Transcript Isosceles Triangle Theorem - Mustang-Math

(4.5) Isosceles and Equilateral Triangles
It does not do to dwell on dreams… and forget to live.
-Dumbledore
(4.5) Isosceles and Equilateral Triangles
OBJ: To use and apply
properties of isosceles
triangles.
Isosceles Triangles
The congruent sides are the
LEGS.
The third side is the BASE.
The two congruent sides form the
VERTEX ANGLE.
The other two angles are the
BASE ANGLES.
Theorems 
Theorem 4.3: Isosceles Triangle Theorem – If two sides of a
triangle are congruent, then the angles opposite those sides are
congruent.
A  B
Theorems 
Theorem 4.4: CONVERSE of Isosceles Triangle Theorem – If two
angles of a triangle are congruent, then the sides opposite the angles
are congruent.
AC  BC
Theorems 
Theorem 4.5: The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
CD  AB and CD bisects AB
Proof of the Isosceles Triangle Theorem
Given : XY  XZ , XB bisects YXZ
Prove : Y  Z
Statements
Reasons
1. XY  XZ ,
1. Given
XB bisects YXZ
2.
1  2
2. Definition of Perpendicular Bisector
3. XB  XB
3. Reflexive Property of Congruence
4. XYB  XZB
4. SAS
5. Y  Z
5. CPCTC
Applications
Explain why each statement is true.
a. WVS  S
Isosceles Triangle Theorem
b. TR  TS
Since angle TRS is congruent to angle WVS by Corresponding
angles and angle WVS is congruent to angle S, then angle TRS
is also congruent to angle S by transitive property of
congruence.
So by the converse of the isosceles triangle theorem, segment
TR is congruent to segment TS.
Applications
Find the value of y.
Corollaries
corollary – a statement that follows immediately from a theorem
Corollary to Theorem 4.3: If a triangle is
equilateral, then the triangle is equiangular.
X  Y  Z
Corollary to Theorem 4.4: If a triangle is
equiangular, then the triangle is equilateral.
XY  YZ  ZX
4.5: Isosceles and Equilateral Triangles
Homework:
(4.5) Pg. 230;
#1-13, 15, 20,
22, 28, 30
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