Concepts, Theorems and Postulates that can be use to prove that triangles are congruent.
-
Upload
victor-oliver -
Category
Documents
-
view
220 -
download
0
Transcript of 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.
Goal 1
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.
Learning Target
Goal 1
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.
Corresponding Angles Corresponding Sides
A P B Q C R
BC QR
RPCA
AB PQ
For the triangles below, you can write , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence.
ABC PQR
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
Learning Target
Example
Naming Congruent Parts
The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts.
SOLUTION
Angles:
Sides:
D R, E S, F T
, , RS DE TRFD ST EF
The diagram indicates that .
The congruent angles and sides are as follows.
DEF RST
Example
Using Properties of Congruent Figures
In the diagram, NPLM EFGH.
Find the value of x.
SOLUTION
You know that .GHLM
So, LM = GH.
8 = 2 x – 3
11 = 2 x
5.5 = x
Example
Using Properties of Congruent Figures
In the diagram, NPLM EFGH.
Find the value of x.
SOLUTION
You know that .GHLM
So, LM = GH.
8 = 2 x – 3
11 = 2 x
5.5 = x
Find the value of y.
You know that N E.
So, m N = m E.
72˚ = (7y + 9)˚
63 = 7y
9 = y
SOLUTION
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.
TRANSITIVE If A B and B C, then A C
SYMMETRIC If A B, then B A
REFLEX IVE For any angle A, A A
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, PROVE A C
A
B
C
B C
Transitive Property of Angle Congruence
GIVEN A B,
B C
PROVE A C
Statements Reasons
1
2
3
4
m A = m B Definition of congruent angles
5 A C Definition of congruent angles
A B, Given
B C
m B = m C Definition of congruent angles
m A = m C Transitive property of equality
Using the Transitive Property
This two-column proof uses the Transitive Property.
Statements Reasons
2
3
4
m 1 = m 3 Definition of congruent angles
GIVEN m 3 = 40°, 1 2, 2 3
PROVE m 1 = 40°
1
m 1 = 40° Substitution property of equality
1 3 Transitive property of Congruence
Givenm 3 = 40°, 1 2,
2 3
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 2 are right angles
PROVE 1 2
Proving Right Angle Congruence Theorem
Statements Reasons
1
2
3
4
m 1 = 90°, m 2 = 90° Definition of right angles
m 1 = m 2 Transitive property of equality
1 2 Definition of congruent angles
GIVEN 1 and 2 are right angles
PROVE 1 2
1 and 2 are right angles Given
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.
1 233
If m 1 + m 2 = 180°
m 2 + m 3 = 180°
and
1
then
1 3
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.
45
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
If m 4 + m 5 = 90°
m 5 + m 6 = 90°
and
then
4 6
566
4
Proving Congruent Supplements Theorem
Statements Reasons
1
2
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
1 and 2 are supplements Given
3 and 4 are supplements
1 4
m 1 + m 2 = 180° Definition of supplementary anglesm 3 + m 4 = 180°
Proving Congruent Supplements Theorem
Statements Reasons
3
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
4
5 m 1 + m 2 = Substitution property of equalitym 3 + m 1
m 1 + m 2 = Transitive property of equalitym 3 + m 4
m 1 = m 4 Definition of congruent angles
Proving Congruent Supplements Theorem
Statements Reasons
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
6
7
m 2 = m 3 Subtraction property of equality
2 3 Definition of congruent angles
POSTULATE
Linear Pair Postulate
If two angles for m a linear pair, then they are supplementary.
m 1 + m 2 = 180°
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Proving Vertical Angle Theorem
THEOREM
Vertical Angles Theorem
Vertical angles are congruent
1 3, 2 4
Proving Vertical Angle Theorem
PROVE 5 7
GIVEN 5 and 6 are a linear pair,
6 and 7 are a linear pair
1
2
3
Statements Reasons
5 and 6 are a linear pair, Given6 and 7 are a linear pair
5 and 6 are supplementary, Linear Pair Postulate6 and 7 are supplementary
5 7 Congruent Supplements Theorem
Goal 1
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.
Example
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)˚
30 = 2 x
15 = x
Third Angles Theorem
Substitute.
Subtract 30 from each side.
Divide each side by 2.
Goal 2
SOLUTION
Paragraph Proof
From the diagram, you are given that all three corresponding sides are congruent.
, NQPQ ,MNRP QMQR and
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.
Decide whether the triangles are congruent. Justify your reasoning.
So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, .PQR NQM
Proving Triangles are CongruentLearning Target
Example
Proving Two Triangles are Congruent
A B
C D
E
|| , DCAB ,
DCAB E is the midpoint of BC and AD.
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.
GIVEN
PROVE .AEB DEC
Prove that .AEB DEC
Example
Proving Two Triangles are Congruent
Statements Reasons
EAB EDC, ABE DCE
AEB DEC
E is the midpoint of AD,E is the midpoint of BC
,DEAE CEBE
Given
Alternate Interior Angles Theorem
Vertical Angles Theorem
Given
Definition of congruent triangles
Definition of midpoint
|| ,DCAB DCAB
SOLUTION
AEB DEC
A B
C D
E
Prove that .AEB DEC
Goal 2
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
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
D
E
F
A
B
C
J K
L
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles
Transitive Property of Congruent Triangles
If , then .ABC DEF DEF ABC
If and , then .JKLABC DEF DEF ABC JKL
Proving Triangles are Congruent
SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) arecongruent, then the triangles are congruent.
and thenIfSides are congruent
1. AB DE
2. BC EF
3. AC DF
Angles are congruent
4. A D
5. B E
6. C F
Triangles are congruent
ABC DEF
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE: Side - Side - Side (SSS) Congruence Postulate
Side MN QR
Side PM SQ
Side NP RS
If
If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.
then MNP QRS
S
S
S
Using the SSS Congruence Postulate
Prove that PQW TSW.
Paragraph Proof
SOLUTION
So by the SSS Congruence Postulate, you
know that
PQW TSW.
The marks on the diagram show that PQ
TS,
PW TW, and QW SW.
POSTULATE
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE: Side-Angle-Side (SAS) Congruence Postulate
Side PQ WX
Side QS XY
then PQS WXYAngle Q X
If
If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
A
S
S
1
Using the SAS Congruence Postulate
Prove that AEB DEC.
2
3 AEB DEC SAS Congruence Postulate
21
AE DE, BE CE Given
1 2 Vertical Angles Theorem
Statements Reasons
D
GA R
Proving Triangles Congruent
MODELING A REAL-LIFE SITUATION
PROVE DRA DRG
SOLUTION
ARCHITECTURE You are designing the window shown in the drawing. Youwant to make DRA congruent to DRG. You design the window so that DR AG and RA RG.Can you conclude that DRA DRG ?
GIVEN DR AG
RA RG
2
3
4
5
6 SAS Congruence Postulate DRA DRG
1
Proving Triangles Congruent
GivenDR AG
If 2 lines are , then they form 4 right angles.
DRA and DRGare right angles.
Right Angle Congruence Theorem DRA DRG
GivenRA RG
Reflexive Property of CongruenceDR DR
Statements Reasons
D
GA R
GIVEN
PROVE DRA DRG
DR AG
RA RG
Congruent Triangles in a Coordinate Plane
AC FH
AB FGAB = 5 and FG = 5
SOLUTION
Use the SSS Congruence Postulate to show that ABC
FGH.
AC = 3 and FH = 3
Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 3 2 + 5
2
= 34
BC = (– 4 – (– 7)) 2 + (5 – 0 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 5 2 + 3
2
= 34
GH = (6 – 1) 2 + (5 – 2 )
2
Use the distance formula to find lengths BC and GH.