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Isosceles, Equilateral, Right
Triangles
I. Isosceles Triangle (at least two sides are congruent)
vertex 
leg
leg
base ’s
Legs: the two congruent sides of the triangle.
Base: the remaining side of the triangle.
Base Angles: the angles at the base.
Vertex Angle: the angle opposite the base.
base
Base Angles Theorem
If 2 sides of a triangle are congruent, then the angles
opposite them are congruent.
If AB  AC, then B  C
A
B
C
Ex. 1 Find the measure of F and G.
F
G
m F = 65o base angles thm
65o
H
Converse of Base Angles Theorem
If 2 angles of a triangle are congruent, then the sides
opposite them are congruent.
If B  C , then AB  AC
A
B
C
Ex. 2 Find the value of x.
4x + 4 = 32 converse of base angles thm
R
4x + 4
Q
S
32
4x = 32 - 4
4x = 28
x=7
ex. 3
The vertex of an isosceles triangle is 60o. What is
the measure of the base angles.
A
mB = mC = x
60o
x + x + 60 = 180
2x = 180 - 60
2x = 120
B
x
x
C
x = _120_
2
x = 60
mB = mC = 60o
Base angles thm
∆ sum thm
Ex. 4 Find the mB.
A
68o
C
xo
B
II. Equilateral Triangle (all sides are congruent)
All equilateral triangles are equiangular and all
equiangular triangles are equilateral.
All 3 sides are congruent and all 3 angles are
congruent.
Ex. 3 Find the length of each side of the equiangular triangle.
6x + 3 = 7x – 1
6x – 7x = - 1 – 3
-x = -4
x= 4
Determine the length of the sides by
substituting x = 4 into the equation.
7(4) – 1
=28 – 1
= 27 units
equiangular equilateral
triangle
Open your Textbooks
Pg 188 #8-14 (even)
Pg 189 #18-24 (even)
Pg 190 #33-34
Pythagorean Theorem and
the
Distance Formula
Hypotenuse
(opposite the right angle)
Side
c
a
Side
b
2 = a2 + b2
c
Pythagorean Theorem is written as
when a and b represent the lengths of the
sides and c represents the length of the
hypotenuse.
Pythagorean Theorem
(hypotenuse)2 = (side)2 + (side)2
2
c
=
2
a
+
2
b
Ex. 1 Use Pythagorean Theorem to find
the missing side.
8 cm
c2 = a2 + b2
c2 = 82 + 62
c2 = 64 + 36
c2 = 100
c = 100
c = 10 cm
6 cm
c
Ex. 2 Find the unknown side.
c2 = a2 + b2
132 = x2 + 32
169 = x2 + 9
169 – 9 = x2
160 = x2
x= 160
x
13 cm
3 cm
Ex. 3 Find the length of the hypotenuse of the right
triangle. Do the lengths of the sides of the triangle form a
Pythagorean triple?
c2 = a2 + b2
x
21cm
x2 = 202 + 212
x2 = 400 + 441
x2 = 841
x = 841
x = 29 cm
20cm
20, 21, and 29 form a
Pythagorean triple,
because they are integers
that satisfy c2 = a2 + b2
Open your Textbooks
• Pg. 195 #2, 4, 10, 14
• Pg 197 #35, 36
Converse of Pythagorean Theorem
We can use the Converse of the Pythagorean theorem to determine
whether a triangle is an acute, obtuse, and right triangle.
*Note: c represents the largest side measure.
a
c
b
• If c2 < a2 + b2, then ΔABC is an acute triangle.
• If c2 = a2 + b2, then ΔABC is a right triangle.
• If c2 > a2 + b2, then ΔABC is an obtuse triangle.
Ex1. Classify the triangles as acute, obtuse, or right.
X
Do the Pythagorean Theorem
c2 = a2 + b2
7cm
11cm
112 = 72 + 92
W
121 = 49 + 81
Y
9cm
121 < 130 (121 is less than 130)
∆WXY is an acute triangle
Ex 2. Classify the triangles as acute, obtuse, or right.
A
c2 = a2 + b2
372 = 122 + 352
1369 = 144 + 1225
12
37
c
35
1369 = 1369
∆ABC is a right triangle
B
Ex 3. Classify the triangles as acute, obtuse, or right.
c2 = a2 + b2
352 = 122 + 302
12cm
35cm
1225 = 144 + 900
30cm
1225 > 1044
∆WXY is an obtuse triangle
Distance Formula: the distance d between two points A(x1, y1) and B(x2, y2) is
d=
Ex. 4 Find the distance between T(5, 2) and R(-4, -1) to the nearest
tenth (one decimal place).
d = √90
Class work
Pg. 195 #5
Pg. 197 #28
Pg. 203 #2, 6, 14,
Pg 204 # 18-20, 25