Download Classifying Triangles

Classifying Triangles

Angle classification of triangles
o Acute triangle-all the angles are acute
o Obtuse triangle-one angle is obtuse
o Right triangle-one angle is right or 90°
o Equiangular triangle-an acute triangle in which all angles are congruent

Side classification of triangles
o Scalene triangle-no two sides are congruent
o Isosceles triangle-at least two sides are congruent
o Equilateral triangle-all sides are congruent

Parts of an isosceles triangle
o Legs-congruent sides
o Vertex angle-angle formed by the congruent sides
o Base angles-two angles formed by the base and one of the congruent sides
Vertex angle
Leg
Leg
Base angle ->
<- Base angle
Base
Example:
Keneshee says the figure below is an isosceles triangle, but Norma says it is a
right triangle. Who is correct? Justify your answer.
Answer: Both girls are right. Referring to the side classification of triangles, the
figure is an isosceles triangle because two sides are congruent. Referring to the
angle classification of triangles, the figure is a right triangle because it has one
right angle. Every triangle can be classified by its angles as well as its sides.
Example:
Given  DAR with vertices D(2,6), A(4,-5), and R(-3,0), use the distance formula
to show that  DAR is scalene.
Answer: According to the distance formula, the distance between the points at
( x1 , y 1 ) and ( x 2 , y 2 ) is
( x 2  x1 ) 2  ( y 2  y 1 ) 2
DR  ( 3  2 ) 2  ( 0  6 ) 2  25  36  61
AD  ( 2  4 ) 2  ( 6  ( 5 )) 2  4  121  125
RA  ( 4  ( 3 )) 2  ( 5  0 ) 2  49  25  74
Since no two sides have the same length, the triangle is scalene.
Example:
Triangle RST is an isosceles triangle. <R is the vertex angle, RS = x + 7,
ST = x-1, and RT = 3x-5. Find x, RS, ST, and RT.
S
x+7
R
x-1
3x - 5
T
Answer: Since <R is the vertex angle, the side opposite <R, ST, is the base of the
triangle. The congruent legs are RS and RT. So, RS = RT.
RS  RT
RS  RT
x  7  3x  5
12  2 x
6x
congruent sides
equal lengths of sides
by substitution
isolate the variable x
divide by 2
If x = 6, then RS = 6 + 7 or 13. Since RS = ST, RT = 13. Since ST = x – 1,
ST = 6 – 1 or 5. The legs of the isosceles triangle are each 13 units long and the
base is 5 units long.