Download Lesson 4.3 Powerpoint - peacock

Isosceles and Equilateral
Triangles
Section 4.3
Objectives
• Use properties of isosceles triangles
• Use properties of equilateral triangles
Key Vocabulary
•
•
•
•
Legs of an Isosceles Triangle
Base of an Isosceles Triangle
Vertex Angle
Base Angles
Theorems
•
•
•
•
4.3 Base Angles Theorem
4.4 Converse of Base Angles Theorem
4.5 Equilateral Theorem
4.6 Equiangular Theorem
Definitions Review
• Isosceles Triangle
– At least 2 congruent sides
– From Greek: Isos – means “equal,” and – sceles
means “leg.” So, isosceles means equal legs.
• Equilateral Triangle
– 3 congruent sides
– From Latin: Equi – means “equal,” and – lateral
means “side.” So, equilateral means equal sides.
– An equilateral triangle is a special case of an
isosceles triangle having not just two, but all three
sides equal.
Properties of Isosceles Triangles
• The  formed by the ≅ sides is
called the vertex angle.
• The two ≅ sides are called legs.
The third side is called the base.
vertex
leg
leg
• The two s formed by the base
and the legs are called the
base angles.
base
Definitions - Review
VABC is an isosceles triangle.
A
B
C
Name each item(s):
Vertex Angle B
Base
AC
Legs
AB, CB
Base Angles
A, C
Side opposite  C
AB
Angle opposite BC
A
Base Angles Theorem
• Theorem 4.3
If two sides of a triangle are congruent, then the
angles opposite those sides are congruent ().
A
• If AC ≅ AB, then B ≅ C.
B
C
The Converse of Base Angles Theorem
• Theorem 4.4
If two angles of a triangle are congruent,
then the sides opposite those angles are
congruent.
A
• If B ≅ C, then AC ≅ AB.
B
C
Example 1
M
Find the measure of L.
SOLUTION
L
?
Angle L is a base angle of an
isosceles triangle. From the Base
Angles Theorem, L and N have
the same measure.
ANSWER
The measure of L is 52°.
52
N
Example 2
Find the value of x.
SOLUTION
By the Converse of the Base Angles Theorem, the legs have
the same length.
DE = DF
x + 3 = 12
x=9
ANSWER
Converse of the Base Angles Theorem
Substitute x + 3 for DE and 12 for DF.
Subtract 3 from each side.
The value of x is 9.
Your Turn:
Find the value of y.
1.
ANSWER
50
ANSWER
9
ANSWER
12
2.
3.
Example 3a:
Name two congruent angles.
Answer:
Example 3b:
Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the
sides opposite congruent angles are congruent. So,
Answer:
Your Turn:
a. Name two congruent angles.
Answer:
b. Name two congruent
segments.
Answer:
More Practice
∠1≅∠3
∠11≅∠8
RG  HG
TN and HN
∠G
Solve for x and y
x = 72
y + 72 +72 = 180
y + 144 = 180
y = 180 - 144
y = 36
Solve for x and y
∠x≅∠1
m∠x + m∠1 = 90
2x = 90
x = 45
x + y = 180
y = 180 - 45
y = 135
1
Solve for x
 3x  8   3x  8    2 x  20   180
8 x  36  180
8 x  144
x  18
Solve for x
4 x 10 180  80
4 x  90
x  22.5
Solve for x
x = 63
Your Turn - Find the missing measures
(not drawn to scale)
• 1.
• 2.
44°
?
?
30°
?
?
Find the missing measures
(not drawn to scale)
• 1.
• The two base angles
are = to each other
b/c they are opposite
congruent sides
• 180 – 44 = 136°
• 136/2 = 68°
44°
? 68°
68°?
Find the missing measures
(not drawn to scale)
• 2.
?
?
30°
Find the missing measures
(not drawn to scale)
• The other base angle
must be 30° b/c its
opposite from a
congruent side
• 180 – (30+30) = 120
• 2.
?
120°
30°
30° ?
Properties of Equilateral ∆’s
Equilateral Triangle – a triangle
with three congruent sides.
Equilateral Theorem
• Theorem 4.5
If a triangle is equilateral, then it is
equiangular.
A
If AB  BC  AC, then
mA  mB  mC
B
C
Equiangular Theorem
• Theorem 4.6
If a triangle is equiangular, then it is
equilateral.
A
If A  B  C , then AB  BC  CA
B
60˚
60˚
60˚
C
Equilateral and Equiangular
Theorems
• What these theorems mean.
• In a Triangle;
1) If all 3 sides are equal, then all 3 angles
measure 60˚.
2) If all 3 angles measure 60˚, then all 3 sides
are equal.
• An equilateral triangle is an equiangular
triangle and vice versa.
Example 4
Find the length of each side of the equiangular triangle.
SOLUTION
The angle marks show that ∆QRT is equiangular. So, ∆QRT is also
equilateral.
3x = 2x + 10
x = 10
3(10) = 30
ANSWER
Sides of an equilateral ∆ are congruent.
Subtract 2x from each side.
Substitute 10 for x.
Each side of ∆QRT is 30.
Example 5a:
EFG is equilateral, and
Find
and
bisects
bisects
Each angle of an equilateral triangle measures 60°.
Since the angle was bisected,
Example 5a:
is an exterior angle of EGJ.
Exterior Angle Theorem
Substitution
Add.
Answer:
Example 5b:
EFG is equilateral, and
Find
bisects
bisects
Linear pairs are supplementary.
Substitution
Subtract 75 from each side.
Answer: 105
Your Turn:
ABC is an equilateral triangle.
a. Find x.
Answer: 30
b.
Answer: 90
bisects
Your Turn:
Solve for x and y
x = 60
y = 120
Joke Time
• What has four legs and one arm?
• A happy pit bull.
• What's the difference between chopped beef and pea
soup?
• Everyone can chop beef, but not everyone can pea
soup!
• What do you get when you cross an elephant and a
rhino?
• el-if-i-no
Assignment
• Sec 4.3, Pg. 188-190: #1 – 25 odd, 29 –
39 odd