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Unit 11
CCGPS Analytic Geometry
Proving Triangles Congruent
(SSS, SAS, ASA, AAS, HL)
Unit 1 : SSS, SAS, ASA 2
SSS
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
A
B C
D
E F
Side-Side-SideSSS
• If all 3 sides of 2 triangles are congruent, the triangles are congruent.
If AB ED,
Make sure that you write the congruency statement so that the corresponding vertices (and thus the corresponding sides) are in the same position in thecongruency statement.
∆ABC ∆EDF
BC DF, and AC EF, then the 2 triangles are congruent
not ∆ABC ∆DEF
Lesson 4-3: SSS, SAS, ASA 4
Included Angles
& .A is the included angle for AB AC
& .B is the included angle for BA BC
& .C is the included angle for CA CB
A
B C
Included Angle:
** *
In a triangle, the angle formed by two sides is the included angle for the two sides.
Included Angle
Lesson 4-3: SSS, SAS, ASA 5
Included Sides
A
B C
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
** *
Included Side: The side of a triangle that forms a side of two given angles.
Lesson 4-3: SSS, SAS, ASA6
ASAAngle Side Angle
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 C
D
E F
EA Sides AB = ED
DB
A
S
A
EDFABC
Lesson 4-3: SSS, SAS, ASA 7
SAS Side Angle Side
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.
S
S
A
Lesson 4-3: SSS, SAS, ASA 8
Steps for Proving Triangles Congruent
1. Mark the Given.
2. Mark … Reflexive Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts … in the order of the method.
5. Fill in the Reasons … why you marked the parts.
6. Is there more?
Lesson 4-3: SSS, SAS, ASA 9
Problem 1 - Given: AB CD BC DAProve: ABC CDA
Statements Reasons
Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?
A B
D C
SSS
Given
Given
Reflexive Property
SSS Postulate4. ABC CDA
1. AB CD2. BC DA3. AC CA
Unit 1 CCGPS Analytic Geom. SSS, SAS, ASA
10
Problem 2Step 1: Mark the Given Step 2: Mark vertical angles congruentStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?
SAS
Given
Given
Vertical Angles.
SAS Postulate
: ;
Pr :
Given AB CB EB DB
ove ABE CBD
E
C
D
AB
1. AB CB2. ABE CBD
3. EB DB4. ABE CBD
Statements Reasons
Lesson 4-3: SSS, SAS, ASA 11
Problem 3
Statements Reasons
Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?
ASA
Given
Given
Reflexive Postulate
ASA Postulate
: ;
Pr :
Given XWY ZWY XYW ZYW
ove WXY WZY
Z
W Y
X 1. XWY ZWY
2. WY WY3. XYW ZYW
4. WXY WZY
AAS Angle Angle Side (corresponding)
DEFABC
HGJDEFABC
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.
IS NOT CONGRUENT WITH EITHER OF THE OTHER 2 Congruent Angles and side DO NOT correspond..
Hypotenuse Leg HL
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.
A
B C
D
E F
Problem 1
Statements Reasons
Step 1: Mark the Given Step 2: Mark vertical anglesStep 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?
AAS
Given
Given
Vertical Angle Thm
AAS Postulate
Given: A C BE BDProve: ABE CBD
E
C
D
AB
1. A C2. ABE CBD
3. BE BD
4. ABE CBD
Problem 2
3. AC AC2. AB AD
1. ,ABC ADCright s
Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?
HL
Given
Given
Reflexive Property
HL Postulate
Given: ABC, ADC right s AB ADProve: ABC ADC
CB D
A
4. ABC ADC
Statements Reasons