Download Isosceles and Equilateral Triangles

Isosceles and Equilateral
Triangles
Academic Geometry
Isosceles and Equilateral Triangles
Draw a large isosceles triangle ABC, with
exactly two congruent sides, AB and AC.
What is symmetry?
How many lines of symmetry does it have?
Label the point of intersection D.
Isosceles and Equilateral Triangles
What is the relationship between AD and
BC?
Isosceles and Equilateral Triangles
Draw a Triangle XYX with exactly two
congruent angles, <Y and <Z. Find the
line of symmetry.
What can you conclude about the sides?
Isosceles Triangle Theorems
The congruent sides of an isosceles trianlge are its
legs.
The third side is the base.
The two congruent sides form the vertex angle.
The other two angles are base angles.
Theorem 4-3
Isosceles Triangle Theorem
The base angles of an isosceles triangle are
congruent.
If the two sides of a triangle are congruent, then
the angles opposite those sides are congruent.
c
a
b
Theorem 4-4
Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent,
then the sides opposite those angles are
congruent.
c
a
b
Theorem 4-5
The line of symmetry for an isosceles
triangle bisects the vertex angle and is the
perpendicular bisector of the base.
c
CD
AB
CD bisects AB
a
b
d
Example
EC is a line of symmetry for isosceles
triangle MCJ.
Draw and label the triangle.
M<MCJ = 72. Find m<MEC, m<CEM and
EJ.
ME = 3
Proof of the Isosceles
Triangle Theorem
Begin with isosceles triangle XYZ. XY is congruent XZ.
Draw XB, the bisector of the vertex angle YXZ
Prove <Y congruent <Z
Statements
Reasons
Using the Isosceles Triangle
Theorems
Why is each statement true?
<WVS congruent <S
t
u
TR congruent TS
w
r
s
v
Can you deduce that Triangle RUV is
isosceles? Explain
Using Algebra
Find the value of y
m
y
63
l
o
n
Equilateral Triangles
Draw a large equilateral triangle, EFG.
Find all the lines of symmetry. How many
are there?
What do we know about the sides?
The angles?
Isosceles and Equilateral Triangles
We learned in the last chapter that
equilateral triangles are also isosceles.
A corollary is a statement that immediate
follows from a theorem.
Corollary to Theorem 4-3
If a triangle is equilateral, then the triangle is
equiangular.
y
x
z
<X is congruent to <Y is congruent to <Z
Corollary to Theorem 4-4
If the triangle is equiangular, then the
triangle is equilateral.
y
x
z
XY is congruent to YZ is congruent to ZX