Download Steps for Proving Triangles Congruent Mark the Given.

1-9-15
Unit 7
Congruency and Similarity
Proving Triangles
Congruent
(SSS, SAS, ASA, AAS, and HL)
1
Congruent Figures
Congruent figures are two figures that have the same
size and shape.
IF two figures are congruent THEN they have the
same size and shape.
IF two figures have the same size and shape THEN
they are congruent.
Two figures have the same size and shape IF they are
congruent.
2
Methods of Proving
Triangles Congruent
SSS
If three sides of one triangle are congruent to three sides of
a second triangle, then the triangles are congruent.
A
B
D
C E
F
Included Angle: In a triangle, the angle formed by two sides is the
included angle for the two sides, where the sides
meet.
Included Side: The side of a triangle that is shared by the two
given angles.
3
Included Angles & Sides
Included Angle:
A is the included angle for AB & AC.
B is the included angle for BA & BC.
A
*
C is the included angle for CA & CB.
B
Included Side:
AB is the included side for A & B.
*
*
BC is the included side for B & C .
AC is the included side for A & C.
4
C
Methods for Proving Triangles
Congruent
ASA If two angles and the included side of one triangle are
congruent to the two angles and the included side of another
triangle, then the triangles are congruent.
A
B
SAS
A
D
C
E
F
B
D
C
F
E
If two sides and the included angle of one triangle are
congruent to the two sides and the included angle of another
triangle, then the triangles are congruent.
5
Methods of Proving Triangles
Congruent
AAS If two angles and a non included side of one triangle are
congruent to the corresponding two angles and side of a
second triangle, then the two triangles are congruent.
A
B
HL
C
E
D
A
D
F
B
C
F
E
If the hypotenuse and a leg of one right triangle are
congruent to the hypotenuse and corresponding leg of
another right triangle, then the triangles are congruent.
6
Steps for Proving Triangles Congruent
1. Mark the Given. ( What have I been told is
congruent)
2. Mark anything else I know to be congruent … For
ex. Common Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA, AAS, and HL)
4. Is there more than one possible answer ?? …
5. Fill in the Reasons … why you marked the parts.
All statements must have a VALID reason.
7
Congruent Triangles - CPCTC
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Two triangles are congruent IF their corresponding parts
(angles and sides) are congruent.
A
If
ABC  PQR
A ↔ P; B ↔ Q; C ↔ R
B
C
≡
Vertices of the 2 triangles correspond in the same order
as the triangles are named.
P
Corresponding sides and angles of the two congruent triangles:
AB  PQ
B  Q
BC  QR
C  R
AC  PR
Q
≡
A  P
8
R
Problem 1 
Given: AB  CD
BC  DA
Prove: ABC  CDA
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA )
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
A
B
1. AB  CD
2. BC  DA
SSS
Reasons
Given
Given
3. AC  CA Reflexive Property
D
C
4. ABC  CDA
SSS Postulate
9
Given : AB  CB ; EB  DB
Problem 2 
Pr ove:
ABE  CBD
Step 1: Mark the Given
Step 2: Mark vertical angles
Step 3: Choose a Method (SSS /SAS/ASA)
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
A
C
B
E
1. AB  CB
2. ABE  CBD
3. EB  DB
D
4. ABE  CBD
SAS
Reasons
Given
Vertical Angles.
Given
SAS Postulate
10
Given : XWY  ZWY ; XYW  ZYW
Problem 3
Pr ove: WXY  WZY
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA)
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
1. XWY  ZWY
X
W
Y
Z
2. WY  WY
3. XYW  ZYW
4. WXY  WZY
ASA
Reasons
Given
Reflexive Postulate
Given
ASA Postulate
11
Problem 4 
Given: A  C
BE  BD
Prove: ABE  CBD
Step 1: Mark the Given
AAS
Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )
Step 4: List the Parts in the order of the method
Step6:5:IsFill
in more?
the reasons
Step
there
Statements
Reasons
Given
C 1. A  C
A
2. ABE  CBD
B vertical angles
Step 2: Mark
E
D
3. BE  BD
Vertical Angle Thm
Given
4. ABE  CBD AAS Postulate
Lesson 4-4: AAS & HL Postulate
12
Given:
Problem 5 
ABC, ADC right
AB  AD
Prove: ABC  ADC
s
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Reasons
Step 6: Is there more?
1.
ABC , ADC
Given
HL
A
B
C
right
s
2. AB  AD
D
3. AC  AC
Given
Reflexive Property
4. ABC  ADC HL Postulate
Lesson 4-4: AAS & HL Postulate
13