Download Notes for Isosceles Triangle Theorem

Notes for Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Given: AC  BC
Prove: A  B
B
A
Given
AC  BC
C
C
B
A
Reflexive
Auxiliary Line
CD  CD
Draw CD , the bisector
of  ACB
Def. of angle
bisector
ACD  BCD
SAS
∆ACD  ∆BCD
CPCTC
A   B
Converse of Isosceles Triangle Theorem – If two angles of a triangle
are congruent, then the sides opposite those angles are congruent.
A
Given: A  B
Prove: AC  BC
Given
A  B
C
C
Auxiliary Line
Draw CD , the bisector
of  ACB
Def. of angle
bisector
ACD  BCD
AAS
∆ACD  ∆BCD
CPCTC
AC  BC
B
B
A
Reflexive
CD  CD
Theorem – The bisector of the vertex angle of an isosceles triangle is the perpendicular
bisector of the base.
If-then form: If a segment is the bisector of the vertex angle of an isosceles triangle, then
that segment is the perpendicular bisector of the base of the isosceles triangle.
C
Given: ∆ABC is isosceles with vertex  C
CD bisects  ACB
Prove: CD is the perpendicular bisector of AB
A
Given
Given
∆ABC is isosceles
with vertex  C
Reflexive
CD bisects  ACB
CD  CD
Def. of angle
bisector
Def. of isos. ∆
AC  BC
ACD  BCD
SAS
∆ACD  ∆BCD
CPCTC
AD  BD
and
ADC  BDC
Def.
segment
bisector
CD bisects AB
 linear  s →  lines
CD  AB
Def. of  bisector
CD is  bisector of AB
D
B