Transcript Geometry 4.4 The Isosceles Triangle Theorems

Geometry
4.4 The Isosceles Triangle
Theorems
Isosceles Triangle
Vertex Angle
Leg
Leg
Base Angle
Base
Base Angle
The Isosceles Triangle Theorem
If two sides of a triangle are congruent, then
the angles opposite those sides are
congruent.
Iso.
Thm.
Corollary 1
An equilateral triangle is also equiangular.
Why does this come from the Isosceles Triangle Theorem?
Corollary 2
An equilateral triangle has three 60o angles.
Corollary 3
The bisector of the vertex angle of an
isosceles triangle is perpendicular to the
base at its midpoint.
Proof Plan:
Converse to Isosceles Triangle Theorem
If two angles of a triangle are congruent, then
the sides opposite those angles are
congruent.
Converse to Iso.
Thm.
2 ways to remember which is which…
Corollary
An equiangular triangle is also equilateral.
Use the given information to name an isosceles triangle and its congruent base angles.
1. AB  AE
2.
A
A
AC  AD
C
C
BB
D
D
FF
EE
GG
Use the given information to name an isosceles triangle and its congruent legs.ZYXWVUTS
2.
2. STU  SUT
3.
S
S
T
T
3.
V  TWV
VV
U
U
W
W
XX
Z
Z
YY
5.
6.
x
x
3x
3x
x
xx
x
8.
7.
100
9.
70
70
50
50
x
x
62
62
xx
x
x
100
E
Given: 4  3
Prove: DE  FE
D
4
2. vert
1.
4.
3.
Statements
Reasons
4  3
2. 4

3 
3.
DE  FE
Given
Given:
Vert. angles congruent
Prove:
Transitive
4.
1
2
F
3
Given: j // k
AB  AC
A
Prove: AD  AE
Statements
1.then
,2.
6.
5.
4.
s3.
j lines
If
corr
j // k
Reasons
1
D
E
j
Given
3
3 
4 
2
// implies corr. angles
congruent
AB  AC
Given
3  4
4.
1  2
5.
AD  AE
6.
B
4
C
k
HW
P. 137-138 #1-10,13,14, 23-25, 27, 28
A Few from the HW Together
23) a) If m<1 = 20, then m<3 = ____.
m<4 = ____ and m<5 = ____.
b) If m<1 = x, then m<3 = ____.
m<4 = ____ and m<5 = ____.
4
3
5
2
1
A Few From the HW
3) Find x.
74o
10
10
x
12
A Few From the HW Together
5) Solve for x.
40o
40o
5x - 8
2x + 7
A Few from the HW Together
13) Given: M is the midpoint of JK
1~
= 2
J
Prove: JG ~
= MK
7
1
2
M
G
K
7