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4-6 Isosceles And Equilateral
Triangles
You identified isosceles and equilateral
triangles.
• Use properties of isosceles triangles.
• Use properties of equilateral triangles.
Isosceles Triangles
Parts
vertex
Vertex angle
leg
leg
Base angles
base
The Isosceles Have It!
An isosceles triangle has been drawn
on a piece of paper and then cut out.
(How do you draw an isosceles
triangle on a piece of paper?)
If the triangle is folded in half, what can
be said about the base angles?
What can be said about the sides?
Isosceles Triangle Theorem
If two side of a triangle are congruent, then the
angles opposite those sides are congruent.
Converse of the Isosceles
Triangle Theorem
If two angles of a triangle are congruent, then
the sides opposite those angles are congruent.
Page 285
A. Name two unmarked congruent angles.
__
BCA is opposite BA
____
and A is opposite BC,
so BCA  A.
Answer: BCA and A
B. Name two unmarked congruent segments.
Answer: BC  BD
Page 286
Page 286
A. Find mR.
Since QP = QR, QP  QR. By the
Isosceles Triangle Theorem, base
angles P and R are congruent, so
mP = mR . Use the Triangle Sum
Theorem to write and solve an equation
to find mR.
Triangle Sum Theorem
mQ = 60, mP = mR
Simplify.
Answer:
mR = 60
Subtract 60 from each side.
Divide each side by 2.
B. Find PR.
Since all three angles measure
60, the triangle is equiangular.
Because an equiangular triangle is
also equilateral, QP = QR = PR.
Since QP = 5, PR = 5 by
substitution.
Answer: PR = 5 cm
A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°
ALGEBRA Find the value of each variable.
Since E = F, DE  FE by the Converse of
the Isosceles Triangle Theorem. DF  FE, so all
of the sides of the triangle are congruent. The
triangle is equilateral. Each angle of an
equilateral triangle measures 60°.
mDFE = 60
4x – 8 = 60
4x = 68
x = 17
Definition of equilateral triangle
Substitution
Add 8 to each side.
Divide each side by 4.
The triangle is equilateral, so all the sides are
congruent, and the lengths of all of the sides are
equal.
DF = FE
Definition of equilateral triangle
6y + 3 = 8y – 5 Substitution
3 = 2y – 5 Subtract 6y from each side.
8 = 2y
Add 5 to each side.
4=y
Divide each side by 2.
Answer: x = 17, y = 4
Try It
B
In isosceles triangle ABC , AB  BC .
What else must be true?
C
A
O
In MNO, M  N . Find the
lengths of sides MO and NO.
3x+8
M
4x−10
N
What makes an isosceles unique?
An isosceles triangle has two congruent
sides and two congruent base angles.
What is an auxiliary line?
Auxiliary line is a line (or part of a line) added
to a figure.
4-6 Assignment
Page 289, 1-2, 1522, 29-32