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Teacher: Mr. Joseph
Triangle Proofs
Name:__________________________
Date: 11/16/16
Solve:
1)
2)
Given: YQL
GDP, write a
congruence statement for corresponding
angles.
3)
Given F
G, S
N, L
K, and the fact that the triangles are
congruent, write the triangle
congruence statement.
4)
Given AT = UG, TR = UJ, and the fact
that the triangles are congruent, write
the triangle congruence statement.
5)
Given T
S, BT = WS, YT = SP,
and the fact that the triangles are
congruent, write the triangle
congruence statement.
6)
Given: YVP
NSC, write a
congruence statement for corresponding
sides.
Given T
N, Q
R, WT =
NS, and the fact that the triangles are
congruent, write the triangle
congruence statement.
Name the Postulate or Theorem Used to Prove the Triangles Congruent (if
any):
7)
Given:
__ __ __ __
GA AP, BG PR,
8)
GAP
PAR
Given:
__ __ __ __
TA NA, BA MA
9)
Given:
10)
Given:
__ __
WP GF
11)
Given:
12)
Given:
__ __ __ __
AC OD, CT GD,
C
D
Fill in the reasons in the proof:
13)
14)
Given:
P
D,
PGI
DGI
Prove:
PIG
DIG.
Statement
Reason
P
PGI
__ __
JA RI,
Prove:
D
DGI
__ __
IG IG
PIG
Given:
DIG
J
JAM
Statement
__ __
JA RI
J
R
R
RIM.
Reason
RMI and AMJ are
vertical angles
RMI
AMJ
JAM
15)
16)
Given:
RIM
Given:
__ __ __ __
CO PI , OW GI,
Prove:
SIT
Statement
AND.
Reason
Prove:
COW
Statement
__ __
CO PI
__ __
SI AN
S
A
T
D
SIT
O
AND
18)
Given:
RFP
LFP,
Prove:
PLF
Statement
RFP
LFP
RPF
LPF
__ __
PF PF
PLF
I
PIG.
Reason
__ __
OW GI
I
COW
17)
O
RPF
PRF.
Reason
LPF
PIG
Given:
__ __ __ __
NM NT, AN EN
Prove:
MEN
Statement
__ __
NM NT
__ __
AN EN
PRF
N
N
TAN.
Reason
MEN
TAN
Write a Proof:
19)
20)
Given:
SEAT is a square.
U
C
Prove:
USE
Given:
CTA.
Prove:
21)
22)
Given:
SIT
AND.
SIT
AND.
Given:
__ __ __ __
TA NA, BA MA
Prove:
BAT
MAN.
Prove:
Fill in the reasons in the proof:
23)
24)
Given:
Given:
__ __ __ __ __ __
IT ER, LT TR, IT // ER
Prove:
__ __
LI // ET
.
Statement
__ __
IT ER
R
TER
ETR
L and ETR are
corresponding interior
angles
__ __
LI // ET
__ __
SI AN
A
LTI and R are
corresponding
L
Reason
SIT
__ __
IT // ER
LIT
__ __
ST AD
__ __
IT DN
__ __
LT TR
LTI
Prove:
A
Statement
AND
S
S.
Reason
25)
26)
Given:
Given:
__ __
BA MA,
Prove:
__ __
IT ND
.
Statement
__ __
SI AN
A
I
N
__ __
IT ND
M
Prove:
__ __
MN BT
.
Statement
__ __
BA MA
S
SIT
Reason
B
B
AND
M
BAT and MAN are
vertical angles
BAT
BAT
__ __
MN BT
MAN
MAN
Reason
EasyWorksheet Step By Step Answers
User Name: J. Joseph
Form #112511235638
Practice Problems
1) Given: YQL
GDP, write a congruence statement for corresponding
angles.
Don't forget, since the order of the congruence explains the order of the corresponding
pieces, that we can substitute in G in for Y everywhere, D in for Q everywhere, and P
in for L.
All we have to do is list out the angles:
Y
G, Q
D, L
P
2) Given T
N, Q
R, WT = NS, and the fact that the triangles are
congruent, write the triangle congruence statement.
All we have to do is write the letters in the same order for the first two pieces. Since
the remaining side gives us the last set of letters, we get:
TQW
NRS
3) Given F
G, S
N, L
K, and the fact that the triangles are
congruent, write the triangle congruence statement.
All we have to do is write the letters in the same order:
FSL
GNK
4) Given AT = UG, TR = UJ, and the fact that the triangles are congruent, write
the triangle congruence statement.
First, we need to figure out which angles are the same.
In this case, both statements have T and U in common, so those must be the same
angle.
All we have to do is write the first letters in order: T __ __
U __ __
Next, we'll use the fact that AT = UG to fill in the second spot: TA __
UG __
And finally, we can fill in the last piece of information:
TAR
UGJ
5) Given T
S, BT = WS, YT = SP, and the fact that the triangles are
congruent, write the triangle congruence statement.
All we have to do is write the first letters in order: T __ __
Next, we'll use the fact that BT = WS to fill in the second spot:
And finally, we can fill in the last piece of information:
TBY
SWP
S __ __
TB __
SW __
6) Given: YVP
NSC, write a congruence statement for corresponding sides.
Don't forget, since the order of the congruence explains the order of the corresponding
pieces, that we can substitute in N in for Y everywhere, S in for V everywhere, and C
in for P.
All we have to do is list out the sides:
__ __ __ __ __ __
YV NS, VP SC, PY CN
7) Given:
__ __ __ __
GA AP, BG PR,
GAP
PAR
Since GAP and PAR are a linear pair, they are supplementary.
Because they are also congruent, they must each be a right angle!
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or HL (HL is only valid in the case of right
triangles!)
Since the angle does not touch both segments, we would use HL (Hypotenuse-Leg).
And we can write the congruence as BAG
RAP.
8) Given:
__ __ __ __
TA NA, BA MA
First, notice that BAT and MAN are vertical angles, so they are congruent!
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as BAT
MAN.
9) Given:
In this case, we have two angles congruent. That's not enough information.
(Remember, we need a side!)
So the triangles are not necessarily congruent.
10) Given:
__ __
WP GF
First, since l // m, we have two sets of alternate interior angles that are congruent:
OPW
OFG and OWP
OGF
And, because they are vertical angles, POW
FOG
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
In this case, we know all three angles are congruent, so either theorem would work!
And we can write the congruence as POW
FOG.
11) Given:
Watch out! In this case the only thing the triangles have congruent to each other is one
side length! That's not enough information.
So the triangles are not necessarily congruent.
12) Given:
__ __ __ __
AC OD, CT GD,
C
D
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as CAT
DOG.
13) Given:
P
D, PGI
DGI
Prove:
PIG
DIG.
Statement
P
PGI
Reason
D
DGI
__ __
IG IG
PIG
DIG
__
__
First, notice that IG
IG by the Reflexive Property.
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show
the two triangles are congruent.
And we can write the congruence as PIG
DIG.
So we have the following proof when we are done:
Statement
P
Reason
Given
D
DGI Given
PGI
__ __
IG IG
Reflexive Property
DIG AAS
PIG
14) Given:
__ __
JA RI,
J
Prove:
R
JAM
RIM.
Statement
Reason
__ __
JA RI
J
R
RMI and
AMJ are vertical angles
RMI
AMJ
JAM
RIM
First, notice that AMJ and RMI are vertical angles, so they are congruent!
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show
the two triangles are congruent.
And we can write the congruence as JAM
RIM.
So we have the following proof when we are done:
Statement
__ __
JA RI
J
Reason
Given
Given
R
RMI and
AMJ are vertical angles Given
RMI
AMJ
Vertical Angle Theorem
JAM
RIM
AAS
15) Given:
Prove:
SIT
Statement
__ __
SI AN
S
A
T
D
SIT
AND.
Reason
AND
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show
the two triangles are congruent.
And we can write the congruence as SIT
AND.
So we have the following proof when we are done:
Statement
__ __
SI AN
Reason
Given
S
A
Given
T
D
Given
AND AAS
SIT
16) Given:
__ __ __ __
CO PI , OW GI,
Prove:
COW
Statement
O
I
PIG.
Reason
__ __
CO PI
__ __
OW GI
O
COW
I
PIG
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as COW
PIG.
So we have the following proof when we are done:
Statement
Reason
__ __
CO PI
Given
__ __
OW GI
Given
O
Given
I
PIG SAS
COW
17) Given:
RFP
Prove:
LFP,
PLF
Statement
RFP
LFP
RPF
LPF
RPF
PRF.
LPF
Reason
__ __
PF PF
PLF
PRF
First, notice that we must be using the smallest triangles -- since we are given sides
and/or angles that don't appear in the larger triangles.
__
First, notice that PF
__
PF by the Reflexive Property.
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent.
And we can write the congruence as PLF
PRF.
So we have the following proof when we are done:
Statement
RFP
Reason
LFP Given
LPF Given
RPF
__ __
PF PF
Reflexive Property
PRF ASA
PLF
18) Given:
__ __ __ __
NM NT, AN EN
Prove:
MEN
Statement
__ __
NM NT
TAN.
Reason
__ __
AN EN
N
N
MEN
TAN
First, notice that we must be using the larger triangles -- since we are given sides
and/or angles that don't appear in the smaller triangle.
Also, N
N by the Reflexive Property.
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as MEN
TAN.
So we have the following proof when we are done:
Statement
__ __
NM NT
Reason
Given
__ __
AN EN
N
Given
Reflexive Property
N
TAN SAS
MEN
19) Given:
SEAT is a square.
U
C
Prove:
USE
CTA.
First, since SEAT is a square, we know that ESU
angles).
ATC (they are both right
__ __
And we know that SE AT.
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show
the two triangles are congruent.
And we can write the congruence as USE
CTA.
So we have the following proof when we are done:
Statement
U
Given
C
SEAT is a square
ESU and
ESU
20) Given:
Given
ATC are right angles Definition of a Square
ATC
__ __
SE AT.
USE
Reason
All Right Angles are Congruent
Definition of a Square
CTA
AAS
Prove:
SIT
AND.
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side doesn't touching both angles, we use AAS (Angle-Angle-Side) to show
the two triangles are congruent.
And we can write the congruence as SIT
AND.
So we have the following proof when we are done:
Statement
__ __
SI AN
Reason
Given
S
A
Given
T
D
Given
SIT
AND AAS
21) Given:
__ __ __ __
TA NA, BA MA
Prove:
BAT
MAN.
First, notice that BAT and MAN are vertical angles, so they are congruent!
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as BAT
MAN.
So we have the following proof when we are done:
Statement
Reason
__ __
TA NA
Given
__ __
BA MA
Given
BAT and
MAN are vertical angles Given
BAT
MAN
Vertical Angle Theorem
BAT
MAN
SAS
22) Given:
Prove:
SIT
AND.
In this case, all sides of one triangle are congruent to all sides of the other triangle, so
we use SSS (Side-Side-Side) to show the two triangles are congruent.
And we can write the congruence as SIT
AND.
So we have the following proof when we are done:
Statement
__ __
ST AD
Reason
Given
__ __
SI AN
Given
__ __
IT DN
Given
SIT
AND SSS
23) Given:
__ __ __ __ __ __
IT ER, LT TR, IT // ER
Prove:
__ __
LI // ET
.
Statement
Reason
__ __
IT ER
__ __
LT TR
__ __
IT // ER
LTI and
R are corresponding
LTI
R
LIT
TER
L
ETR
L and
ETR are corresponding interior angles
__ __
LI // ET
__ __
First, notice we have IT //ER, so we have corresponding angles:
So LTI
R.
In this case, we have two sides congruent and one angle. This means we are either
going to use SAS (which IS a theorem) or ASS (which is NOT a theorem.)
Since the angle touching both segments is included, we use SAS (Side-Angle-Side) to
show the two triangles are congruent.
And we can write the congruence as LIT
TER.
So we have the following proof when we are done:
Statement
Reason
__ __
IT ER
Given
__ __
LT TR
Given
__ __
IT // ER
Given
LTI and
R are corresponding
Given
LTI
R
Corresponding Angles Theorem
LIT
TER
SAS
L
L and
CPCTC
ETR
ETR are corresponding interior angles Given
__ __
LI // ET
Corresponding Angle Converse
24) Given:
Prove:
A
Statement
__ __
ST AD
__ __
SI AN
__ __
IT DN
SIT
AND
S.
Reason
A
S
In this case, all sides of one triangle are congruent to all sides of the other triangle, so
we use SSS (Side-Side-Side) to show the two triangles are congruent.
And we can write the congruence as SIT
AND.
So we have the following proof when we are done:
Statement
Reason
__ __
ST AD
Given
__ __
SI AN
Given
__ __
IT DN
Given
AND SSS
SIT
A
CPCTC
S
25) Given:
Prove:
__ __
IT ND
.
Statement
__ __
SI AN
S
A
I
N
SIT
AND
Reason
__ __
IT ND
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent.
And we can write the congruence as SIT
AND.
So we have the following proof when we are done:
Statement
Reason
__ __
SI AN
Given
S
A
Given
I
N
Given
AND ASA
SIT
__ __
IT ND
CPCTC
26) Given:
__ __
BA MA,
B
M
Prove:
__ __
MN BT
.
Statement
__ __
BA MA
B
M
BAT and
BAT
MAN are vertical angles
MAN
Reason
BAT
MAN
__ __
MN BT
First, notice that BAT and MAN are vertical angles, so they are congruent!
In this case, we have two angles congruent and one side. This means we are either
going to use AAS or ASA.
Since the side is touching both angles (it's the included side!), we use ASA (AngleSide-Angle) to show the two triangles are congruent.
And we can write the congruence as BAT
MAN.
So we have the following proof when we are done:
Statement
__ __
BA MA
Reason
Given
M
Given
BAT and
MAN are vertical angles Given
B
BAT
MAN
Vertical Angle Theorem
BAT
MAN
ASA
__ __
MN BT
CPCTC
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