Transcript 4.5 Using Congruent Triangles Geometry Mrs. Spitz Fall 2004 Objectives: Use congruent triangles to plan and write proofs. Use congruent triangles to prove constructions are valid.

4.5 Using Congruent
Triangles
Geometry
Mrs. Spitz
Fall 2004
Objectives:
Use congruent triangles to plan and write
proofs.
Use congruent triangles to prove
constructions are valid.
Assignment:
Pgs. 232-234 #1-21 all
Planning a proof
Knowing that all pairs of corresponding
parts of congruent triangles are congruent
can help you reach conclusions about
congruent figures.
Planning a proof
For example, suppose you want
to prove that PQS ≅ RQS
in the diagram shown at the
right. One way to do this is
to show that ∆PQS ≅ ∆RQS
by the SSS Congruence
Postulate. Then you can
use the fact that
corresponding parts of
congruent triangles are
congruent to conclude that
PQS ≅ RQS.
Q
R
P
S
Ex. 1: Planning & Writing a Proof
Given: AB ║ CD, BC ║
DA
Prove: AB≅CD
Plan for proof: Show
that ∆ABD ≅ ∆CDB.
Then use the fact that
corresponding parts of
congruent triangles are
congruent.
B
A
C
D
Ex. 1: Planning & Writing a Proof
Solution: First copy the
diagram and mark it with the
given information. Then mark
any additional information you
can deduce. Because AB and
CD are parallel segments
intersected by a transversal, and
BC and DA are parallel
segments intersected by a
transversal, you can deduce that
two pairs of alternate interior
angles are congruent.
B
A
C
D
Ex. 1: Paragraph Proof
Because AD ║CD, it follows
from the Alternate Interior
Angles Theorem that ABD
≅CDB. For the same
reason, ADB ≅CBD
because BC║DA. By the
Reflexive property of
Congruence, BD ≅ BD. You
can use the ASA
Congruence Postulate to
conclude that ∆ABD ≅
∆CDB. Finally because
corresponding parts of
congruent triangles are
congruent, it follows that AB
≅ CD.
B
A
C
D
Ex. 2: Planning & Writing a Proof
Given: A is the midpoint of
MT, A is the midpoint of SR.
Prove: MS ║TR.
Plan for proof: Prove that
∆MAS ≅ ∆TAR. Then use
the fact that corresponding
parts of congruent triangles
are congruent to show that
M ≅ T. Because these
angles are formed by two
segments intersected by a
transversal, you can
conclude that MS ║ TR.
M
R
A
S
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
A
Statements:
Reasons:
1.
1.
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
R
Given
S
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
Statements:
Reasons:
1.
1.
Given
2.
Definition of a midpoint
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
Statements:
Reasons:
1.
1.
Given
2.
3.
Definition of a midpoint
Vertical Angles Theorem
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
5.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.
R
A
S
T
Statements:
Reasons:
1.
1.
Given
2.
3.
4.
5.
6.
Definition of a midpoint
Vertical Angles Theorem
SAS Congruence Postulate
Corres. parts of ≅ ∆’s are ≅
Alternate Interior Angles
Converse.
2.
3.
4.
5.
6.
A is the midpoint of MT, A
is the midpoint of SR.
MA ≅ TA, SA ≅ RA
MAS ≅ TAR
∆MAS ≅ ∆TAR
M ≅ T
MS ║ TR
Ex. 3: Using more than one pair of
triangles.
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
Plan for proof: The only
information you have about
∆BCE and ∆DCE is that 1≅2
and that CE ≅CE. Notice,
however, that sides BC and DC
are also sides of ∆ABC and
∆ADC. If you can prove that
∆ABC≅∆ADC, you can use the
fact that corresponding parts of
congruent triangles are
congruent to get a third piece of
information about ∆BCE and
∆DCE.
D
C
2
1
4
E
B
3
A
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
B
Statements:
Reasons:
1.
2.
3.
4.
5.
6.
1. Given
1≅2, 3≅4
AC ≅ AC
∆ABC ≅ ∆ADC
BC ≅ DC
CE ≅ CE
∆BCE≅∆DCE
4
3
A
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3.
4.
5.
6.
∆ABC ≅ ∆ADC
BC ≅ DC
CE ≅ CE
∆BCE≅∆DCE
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
C
2
1
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
Corres. parts of ≅ ∆’s are ≅
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
5.
Corres. parts of ≅ ∆’s are ≅
Reflexive Property of
Congruence
Given: 1≅2, 3≅4.
Prove ∆BCE≅∆DCE
2
1
C
E
4
3
A
B
Statements:
Reasons:
1. 1≅2, 3≅4
2. AC ≅ AC
1. Given
2. Reflexive property of
Congruence
3. ASA Congruence
Postulate
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC
5. CE ≅ CE
6. ∆BCE≅∆DCE
4.
5.
6.
Corres. parts of ≅ ∆’s are ≅
Reflexive Property of
Congruence
SAS Congruence Postulate
Ex. 4: Proving constructions are
valid
In Lesson 3.5 – you learned to copy an
angle using a compass and a straight edge.
The construction is summarized on pg. 159
and on pg. 231.
Using the construction summarized above,
you can copy CAB to form FDE. Write
a proof to verify the construction is valid.
Plan for proof
Show that ∆CAB ≅ ∆FDE.
Then use the fact that
corresponding parts of
congruent triangles are
congruent to conclude that
CAB ≅ FDE. By
construction, you can
assume the following
statements:
– AB ≅ DE Same compass
setting is used
– AC ≅ DF Same compass
setting is used
– BC ≅ EF Same compass
setting is used
C
A
B
F
D
E
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
4
3
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
4
3
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
4
3
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
4.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
SSS Congruence Post
C
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF
Prove CAB≅FDE
2
1
A
B
4
3
F
D
E
Statements:
Reasons:
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
AB ≅ DE
AC ≅ DF
BC ≅ EF
∆CAB ≅ ∆FDE
CAB ≅ FDE
Given
Given
Given
SSS Congruence Post
Corres. parts of ≅ ∆’s
are ≅.
Q
Given: QSRP, PT≅RT
Prove PS≅ RS
2
1
4
3
P
T
Statements:
Reasons:
1. QS  RP
2. PT ≅ RT
1. Given
2. Given
S
R