Download proof of theorem 4-1

Review
Definition of Isosceles Triangle:
A triangle with at least two sides congruent.
Theorem 4-1: If a triangle has two congruent
sides, the angles opposite those sides are
congruent.
B
So, A  C
A
C
PROOF OF THEOREM 4-1:
B
Given: BD bisects ABC
AB  BC
A
Prove: A  C
Statements
D
C
Reasons
1. BD bisects ABC
1.
2. ABD  CBD
2. Def. of Angle Bisector
3.
4.
3.
Given
4. Reflexive Property
5.
6.
AB  BC
BD  BD
ΔABD  ΔCBD
A  C
Given
5. SAS Postulate
6.
CPCTC
Theorem 4-2: If two angles of a
triangle are congruent, the sides
opposite those angles are congruent.
B
C
A
So, AB  BC
PROOF EXAMPLE 1:
Given:
XY  XZ
Prove:
1  3
X
Y
Statements
1.
XY  XZ
1
2
Z
3
Reasons
1.
Given
2. If two sides of a
2.
1  2
triangle are congruent,
the angles opposite those
sides are congruent.
3.
4.
2  3
1  3
3. Vertical Angle Theorem
4. Substitution
PROOF EXAMPLE 2:
Given:
RS  RT
Prove:
3  4
R
S
3
Statements
1.
2.
1
2
T
4
Reasons
RS  RT
1.
Given
1  2
2. If two sides of a triangle are
congruent, the angles opposite
those sides are congruent.
3. 1  3, 2  4
3.
Vertical Angles Theorem
4.
2  3
4.
Substitution
5.
3  4
5.
Substitution
PROOF EXAMPLE 3:
Given:
XY  XZ
OY  OZ
Prove:
m1 = m4
X
1
Y
2
O
4
3
Z
Statements
1.
XY  XZ
2.
XYZ  XZY
mXYZ = mXZY
3.
OY  OZ
4. 2  3; m2 = m3
5. m1 + m2 = mXYZ
m3+ m4 = mXZY
Reasons
1.
Given
2. If two sides of a triangle are
congruent, the angles opposite
those sides are congruent.
3.
Given
4. If two sides of a triangle are
congruent, the angles opposite
those sides are congruent.
5. Angle Addition Postulate
6. m1 + m2 = m3+ m4
6.
Substitution
7.
7.
Subtraction
m1 = m4