Proving Triangles are Congruent
For each of the following, write a two-column proof.
1.
2.
A
Given: AB  DF
AC  DE
BC  EF
Prove: ABC  DFE
B
C
E
Given: T  N
TAB  NAB
Prove: TAB  NAB
F
4.
A
Given: AT bisects LAS
LA  AS
Prove: ∆ATL  ∆ATS
T
L
Given: B  C
A  O
AC  BO
Prove: CAR  BOX
A
S
7.
X
Given: M  P
MA  PE
MN  PT
Prove: MAN  PET
A
E
N
P
M
N
I
X
T
8.
Given: PT  AE
AT  TE
C
R
Prove: ∆PAT  ∆PET
A
9.
10.
L
Given: PA bisects LAN
LA  AN
Prove: PLA  PNA
N
T
M
Given: X is the midpoint of AI
A  I
MA  IN
Prove: MAX  NIX
A
O
B
E
Given: RA  RE
EC  AC
Prove: REC  RAC
A
6.
C
R
T
D
3.
5.
B
A
P
Given: KWL  ALW
AWL  WLK
Prove: ∆KWL  ∆ALW
E
K
W
A
N
P
A
L
Proofs with CPCTC
For each of the following, write a two-column proof.
1.
2.
A
Given: AD  BC
BD  CD
Prove: AB  AC
Given: CL  PA
1  2
Prove: LA  CP
1
2
L
A
C
D
B
3.
4.
L
Given: PA bisects LPN
PA bisects LAN P
Prove: N  L
Given: I is the midpoint of CM
I is the midpoint of BL
Prove: CL  MB
A
N
5.
P
6.
A
Given: PA || TN
PN bisects AT
Prove: PI  IN
P
C
Given: SE  SU
E  U
Prove: MS  OS
I
T
N
B
C
1
2
I
M
L
U
E
1
S
2
O
M
Z
T
7.
8.
P
Given: PT  PA
O  A
PR  AT
Prove: R is the midpoint of AT
Prove: ∆TOP  ∆ZAP
T
9.
P
R
A
PA  TR
A
O
A
10.
E
Given: SA  NE
Given: PA || TR
Prove: PT  AR
P
Given: TO  AZ
N
SE || NA
T
R
Prove: SA  NE
S
A
Proving Triangle Theorems
Complete a two-column proof for each of the following theorems.
Third Angle Theorem: If two angles in one triangle are equal
in measure to two angles of another triangle, then the third
angle in each triangle is equal in measure to the third angle in
the other triangle.
Given: A  D
B  E
Prove: C  F
Exterior Angle Theorem: The measure of an exterior angle
of a triangle is equal to the sum of the two remote interior
angles.
A
C
D
B
2
E
B
A
F
Isosceles Triangle Theorem: If two sides of a triangle are
congruent, then the angles opposite those sides are congruent.
T
Given: TW  TN
TE is the median of TWN
Prove: W  N
W
B
N
W
C
S
E
Corollary: If the three angles of a triangle are congruent, then
the triangle is equilateral.
Given: A  B  C
Prove: ABC is equilateral.
A
Given: ABC is equilateral
Prove: A  B  C
Isosceles Triangle Converse: If two angles of a triangle are
congruent, then the triangle is isosceles.
N
Corollary: If a triangle is equilateral, then the angles are
congruent.
D
Prove: m1  mA  mB
Given: W  E
NS bisects WNE
Prove: NEW is isosceles
E
1
C
A
B
C