measures to classify ∆𝐽𝑀𝐿. ∠JLM and ∠JLK form a linear pair,
so they are supplementary (must add to 180˚).
Using angle addition we know m ∠JLM + m ∠JLK = 180
m∠JLM + 120 = 180
m∠JLM = 60˚
Definition of Supplementary angles
Substitution Property
Subtraction Property of Equality
Since all the angles in ∆𝐽𝑀𝐿 are congruent, ∆𝐽𝑀𝐿 is an equiangular triangle.
b. You can use triangle classification to find the side lengths of a triangle
You only need two sides to make an equation. Even though this is equilateral, we only
need two sides to solve for x, then plug it in to find the length of all three sides.
̅̅̅̅ = 𝑅𝑆
̅̅̅̅
𝑄𝑅
Definition of Congruent Segments
4x = 3x + 5
Substitution Property of Equality
x =5
Subtraction Property of Equality
Substitute 5 back into 4x and 3x +5 and you will see both sides 4(5) = 3(5) +5 = 20.
Because this is an equilateral Triangle by the tick marks, you know all three sides equal
30 units.