Download Ways to prove Triangles Congruant

Ways to prove
Triangles Congruent
(SSS), (SAS), (ASA)
4-2 to 4-4
EXAMPLE 4
Find m
Use the Third Angles Theorem
BDC.
SOLUTION
A
B and ADC
BCD, so by the Third
Angles Theorem, ACD
BDC. By the Triangle
Sum Theorem, m ACD = 180° – 45° – 30° = 105° .
ANSWER
So, m ACD = m BDC = 105° by the definition of
congruent angles.
EXAMPLE 5
Prove that triangles are congruent
Write a proof.
GIVEN
AD
CB, DC
ACD
PROVE
AB
CAB,
ACD
CAD
ACB
CAB
Plan for Proof
a. Use the Reflexive Property to show that
AC AC.
b. Use the Third Angles Theorem to show that
B
D
EXAMPLE 5
Prove that triangles are congruent
Plan in Action
STATEMENTS
REASONS
1.
AD
CB, DC
a. 2.
AC
AC.
BA
1. Given
2. Reflexive Property of
Congruence
3.
b. 4.
5.
ACD
CAD
B
ACD
CAB,
ACB
D
3. Given
4. Third Angles Theorem
CAB
5. Definition of
GUIDED PRACTICE
for Examples 4 and 5
4. In the diagram, what is m
DCN.
SOLUTION
CDN
NSR, DNC
SNR then the third
angles are also congruent NRS
DCN = 75°
GUIDED PRACTICE
for Examples 4 and 5
5. By the definition of congruence, what
additional information is needed to
know that
NDC
NSR.
SOLUTION
Given : CN
NR,
DCN
Proved :
NDC
CDN
NRS
NSR.
NSR, (Proved from above
sum)
GUIDED PRACTICE
for Examples 4 and 5
STATEMENT
REASON
CDN
NSR
Given
DCN
NRS
Given
Therefore DC RS, DN SN as angles are congruent
their sides are congruent.
EXAMPLE 1
Identify congruent parts
Write a congruence statement for the
triangles. Identify all pairs of congruent
corresponding parts.
SOLUTION
The diagram indicates that
Corresponding angles
J
Corresponding sides JK
JKL
T,  K
TS, KL
TSR.
S,
SR, LJ
L
R
RT
EXAMPLE 2
Use properties of congruent figures
In the diagram, DEFG
SPQR.
a.
Find the value of x.
b.
Find the value of y.
SOLUTION
a.
You know that FG
FG = QR
12 = 2x – 4
16 = 2x
8=x
QR.
EXAMPLE 2
b.
Use properties of congruent figures
You know that  F
m
F=m
Q
68o = (6y + x)o
68 = 6y + 8
10 = y
Q.
EXAMPLE 3
Show that figures are congruent
PAINTING
If you divide the wall into
orange and blue sections
along JK , will the sections
of the wall be the same size
and shape?Explain.
SOLUTION
From the diagram, A
C and D
B because all
right angles are congruent. Also, by the Lines
Perpendicular to a Transversal Theorem, AB DC .
EXAMPLE 3
Show that figures are congruent
Then, 1
4 and 2
3 by the Alternate Interior
Angles Theorem. So, all pairs of corresponding angles
are congruent.
The diagram shows AJ CK , KD
JB , and DA BC .
By the Reflexive Property, JK KJ . All corresponding
parts are congruent, so AJKD
CKJB.
GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
CDEF.
1. Identify all pairs of congruent
corresponding parts.
SOLUTION
Corresponding sides:
Corresponding angles:
AB CD, BG DE,
GH FE, HA FC
A
G
C, B
E, H
D,
F.
GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
2. Find the value of x and find m
SOLUTION
(a)
You know that
(4x+ 5)° = 105°
4x = 100
x = 25
(b)
You know that H
m H m F =105°
H
F
F
CDEF.
H.
GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
3. Show that
PTS
CDEF.
RTQ.
SOLUTION
In the given diagram
PS
QR, PT TR, ST TQ and
Similarly all angles are to each other, therefore all
of the corresponding points of PTS are congruent to
those of RTQ by the indicated markings, the Vertical
Angle Theorem and the Alternate Interior Angle
theorem.
EXAMPLE 1
Use the SSS Congruence Postulate
Write a proof.
GIVEN
PROVE
Proof
KL
NL, KM
KLM
NM
NLM
It is given that KL
NL and KM
By the Reflexive Property, LM
So, by the SSS Congruence
Postulate,
KLM
NLM
NM
LN.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
1.
DFG
HJK
SOLUTION
Three sides of one triangle are congruent to three
sides of second triangle then the two triangle are
congruent.
Side DG
HK, Side DF
JH,and Side FG JK.
So by the SSS Congruence postulate,
Yes. The statement is true.
DFG
HJK.
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
2.
ACB
CAD
SOLUTION
GIVEN : BC
PROVE :
PROOF:
AD
ACB
CAD
It is given that BC AD By Reflexive property
AC AC, But AB is not congruent CD.
GUIDED PRACTICE
for Example 1
Therefore the given statement is false and
ABC is not
Congruent to CAD because corresponding sides
are not congruent
GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true.
Explain your reasoning.
3.
QPT
RST
SOLUTION
GIVEN : QT TR , PQ
SR, PT TS
PROVE :
RST
PROOF:
QPT
It is given that QT TR, PQ SR, PT TS. So by
SSS congruence postulate, QPT
RST.
Yes the statement is true
EXAMPLE 1
Use the SAS Congruence Postulate
Write a proof.
GIVEN
BC
DA, BC AD
ABC
PROVE
CDA
STATEMENTS
S
REASONS
1.
BC
DA
1. Given
2.
BC
AD
2. Given
A 3.
S 4.
BCA
AC
DAC
CA
3. Alternate Interior
Angles Theorem
4. Reflexive Property of
Congruence
EXAMPLE 1
Use the SAS Congruence Postulate
STATEMENTS
5.
ABC
CDA
REASONS
5. SAS Congruence
Postulate
EXAMPLE 2
Use SAS and properties of shapes
In the diagram, QS and RP pass through
the center M of the circle. What can you
conclude about
MRS and
MPQ?
SOLUTION
Because they are vertical angles, PMQ
RMS. All
points on a circle are the same distance from the center,
so MP, MQ, MR, and MS are all equal.
ANSWER
MRS and
MPQ are congruent by the SAS
Congruence Postulate.
for Examples 1 and 2
GUIDED PRACTICE
In the diagram, ABCD is a square with four
congruent sides and four right angles. R,
S, T, and U are the midpoints of the sides
VU .
of ABCD. Also, RT SU and SU
1.
Prove that
SVR
UVR
STATEMENTS
REASONS
1.
1. Given
2.
3.
4.
SV
VU
SVR
RV
RVU
VR
SVR
UVR
2. Definition of
line
3. Reflexive Property of
Congruence
4. SAS Congruence
Postulate
GUIDED PRACTICE
2.
Prove that
for Examples 1 and 2
BSR
DUT
STATEMENTS
REASONS
1.
1. Given
2.
BS
RBS
3. RS
4.
DU
BSR
TDU
2. Definition of
line
3. Given
UT
DUT
4. SAS Congruence
Postulate
EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
Write a proof.
GIVEN
PROVE
WY
XZ, WZ ZY, XY ZY
WYZ
XZY
SOLUTION
Redraw the triangles so they are
side by side with corresponding
parts in the same position. Mark
the given information in the
diagram.
EXAMPLE 3
Use the Hypotenuse-Leg Congruence Theorem
STATEMENTS
H 1.
WY
4.
ZY
2. Given
3. Definition of
Z and Y are
lines
right angles
WYZ and XZY are 4. Definition of a right
triangle
right triangles.
L 5. ZY
6.
1. Given
XZ
2. WZ ZY, XY
3.
REASONS
WYZ
YZ
5. Reflexive Property of
Congruence
XZY
6. HL Congruence
Theorem
EXAMPLE 4
Choose a postulate or theorem
Sign Making
You are making a canvas sign to hang on the triangular
wall over the door to the barn shown in the picture. You
think you can use two identical triangular sheets of
canvas. You know that RP QS and PQ
PS . What
postulate or theorem can you use to conclude that
PQR
PSR?
EXAMPLE 4
Choose a postulate or theorem
SOLUTION
You are given that PQ PS . By the Reflexive Property, RP
RP . By the definition of perpendicular lines, both
RPQ and RPS are right angles, so they are congruent.
So, two sides and their included angle are congruent.
ANSWER
You can use the SAS Congruence Postulate to conclude
PQR
PSR
that
.
GUIDED PRACTICE
for Examples 3 and 4
Use the diagram at the right.
3.
Redraw
ACB and
DBC side by
side with corresponding parts in the
same position.
GUIDED PRACTICE
for Examples 3 and 4
Use the diagram at the right.
4.
Use the information in the diagram to
ACB
DBC
prove that
STATEMENTS
H 1.
AB
2. AC
DC
BC, DB BC
B
REASONS
1. Given
2. Given
3. Definition of
lines
3.
C
4.
ACB and DBC are 4. Definition of a right
triangle
right triangles.
GUIDED PRACTICE
for Examples 3 and 4
STATEMENTS
REASONS
L 5. BC
5. Reflexive Property of
Congruence
6.
ACB
CB
DBC
6. HL Congruence
Theorem