Triangle Congruence

1
Triangle Congruence

• Geometry
• High School

2
Teacher Slide

• Content 5 methods to prove triangles are
  congruent.
• Grade Level Geometry
• Creator Don Sutcliffe
• Other Print slides 25-45 as a handout for
  students to take notes.

3
Teacher Slide

• Curriculum Objectives
• Geometry II.A - Determine whether polygons are
  congruent using given information.
• Geometry II.B Develop or complete deductive
  proofs concerning congruent and similar figures.
• Geometry II.C Discover and use the five
  triangle congruency shortcut theorems.
• MAP Objectives
• Missouri Show-Me Standard Math MA2, MA3, MA4
• Missouri Show-Me Standard Goal G3.4, G3.5,
  G3.6

4
Congruent Triangles

• Triangles are congruent if corresponding parts
  are congruent.

B
Y
95
95
30
30
55
55
A
X
Z
C
ltA?ltX
AB?XY
ltB?ltY
BC?YZ
?ABC ? ?XYZ
____________
ltC?ltZ
AC?XZ
5
Solve each equality
40
A
11
Y
Z
80
13
7
3x4
80
60
40
B
X
4y-9
C
3x413
?ABC? ?XYZ
4y-911
mltC
40
3x9
4y20
mltX
80
x3
y5
mltY
60
mltB
60
6
Name all triangles that appear to be congruent.
R
T
S
P
Q
?PQR ?
?QPT
?PST ?
?QSR
7
Side-Side-Side Postulate

• If three sides of a ? are ? to three sides of
  another ?, then the ?s are ?.

A
X
C
B
Y
Z
If AB?XY, BC?YZ, AC?XZ,
?ABC??XYZ
then ___________
8
Given AB?CB, AD?CDProve ?ABD??CBD
A
C
B
D
Statements
Reasons
1. AB?CB, AD?CD
1. Given
2. BD?BD
2. Reflexive
3. ?ABD??CBD
3. SSS Post.
9
Given Quad ABCD is a Prove ?ABC??CDA
C
B
D
A
Statements
Reasons
1. ABCD is a
1. Given
2. AB?CD, BC?AD
2. Def. of a
3. AC?AC
3. Reflexive
4. ?ABD??CBD
4. SSS Post.
10
Side-Angle-Side Postulate

• If two sides and the included angle of a ? are ?
  to two sides and the included angle of another ?,
  then the ?s are ?.

A
X
B
C
Y
Z
If AB?XY, BC?YZ, ltB?ltY
then ___________
?ABC??XYZ
11
Given HJ ? ML, ltJ and ltL are rt. lts, K is the
midpoint of JLProve ?HJK??MLK
J
L
K
H
M
Statements
Reasons
1. HJ ? ML, ltJ and ltL are rt. lts, K is the
midpoint of JL
1. Given
2. ltJ?ltL
2. All rt. lts are ?
3. JK?LK
3. Def. of mdpt.
4. ?HJK??MLK
4. SAS Post.
12
Given NO?NQ, NP bisects ltONQProve
?NOP??NQP
O
N
P
Q
Statements
Reasons

• NO?NQ, NP
• bisects ltONQ

1. Given
2. ltONP?ltQNP
2. Def. of lt Bisector
3. NP?NP
3. Reflexive
4. ?NOP??NQP
4. SAS Post.
13
Angle-Side-Angle Postulate

• If 2 lts and the included side of a ? are ? to 2
  lts and the included side of another ?, then the
  ?s are ?.

A
X
B
C
Y
Z
If ltA?ltX, ltB?ltY, AB?XY,
then ___________
?ABC??XYZ
14
Given HK bisects ltJHL KH bisects ltJKLProve
?JHK??LHK
L
H
K
J
Statements
Reasons
1. HK bisects ltJHL KH bisects ltJKL
1. Given
2. ltJHK?ltLHK, ltJKH?ltLKH
2. Def of lt Bisector
3. HK?HK
3. Reflexive
4. ?JHK??LHK
4. ASA Post.
15
Angle-Angle-Side Theorem

• If 2 lts and a non-included side of a ? are ? to
  2 lts and a non-included side of another ?, then
  the ?s are ?.

A
X
B
C
Y
Z
If ltA?ltX, ltB?ltY, and BC?YZ
then ___________
?ABC??XYZ
16
Given ltA?ltD, ltB?ltE AC?DFProve ?ABC??DEF
A
D
B
E
C
F
Statements
Reasons
1. Given
1. ltA?ltD, ltB?ltE, AC?DF
2. mltAmltBmltC180 mltDmltEmltF180
2. ? Sum Thm.
3. Substitution
3. mltAmltBmltC mltDmltEmltF
4. Subtraction
4. mltCmltF
5. ltC?ltF
5. Def of ? lts
6. ?ABC??DEF
6. ASA Post.
17
Hypotenuse-Leg Theorem

• If hypotenuse and a leg of a ? are ? to the
  hypotenuse and a leg of another ?, then the ?s
  are ?.

A
X
B
C
Y
Z
If AC?XZ, and BC?YZ
then ___________
?ABC??XYZ
18
What method is used to prove the triangles are
congruent?
SSS
SAS
AAS
ASA
19
What method is used to prove the triangles are
congruent?
SAS
AAS
HL
SAS
20
Given ltQ ?ltU, QR?UTProve S is the mdpt. of QU
Q
R
S
U
T
Statements
Reasons
1. ltQ ?ltU, QR?UT
1. Given
2. ltQSR?ltUST
2. Vert, lts are ?
3. AAS Thm.
3. ?QSR??UST
4. Def. of ? ?s
4. QS?US
5. QSUS
5. Def. of ? Segments
6. S is the mdpt. of QU
6. Def. of mdpt.
21
Perpendicular Bisector Theorem

• If a point is on the perpendicular bisector of a
  segment, then the point is equidistant from the
  endpoints of the segment.

22
Given n is the ? bisector of ABProve For any
pt. P on n, PAPB
P
A
C
B
Statements
Reasons

• Line n is the ?
• bisector of AB

1. Given
2. AC?BC, ltACP?ltBCP
2. Def. of ? bisector
3. PC?PC
3. Reflexive
4. ?ACP??BCP
4. SAS Post.
5. PA?PB
5. Def. of ? ?s
6. Def. of ? segments
6. PAPB
23
Given HE?HG, ltE and ltG are rt. ltsProve FH
bisects ltEFG
F
G
E
H
Statements
Reasons

• HE?HG, ltE and ltG are rt. lts

1. Given
2. ?HEF and ?HGF are rt. ?s
2. Def. of rt. ?
3. HF?HF
3. Reflexive
4. ?HEF??HGF
4. HL Thm.
5. ltEFH?ltGFH
5. Def. of ? ?s
6. mltEFHmltGFH
6. Def. of ? lts
7. FH bisects ltEFG
7. Def. of lt Bisector
24
End Show
25
Triangle Congruence

• Geometry
• High School

26
Congruent Triangles

• Triangles are congruent if corresponding parts
  are congruent.

B
Y
95
95
30
30
55
55
A
X
Z
C
ltA?
AB?
ltB?
BC?
____________
ltC?
AC?
27
Solve each equality
40
A
11
Y
Z
80
13
7
3x4
80
60
40
B
X
4y-9
C
3x413
?ABC? ?XYZ
4y-911
mltC
mltX
mltY
mltB
28
Name all triangles that appear to be congruent.
R
T
S
P
Q
?PQR ?
?
?PST ?
?
29
Side-Side-Side Postulate

• If three sides of a ? are ? to three sides of
  another ?, then the ?s are ?.

A
X
C
B
Y
Z
If AB?XY, BC?YZ, AC?XZ,
then
___________
30
Given AB?CB, AD?CDProve ?ABD??CBD
A
C
B
D
Statements
Reasons
31
Given Quad ABCD is a Prove ?ABC??CDA
C
B
D
A
Statements
Reasons
32
Side-Angle-Side Postulate

• If two sides and the included angle of a ? are ?
  to two sides and the included angle of another ?,
  then the ?s are ?.

A
X
B
C
Y
Z
If AB?XY, BC?YZ, ltB?ltY
then
___________
33
Given HJ ? ML, ltJ and ltL are rt. lts, K is the
midpoint of JLProve ?HJK??MLK
J
L
K
H
M
Statements
Reasons
34
Given NO?NQ, NP bisects ltONQProve
?NOP??NQP
O
N
P
Q
Statements
Reasons
35
Angle-Side-Angle Postulate

• If 2 lts and the included side of a ? are ? to 2
  lts and the included side of another ?, then the
  ?s are ?.

A
X
B
C
Y
Z
If ltA?ltX, ltB?ltY, AB?XY,
then
___________
36
Given HK bisects ltJHL KH bisects ltJKLProve
?JHK??LHK
L
H
K
J
Statements
Reasons
37
Angle-Angle-Side Theorem

• If 2 lts and a non-included side of a ? are ? to
  2 lts and a non-included side of another ?, then
  the ?s are ?.

A
X
B
C
Y
Z
If ltA?ltX, ltB?ltY, and BC?YZ
then
___________
38
Given ltA?ltD, ltB?ltE AC?DFProve ?ABC??DEF
A
D
B
E
C
F
Statements
Reasons
39
Hypotenuse-Leg Theorem

• If hypotenuse and a leg of a ? are ? to the
  hypotenuse and a leg of another ?, then the ?s
  are ?.

A
X
B
C
Y
Z
If AC?XZ, and BC?YZ
then
___________
40
What method is used to prove the triangles are
congruent?
41
What method is used to prove the triangles are
congruent?
42
Given ltQ ?ltU, QR?UTProve S is the mdpt. of QU
Q
R
S
U
T
Statements
Reasons
43
Perpendicular Bisector Theorem

• If a point is on the perpendicular bisector of a
  segment, then the point is equidistant from the
  endpoints of the segment.

44
Given n is the ? bisector of ABProve For any
pt. P on n, PAPB
P
A
C
B
Statements
Reasons
45
Given HE?HG, ltE and ltG are rt. ltsProve FH
bisects ltEFG
F
G
E
H
Statements
Reasons