Download Document

Concepts, Theorems
and Postulates that can be
use to prove that triangles
are congruent.
Learning Target
• I can identify and use reflexive, symmetric and
transitive property in proving two triangles are
congruent.
• I can use theorems about line and angles in my
proof.
Corresponding Angles
and
Corresponding Sides
Learning
Target
Identifying Congruent Figures
Two geometric figures are congruent if they have exactly the same size and
shape.
Each of the red figures is congruent to the other red figures.
None of the blue figures is congruent to another blue figure.
Goal 1
Learning
Target
Identifying Congruent Figures
When two figures are congruent, there is a correspondence between their
angles and sides such that corresponding angles are congruent and
corresponding sides are congruent.
For the triangles below, you can write ABC  PQR , which reads “triangle
ABC is congruent to triangle PQR.” The notation shows the congruence and the
correspondence.
Corresponding Angles
Corresponding Sides
AP
BQ
CR
AB  PQ
BC  QR
CA  RP
There is more than one way to write a congruence statement, but it is important
to list the corresponding angles in the same order. For example, you can also
write BCA  QRP .
Goal 1
Naming Congruent Parts
The two triangles shown below are congruent. Write a congruence
statement. Identify all pairs of congruent corresponding parts.
SOLUTION
The diagram indicates that
DEF 
RST .
The congruent angles and sides are as follows.
Angles:  D   R,  E   S,  F   T
Sides: DE  RS , EF  ST , FD  TR
Example
Using Properties of Congruent Figures
In the diagram, NPLM  EFGH.
Find the value of x.
SOLUTION
You know that LM  GH .
So, LM = GH.
8 = 2x – 3
11 = 2 x
5.5 = x
Example
Using Properties of Congruent Figures
In the diagram, NPLM  EFGH.
Find the value of x.
Find the value of y.
SOLUTION
SOLUTION
You know that LM  GH .
You know that  N   E.
So, LM = GH.
So, m N = m E.
8 = 2x – 3
11 = 2 x
5.5 = x
72˚ = (7y + 9)˚
63 = 7y
9=y
Example
Theorems and Postulates on
Congruent Angles
(Transitive, Reflexive and Symmetry
Theorem of Angle Congruence)
CONGRUENCE OF ANGLES
THEOREM
THEOREM 2.2 Properties of Angle Congruence
Angle congruence is r ef lex ive, sy mme tric, and transitive.
Here are some examples.
REFLEX IVE
For any angle A,
SYMMETRIC If
A 
B, then
TRANSITIVE If
A 
B and
B 
C, then
A 
A
B 
A
A 
C
Transitive Property of Angle Congruence
Prove the Transitive Property of Congruence for angles.
SOLUTION
To prove the Transitive Property of Congruence for
angles, begin by drawing three congruent angles. Label
the vertices as A, B, and C.
GIVEN
A
B,
B
C
PROVE
A
C
C
B
A
Transitive Property of Angle Congruence
A
B,
B
C
GIVEN
Statements
1
A 
B,
B 
C
PROVE
A
C
Reasons
Given
2
m
A=m
B
Definition of congruent angles
3
m
B=m
C
Definition of congruent angles
4
m
A=m
C
Transitive property of equality
5
A 
C
Definition of congruent angles
Using the Transitive Property
This two-column proof uses the Transitive Property.
GIVEN
m
3 = 40°,
PROVE
m
1 = 40°
1
2,
Statements
1
m
2
1
3
Reasons
3 = 40°,
2
2
1
2,
Given
3
Transitive property of Congruence
3
3
m
1=m
4
m
1 = 40°
3
Definition of congruent angles
Substitution property of equality
Right Angle Theorem
Proving Right Angle Congruence Theorem
THEOREM
Right Angle Congruence Theorem
All right angles are congruent.
You can prove Right Angle CongruenceTheorem as shown.
GIVEN
1 and
PROVE
1
2 are right angles
2
Proving Right Angle Congruence Theorem
GIVEN
1 and
PROVE
1
2 are right angles
2
Statements
1 and
1
2 are right angles
2
m
1 = 90°, m
3
m
1=m
4
1
Reasons
2
2 = 90°
2
Given
Definition of right angles
Transitive property of equality
Definition of congruent angles
Congruent Supplements
Theorem
(Supplementary Angles)
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to
congruent angles) then they are congruent.
1
2
3
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to
congruent angles) then they are congruent.
1
2
3
If m
m
1+m
2+m
1
3
2 = 180°
3 = 180°
then
and
1
Congruent Complements
Theorem
(Complementary Angles)
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
Congruent Complements Theorem
If two angles are complementary to the same angle (or to
congruent angles) then the two angles are congruent.
5
4
6
PROPERTIES OF SPECIAL PAIRS OF ANGLES
THEOREMS
Congruent Complements Theorem
If two angles are complementary to the same angle (or to
congruent angles) then the two angles are congruent.
4
5
4
If m
m
4+m
5+m
4
6
5 = 90° and
6 = 90°
then
6
Proving Congruent Supplements Theorem
GIVEN
PROVE
1 and
2 are supplements
3 and
4 are supplements
1
4
2
3
Statements
1
1 and
2 are supplements
3 and
4 are supplements
1
2
Reasons
Given
4
m
1+m
2 = 180°
m
3+m
4 = 180°
Definition of supplementary angles
Proving Congruent Supplements Theorem
GIVEN
PROVE
Statements
1 and
2 are supplements
3 and
4 are supplements
1
4
2
3
Reasons
Transitive property of equality
m
1+m
2=
m
3+m
4
4
m
1=m
4
Definition of congruent angles
5
m
1+m
2=
Substitution property of equality
m
3+m
1
3
Proving Congruent Supplements Theorem
GIVEN
PROVE
Statements
6
7
m
2=m
2
3
1 and
2 are supplements
3 and
4 are supplements
1
4
2
3
Reasons
3
Subtraction property of equality
Definition of congruent angles
Linear Pair Postulate
PROPERTIES OF SPECIAL PAIRS OF ANGLES
POSTULATE
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
m
1+m
2 = 180°
Proving Vertical Angle Theorem
THEOREM
Vertical Angles Theorem
Vertical angles are congruent
1
3,
2
4
Proving Vertical Angle Theorem
GIVEN
PROVE
5 and
6 are a linear pair,
6 and
7 are a linear pair
5
7
Statements
5 and
6 are a linear pair,
6 and
7 are a linear pair
2
5 and
6 and
6 are supplementary,
7 are supplementary
3
5
1
7
Theorem
Reasons
Given
Linear Pair Postulate
Congruent Supplements
Third Angles Theorem
The Third Angles Theorem below follows from the Triangle Sum Theorem.
THEOREM
Third Angles Theorem
If two angles of one triangle are congruent to
two angles of another triangle, then the third
angles are also congruent.
If  A   D and  B   E, then  C   F.
Goal 1
Using the Third Angles Theorem
Find the value of x.
SOLUTION
In the diagram,  N   R and  L   S.
From the Third Angles Theorem, you know that  M   T. So, m M = m T.
From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚.
m M = m T
60˚ = (2 x + 30)˚
Third Angles Theorem
Substitute.
30 = 2 x
Subtract 30 from each side.
15 = x
Divide each side by 2.
Example
Learning
Target
Proving Triangles are Congruent
Decide whether the triangles are congruent. Justify your reasoning.
SOLUTION
Paragraph Proof
From the diagram, you are given that all three corresponding sides are congruent.
RP  MN, PQ  NQ , and QR  QM
Because  P and  N have the same measures,  P   N.
By the Vertical Angles Theorem, you know that  PQR   NQM.
By the Third Angles Theorem,  R   M.
So, all three pairs of corresponding sides and all three pairs of
corresponding angles are congruent. By the definition of congruent
triangles, PQR  NQM .
Goal 2
Proving Two Triangles are Congruent
Prove that
AEB 
DEC .
A
B
E
C
GIVEN
PROVE
D
AB || DC , AB  DC, E is the midpoint of BC and AD.
AEB 
DEC .
Plan for Proof Use the fact that  AEB and  DEC are vertical
angles to show that those angles are congruent. Use the fact that BC
intersects parallel segments AB and DC to identify other pairs of
angles that are congruent.
Example
Proving Two Triangles are Congruent
AEB 
Prove that
DEC .
A
B
E
SOLUTION
D
C
Statements
Reasons
AB || DC , AB  DC
Given
 EAB   EDC,
 ABE   DCE
Alternate Interior Angles Theorem
 AEB   DEC
Vertical Angles Theorem
E is the midpoint of AD,
E is the midpoint of BC
Given
AE  DE , BE  CE
Definition of midpoint
AEB 
DEC
Definition of congruent triangles
Example
Proving Triangles are Congruent
You have learned to prove that two triangles are congruent by the definition of
congruence – that is, by showing that all pairs of corresponding angles and
corresponding sides are congruent.
THEOREM
B
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles
If ABC  DEF , then DEF  ABC .
Transitive Property of Congruent Triangles
If ABC  DEF and DEF  JKL , then ABC 
A
C
E
D
F
L
JKL . J
K
Goal 2
SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles)
are
congruent, then the triangles are congruent.
If
Sides are
congruent
and
Angles are
congruent
1. AB
DE
4.
A
D
2. BC
EF
5.
B
E
3. AC
DF
6.
C
F
then
Triangles are
congruent
 ABC
 DEF
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE: Side - Side - Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides
of a second triangle, then the two triangles are congruent.
S
If Side
MN
QR
S
Side
NP
RS
S
Side
PM
SQ
then  MNP
 QRS
Using the SSS Congruence Postulate
Prove that
 PQW
 TSW.
SOLUTION
Paragraph Proof
The marks on the diagram show that PQ 
TS,
PW  TW, and QW  SW.
So by the SSS Congruence Postulate, you
know that
 PQW   TSW.
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE: Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, then the two triangles are congruent.
If
S
Side
A
Angle
S
Side
PQ
WX
Q
X
QS
XY
then  PQS
WXY
Using the SAS Congruence Postulate
Prove that
 AEB  DEC.
Statements
1
2
3
AE
 DE,
1
BE  CE
2
 AEB   DEC
1
2
Reasons
Given
Vertical Angles Theorem
SAS Congruence Postulate
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
ARCHITECTURE
You are designing the window shown in the drawing.
You
want to make  DRA congruent to  DRG. You design the window
so that
DR AG and RA  RG.
D
Can you conclude that  DRA   DRG ?
SOLUTION
GIVEN
PROVE
DR
AG
RA
RG
 DRA
 DRG
A
R
G
Proving Triangles Congruent
GIVEN
DR
AG
RA
RG
 DRA
PROVE
D
 DRG
A
Statements
G
Reasons
Given
1
DR
2
If 2 lines are , then they form
4 right angles.
4
DRA and
DRG
are right angles.
DRA 
DRG
RA  RG
5
DR  DR
Reflexive Property of Congruence
6
 DRA   DRG
SAS Congruence Postulate
3
AG
R
Right Angle Congruence Theorem
Given
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  ABC  
FGH.
SOLUTION
AC = 3 and FH = 3
AC  FH
AB = 5 and FG = 5
AB  FG
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths BC and
GH.
d=
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
d=
BC =
(– 4 – (– 7)) 2 + (5 – 0 ) 2
GH =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(6 – 1) 2 + (5 – 2 ) 2
=
32 + 52
=
52 + 32
=
34
=
34
Congruent Triangles in a Coordinate Plane
BC =
34 and GH =
34
BC
All three pairs of corresponding sides are congruent,
 ABC   FGH by the SSS Congruence Postulate.

GH