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Lesson 4.2
Discover the properties of
Isosceles Triangles.
HOMEWORK:
Lesson 4.2/1-11
QUIZ Tomorrow: 4.1 – 4.2
Isosceles Triangle
at least two sides have the same length
5m
5m
5m
9 in
9 in
4 in
3 miles
Properties of an Isosceles Triangle
1
 Has at least 2 equal sides
 Has 2 equal angles
 Has 1 line of symmetry
Equilateral Triangle
Isosceles Triangle with all three sides are
congruent
7 ft
7 ft
7 ft
Parts of an Isosceles Triangle:
The vertex
angle is the
angle between
two congruent
sides
Parts of an Isosceles Triangle:
The base angles
are the angles
opposite the
congruent sides
Parts of an Isosceles Triangle:
The base is
the side
opposite the
vertex angle
Isosceles Triangle Conjecture
If a triangle is isosceles, then base angles
are congruent.
If
then
Converse of Isosceles Triangle Conjecture
If a triangle has two congruent angles, then
it is an isosceles triangle.
If
then
Equilateral Triangle Conjecture
An equilateral triangle is equiangular, and
an equiangular triangle is equilateral.
Find the missing angle measures.
<68° and < a are base angles 
therefore they are congruent
b
ma = 68˚
Triangle sum to find <b
m<b = 180 – 68 - 68
m<b = 180 -136
mb = 44˚
68˚
a
Find the missing angle measures.
Triangle sum = 180°
180 = 119 + c + d
180 – 119 = c + d
61 = c + d
mc = 30.5˚
<c & <d are base angles and are
congruent
<c = ½ (61) = 30.5
<d = ½ (61) = 30.5
119˚
md = 30.5˚
c
d
Find the missing angle measures.
EFG is an equilateral triangle
Therefore <E = <F = <G
180 /3 = 60
E
mE = 60˚
mF = 60˚
mG = 60˚
F
G
Find the missing angle measures.
Find mG.
∆GHJ is isosceles
<G=<J
x + 44 = 3x
44 = 2x
x = 22
Thus m<G = 22 + 44 = 66°
And m<J = 3(22) = 66°
Find the missing angle measures.
Find mN
Base angles are =
6y = 8y – 16
-2y = -16
y= 8
Thus m<N = 6(8) = 48°.
m<P = 8(8) – 16 = 48°
Find the missing angle measures.
Using Properties of Equilateral Triangles
Find the value of x.
∆LKM is equilateral.
m<K = m<L = m<M
180/3 = 60°
2x + 23 = 60
2x = 37
x = 18.5°
Find the missing side measures.
Using Properties of Equilateral Triangles
Find the value of y.
∆NPO is equiangular therefore,
∆NPO is also equilateral.
ft
5y – 6 = 4y +12
y – 6 = 12
y = 18
Side NO = 5(18) – 6 = 90ft
ft
Find the missing angle measures.
Using the symbols describing shapes answer the following questions:
b
45o
36o
a
c
d
Isosceles triangle
Two angles are equal
Equilateral triangle
all angles are equal
Right-angled triangle
a = 36o
b = 180 – (2 × 36)
= 108o
c = 180 ÷ 3 = 60o
d = 180 – (45 + 90)
= 45o
Find the missing angle measures.
Kite - Made up of 2 isosceles triangles
q
36o
p = 36o
s
q = 180 – (2 * 36) = 108o
p
r
56o
56 + (r + s) = 180o
(r + s) = 180 – 56 = 124
Because r = s
r = s = 124 ÷ 2 = 62o
Find the missing angle measures.
a = 64o
b = 180 – (2 ×64o ) = 52o
Equilateral triangle
c=d
c + d = 180 - 72
c + d = 108
c = d = 54o
e = f = g = 60o
h=i
h + i = 180 - 90
h + i = 90
h = i = 45o
p = 50o
q = 180 – (2 ×50o ) = 80o
r = q = 80o vertical angles are equal
Therefore :
s = t = p = 50o
Properties of Triangles
Find the missing
angle measures.
a = b= c =
60o
d = 180 – 60 = 120o
e + 18 = a = 60
exterior angle = sum of remote interior angles
e = 60 – 18 = 42o
p = q = r = 60o
s = t = 180 - 43 = 68.5o
2
Find the missing angle measures.
B
1) Find the value of x
z 50°
2) Find the value of y
1) x is a base angle
180 = x + x + 50
130 = 2x
x = 65°
A
y°
x°
C
2) y & z are remote interior angles
and base angles of an isosceles
triangle
Therefore: y + z = x and y = z
y + z = 80°
y = 40°
D
Find the missing angle measures.
1) Find the value of x
B
2) Find the value of y
1) ∆CDE is equilateral
All angles = 60°
Using Linear Pair
<BCD = 70°
x is the vertex angle
x = 180 – 70 – 70
x = 40°
y
A
D
x
70°
50 60°
C
2) y is the vertex angle
y = 180 – 100
y = 80°
E
Find the missing measures.
Lesson Quiz
Find each angle measure.
1. mR
2. mP
Find each value.
3. x
5. x
4. y
Find the missing angle measures.
Lesson Quiz
con’t
6. The vertex angle of an isosceles triangle
measures (a + 15)°, and one of the base
angles measures 7a°. Find a and each angle
measure. (Make a sketch)
Lesson Quiz Solutions
1. 28°
2. 124°
3. 20
4. 6
5. 26
6. a = 11; 26°; 77°; 77°
Homework
In your textbook:
• Lesson 4.2/ 1-11