Congruent Figures

1
Lesson 4.1

• Congruent Figures

2
Theorem 4-1(2 angles congruent ,then 3rd angles
congruent)

• If two angles of one triangle are congruent to
  two angles of another triangle, then the third
  angles are congruent.

3
Example Naming Congruent Parts

• If ? TJD ? ? RCF List the congruent
  corresponding parts

4
Example Real-World Connection

• The fins of the Space Shuttle suggest congruent
  pentagons. Find m? B

5
Using a two-column proof, show how youcan
conclude that ? CNG ? ? DNG
?CNG ? ?DNG Is called a congruent statement
CG ? DG, CN ? DN
Given
GN ? GN
Reflexive Prop of ?
? C ? ? D
Given
Right Angles are ?
?CNG ? ?DNG
Thm 4-1
?CGN ? ?DGN
?CNG ? ?DNG
Defn of ? triangles
6
Can you conclude that D JKL _at_ D MNL? Justify
your answer.
Multiple choice
A.Yes, because all corresponding angles can be
proved congruent.
B.Yes, because all corresponding angles and sides
can be proved congruent.
C.No, because corresponding angles and sides are
not necessarily congruent.
D.No, because corresponding sides are not
necessarily congruent.
Answer D
7
You try

• Each pair of polygons is congruent. Find the
  measures of the numbered angles

1
1
110
2
2
1
110
135
2
110
120
3
4
8
Are _________ congruent??
?GJH and ?IJH?
These ?s congruent?
G
95
H
J
95
I
9
Develop a Proof
P
L
Q
N
M

• ? L ?? Q
• ? LNM ? ? PNQ
• ? M ? ? P
• LM ? QP , LN ? QN, MN ? PN
• ? LNM ? ? QNP

Given Vertical ?s Them 4-1 Given Defn of ?
triangles
10
Closure

• Suppose that two pentagons are congruent. How
  many pairs of congruent corresponding parts are
  there? Explain your answer.

At least 10 pairs, 5 pairs of congruent angles
and 5 pairs of congruent sides
11
Homework

• Page 182-184
• 2-12e, 16, 22, 24, 28, 30, 34, 38
• 18,20,26,31,32,35,39
• 44,45,46

12
Section 4.2

• Triangle Congruence by SSS and SAS

13
Postulate 4-1Side-Side-Side (SSS) Postulate

• If the three sides of one triangle are congruent
  to the three sides of another triangle, then the
  two triangles are congruent.

14
Example Bridges
B
D

• Bridge girders are the same size as marked.
• Given the above picture, is this enough
• information to prove the two triangles are
• congruent? If so write a proof.

15
Proof of Bridge Girders

• Statement Reasons

AB ? CB
Given
BD ? BD
Reflexive
B
AD ? CD
Given
?ABD ? ?CBD
SSS
A
C
D
16
Postulate 4-2Side-Angle-side (SAS) Postulate

• If two sides and the included angle of one
  triangle are congruent to two sides and the
  included angle of another triangle, then the two
  triangles are congruent.

17
Example
R
G
H
S

• From the given information, can we conclude that
  ?RSG ? ?RSH??

Yes, by SAS. Which Side??
RS ? RS Reflexive Property
18
Example
A
B
D
C

• What other information do you need to
• prove ?ADC ? ? BCD??

AC ? BD (using SSS) or ?ADC ? ?BCD (using SAS)
19
Closure

• Explain why the SAS Postulate is not
• written SSA.

Triangles are not necessarily ? if the ? angles
are not included between pairs of ? sides
20
Homework

• Page 189-192
• 2-10e,14,16,18,20,24,26,28,30,33,38
• 3,17,21,22,25,29,39
• 36,42,44

21
Postulate 4-3Angle-Side-Angle (ASA) Postulate

• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of another triangle, then the two
  triangles are congruent.

22
Example
D
N
I
C
T
A
F
O
G

• Suppose that ? F is ? ? C and that ? I is NOT ?
  ? C. Name the triangles that are congruent by
  the ASA Postulate.

Ask yourself, What does the diagram show??
The diagram shows, ? N ? ? A ? ? D, AC ? DG ?
NF, and ? C ? ? G
23
Theorem 4-2Angle-Angle-Side (AAS) Theorem

• If two angles and a non-included side of one
  triangle are congruent to two angles and the
  corresponding non-included side of another
  triangle, then the triangles are congruent.

24
Proof of Theorem 4-2
X
A
Z
Y
C
B
25
Planning A Proof

• Study what you are given and what you
• are to prove. Then plan a proof that uses
• AAS.

What can we use to prove AAS?
A
B
C
D
You can use the sides
DA ? BC or AC ? AC or AB ? CD
?BAC ? ?DCA Alt. Interior Angles
26
Closure

• Explain why the letters of ASA and AAS
• are written in a different order.

ASA compares triangles in which congruent sides
are between the two pairs of congruent
angles. AAS compares triangles in which congruent
sides are NOT between pairs of congruent angles.
27
Homework

• Page 197-200
• 2,6,7,8,10,12,14,18,20,22,24,28,32,34
• 5,11,13,16,19,25,27,29,33
• 38,40,44

28
Section 4.4

• CPCTC

29
Corresponding Parts of Congruent Triangles are
Congruent (CPCTC)
If two triangles are congruent, then each of
their corresponding parts are congruent.
You can conclude
?R ? ?Q
?RSP ? ?QPS
?RPS ? ?QSP
30
Section 4.5

• Isosceles
• Triangles

31
Isosceles Triangles
Vertex Angle
Legs
Base
Base Angles
32
Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then
the angles opposite those sides are congruent.
33
Theorem 4-4 Converse of Isosceles Triangle
Theorem
If two angles of a triangle are congruent, then
the sides opposite the angles are congruent.
34
Theorem 4-5 (Isosceles Triangle Bisector)
The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the
base.
then
and
35
Corollary to Theorem 4-3(Equilateral, then
equiangular)
If a triangle is equilateral,
then the triangle is equiangular.
36
Corollary to Theorem 4-4(Equiangular, then
equilateral)
If a triangle is equiangular,
then the triangle is equilateral.
37
Section 4.6

• Congruence in Right Triangles

38
Theorem 4-6 Hypotenuse-Leg (HL) Theorem
If the hypotenuse and a leg of one right triangle
are congruent to the hypotenuse and a leg of
another right triangle, then the triangles are
congruent.
39
Section 4.7

• Using Corresponding Parts of Congruent Triangles

40
Identify Common Parts

• Separate and redraw the triangles and identify
  common sides and angles

D
C
C
D
E
L
A
B
K
A
B
K
L
41
Identify the common angle
N
L
M
M
M
42
Identify common side
D
E
G
F
43
Name Overlapping Triangles
B
Y
A
X
C
W
Z
D
E
?XWZ and ?YWZ
?ADE and ?BED ?ADB and ?BEA
44
Systems of Linear Equations
45
Example

• Using substitution, solve
• y x 2
• 2x 2y 4
• 2x 2(x 2) 4
• 2x 2x 4 4
• 4x 4 4
• 4x 0
• x 0

y x 2 y 0 2 y 2
Solution is (0 , 2)
46
Example

• Using substitution, solve the following
• m 4n 11
• -6n 8m 36
• -6n 8(4n 11) 36
• -6n 32n 88 36
• 26n 88 36
• 26n - 52
• n - 2

m 4n 11 m 4(-2) 11 m 3
Solution is (3, -2)
47
Example

• Using substitution, solve the following
• y 5x 8
• y -10x 3
• 5x 8 -10x 3
• 5 -15x
• -1/3 x

y 5(-1/3) 8 y -5/3 8 y -5/3 24/3 y
19/3
Solution is (-1/3 , 19/3)
48
You Try!!

• Solve the following using substitution

3x 6y 30 6x y 34
Solution is (6 , -2)
c 3d 27 4d 10c 120
Solution is (126/7, 195/7)
49
Elimination Method

• x y 30
• x y 6
• 2x 36
• x 18 x y 30
• 18 y 30
• y 12

Substitute the value into one of the original
equation, THEN solve for the other variable
50
Elimination Method

• 3x - 8y 32
• -x 8y -16
• 2x 16
• x 8 -x 8y -16
• -8 8y -16
• 8y -8
• y -1

Substitute the value into one of the original
equation, THEN solve for the other variable