Download Section 4.1

§ 4.1
C
1. Prove that the three angle
bisectors of a triangle concur.
E
Given:  CAD   BAD,  ABE   CBE
AND  ACF   BCF
A
Prove: AD, BE AND CF concurr.
Statement
Reason
1. Let AD ∩ BE = I
Two lines intersect.
2. I is in the interior of  A
3. I is in the interior of  B
On the bisector
On the bisector
4. I is in the interior of  C
5. I is equidistant from AC and AB
6. I is equidistant from AB and BC
From 2 & 3
Def. angle bisector
Def. angle bisector
7. So I is equidistant from AC and
BC
Transitive property
 I is on CF and CF is a
bisector of  C
Def of bisector.
I
F
D
B
C
2. Prove that the perpendicular bisectors
O
of a triangle concur.
Given:  CAD   BAD,  ABE   CBE
AND  ACF   BCF
A
Prove: AD, BE AND CF concurr.
Statement
Reason
1. Let l 1 ∩ l 2 = O
Two lines intersect.
2. BO = CO
3. AO = CO
O is on the  bisector of AB
O is on the  bisector of AC
4. BO = AO
5. O is on l 3
Transitive property
Def.  bisector
 l 1 , l 2 and l 3 concur at O
l1
l2
B
l3
Q
U
C T
3. Prove that the altitudes of a
triangle concur.
P
O
A
B
V
R
Note that the altitudes of ∆ABC are
the perpendicular bisectors of the
sides of ∆PQR and using the
previous problem the perpendicular
bisectors concur.
B
4. Complete the proof that the
exterior angle of a triangle is
greater than each of its remote
interior angles.
Given: A – C – D and ∆ ABC
Prove  ACG >  A
A
E
D
C
F
G
Statement
Reason
1. Let E be the midpoint of AC.
2. Choose F on BE so that BE = EF
Construction
Construction.
3.  BEA =  CEF
4. ∆AEB = ∆ECF
Vertical angles
SAS
CPCTE
Arithmetic
Angle Addition
CPCTE & Substitution
Arithmetic
Angle measure postulate
5. m  A = m  ECF
6. m A + m FCG = mECF + m FCG
7. mACG = mECF + m FCG
8. mACG = mA + m FCG
9. mACG > mA
10. ACG > A
6. Given: AD bisects  CAB and CA =
CD. Prove: CD parallel to AB.
Given: AD bisects  CAB and CA = CD.
Prove: CD parallel to AB.
Statement
Reason
1. CAD = DAB
Given AD bisector
2. CA = CD
3. CAD = CDA
4. CDA = DAB
Given
s opposite = sides =
Transitive property
5. CD parallel to AB
s in 4 are alternate interior
B
7. Segments AB and CD bisect each
other at E. Prove that AC is parallel
to BD.
E
C
D
A
Statement
Reason
1. AE = EB & CE = ED
1. Given
2. AEC = BED
2. Vertical angles
3. AEC = BED
3. SAS
4. CAE = DBE
4. CPCTE
5. AC parallel to BD
5. s in 4 are alternate interior
angles.
8. Given two lines cut by a transversal. If a
pair of corresponding angles are congruent,
prove that a pair of alternate interior angles are
congruent.
t
B
C
A
Given: l and m cut by transversal m and
 A =  B.
Prove:  A =  C.
Statement
Reason
1.  A =  B.
1. Given
2.  B =  C.
2. Vertical angles
3.  A =  C.
3. Transitive
l
m
9. Given two lines cut by a transversal. If a pair
of corresponding angles are congruent, prove
that the lines are parallel.
t
B
C
A
Given: l and m cut by transversal m and
 A =  B.
Prove: l and m parallel.
Statement
Reason
1.  A =  C.
1. Previous problem
2. l and m parallel.
2. Definition of parallel
l
m
11. Given triangle ABC with AC = BC and
DC = EC, and  EDC =  EBA, prove DE
is parallel to AB.
Given: AC = BC and DC = EC, and
 EDC =  EBA
Prove: DE is parallel to AB.
Statement
Reason
1. AC = BC and DC = EC
1. Given
2. ABC and AEC isosceles.
2. Definition of isosceles
3.  EDC =  EBA.
3. Given.
4.  EDC =  DEC.
4. Base angles.
5.  DEC =  EBA.
5. Substitution of 3 into 4.
6.  DEC &  EBA are corresponding s.
6. Definition.
7. DE is parallel to AB.
7. Corresponding s. equal. 5 & 6