Download 4.2 Apply Congruence and Triangles

4.2 Apply Congruence and
Triangles
Hubarth
Geometry
Corresponding Parts- are angles and sides that are in the same position of their
respected shape.
Congruent- same size and shape
Congruent
Same size and shape
Not Congruent
Different sizes and shapes
Congruence Statements
When you write a congruence statement for two
polygons, always list the corresponding vertices
in the same order. You can write congruence
statements in more than one way. Two possible
congruence statements for the triangles at the
right. ABC  FDE or BCA  DEF
CorrepondingAngles : A  F , B  D, C  E
Correponding Sides : AB  FD, BC  DE , AC  FE
F
A
B
C
D
E
Ex 1 Identifying Congruent Parts
Write a congruence statement for the triangles. Identify all pairs
of congruent corresponding parts.
JKL  TSR
J  T , K  S , L  R
JK  TS , KL  SR, LJ  RT
Ex 2 Use Properties of Congruent Figures
In the diagram, DEFG
SPQR.
a. Find the value of x.
b. Find the value of y.
a. You know that FG
FG
= QR
QR.
b. You know that
m
F =m
F
Q.
Q
12 = 2x – 4
68 = (6y + x)
16 = 2x
68 = 6y + 8
8 =x
10 = y
Ex 3 Show that Figures are Congruent
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.
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 .
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
All corresponding parts are congruent, so AJKD
BC . By the Reflexive Property, JK
CKJB.
KJ
E
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.
B
A
F
D
C
If A  D and B  E, then C  F
EX 4 Use the Third Angles Theorem
Find mBDC
A  B and ADC  BCD, so by the Third Angles Theorem,
ACD  BDC.
By the Triangle Sum Theorem, mACD  180 - 45 - 30  105
So, mACD  mBDC  105
Ex 5 Prove that Triangles are Congruent
Write a proof.
Given : AD  CB, DC  AB, ACB  CAB, CAD  ACB
Prove : ACD  CAB
Plan for Proof
a. Use the Reflexive Property to show AC  AC
b. Use the Third Angles Theorem to show that B  D.
STATEMENTS
REASONS
1. AD  CB, DC  BA
1. Given
2. AC  AC
2. Reflexive Property of
Congruence
3. ACD  CAB
CAD  ACB
3. Given
4. B  D
4. Third Angles Theorem
5. ACD  CAB
5. Def. of  
Practice
In the diagram at the right, ABGH
CDEF.
1. Identify all pairs of congruent corresponding
parts.
AB  CD, AH  FC , GH  EF , BG  DE
A  C , H  F , G  E , B  D
2. Find the value of x and find m
H.
H  105
You know that H
F
(4x+ 5)° = 105°
4x = 100
x = 25
3. Show that PTS  RTQ
All of the corresponding parts of PTS are congruent to those of RTQ by the
indicated markings, the Vertical Angle Theorem and the Alternat Interior Angle
Theorem.
4. In the diagram, what is mDCN.
DCN  75
5. By the definition of congruence, what additional information is needed to know
that DCN  SRN
DC  RS and DN  SN