Download Isosceles and Equilateral Triangles

Aim: What theorems apply to isosceles and
equilateral triangles?
K
Do Now:
Given: AKC is isosceles with KA  KC
KB bisects AKC
A
Prove: A  C
Statements
B
Reasons
1) KA  KC
2) KB bisects AKC
1) Given
2) Given
3) AKB  CKB
3) Def. angle bisector
4) KB  KB
5) AKB  CKB
4) Reflexive Postulate
5) S.A.S. Postulate
6) A  C
C
6) C.P.C.T.C.
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
1
Isosceles Triangle
Theorem:
angles of an isosceles triangle are congruent
Theorem
#8: Base
Isosceles triangles
E
T
P
Z
G
Q
D
V
M
Corollary #8-1: The bisector of the vertex angle of an
isosceles triangle bisects the base.
Corollary #8-2:
The bisector of the vertex angle of an
isosceles triangle is perpendicular to the base.
x
In other words: The median, altitude and angle bisector from
the vertex of an isosceles triangle are all the same segment.
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
2
Equilateral Triangles
Corollary #8-3: Every equilateral triangle is equiangular.
B
B
B
Or
A
A  C
C
Or
A
A  B
C
A
B  C
B
C
A  B  C
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
3
C
Ex: Isosceles Triangles
If the following pairs of segments are
congruent, which angles are congruent.
L
1) LD  LK Ans: D  K
2) QL  QT Ans : QTL  QLT
3) QT  FT Ans : TQF  TFQ
4) FL  TL Ans : LFT  LTF
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
Q
D
F
K
T
4
V
Ex: Proof w/Isosceles Triangle
Given: Isosceles VRK with VR  VK
M is the midpoint of RK
TMR  AMK
Prove: MT  MA
Statements
A
T
R
M
K
Reasons
1) VR  VK
2) R  K
1) Given
2) Base 's of isosc. 's are  .
3) M is the midpoint of RK 3) Given
4) MR  MK
5) TMR  AMK
TMR  AMK
7) MT  MA
6)
4) Def. Of midpoint
5) Given
6) A.S.A. Postulate
7) C.P.C.T.C
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
5
D
Proofs w/Isosceles Triangles
1) Given: Isosceles RDV with RD  RV
Isosceles TDV with TD  TV
Prove: RDT  RVT
T
R
M V
2) Given: Isosceles MGD with MG  MD
y
GHTD , x  y
Prove: MHT is isosceles
G
H
3) Given: Isosceles ALE with AL  AE
QL  NE , PQ  AL, PN  AE
Prove: QPL  NPE
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
x
D
T
A
Q
L
N
P
E
6
Algebra w/Isosceles Triangles
4) Isosceles ABC has AB  CB. If AB  7 x  2,
and CB  4 x  20, determine AB and CB.
5) Isosceles ABC has AB  AC. If AB  8 x  8,
AC  6 x  38, and BC  3x  24 determine AB,
AC and BC.
Geometry Lesson: Isosceles and
Equilateral Triangle Theorems
7