Download Proving Triangles Congruent

Chapter 4 Review
Proving Triangles
Congruent and Isosceles
Triangles
(SSS, SAS, ASA,AAS)
1
Postulates
SSS
If the sides of one triangle are congruent to the 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.
Included Side:
The side of a triangle that forms a side of two
given angles.
2
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.
3
C
Postulates
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.
4
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?
5
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  AC Reflexive Property
D
C
4. ABC  CDA
SSS Postulate
6
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
7
Given : XWY  ZWY ; XYW  ZTW
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
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Postulates
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
D
C
E
F
9
Problem 1 
Given: A  C
BE  BD
Prove: ABE  CBD
Step 1: Mark the Given
Step 2: Mark vertical angles
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?
Given
AAS
A
C
B
E
D
1. A  C
2. ABE  CBD Vertical Angle Thm
3. BE  BD
Given
4. ABE  CBD AAS Postulate
Lesson 4-4: AAS & HL Postulate
10
Parts of an Isosceles Triangle


An isosceles triangle is a triangle with two congruent
sides.
The congruent sides are called legs and the third side is
called the base.
3
Leg
Leg
1 and2 are base angles
3 is the vertex angle
1
2
Base
11
Isosceles Triangle Theorems
If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
A
If AB  AC , then B  C.
B
C
Example: Find the value of x. By the Isosceles Triangle Theorem,
the third angle must also be x.
Therefore, x + x + 50 = 180
50
2x + 50 = 180
2x = 130
x
x = 65
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Isosceles Triangle Theorems
If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
A
If B  C , then AB  AC.
B
C
Example: Find the value of x. Since two angles are congruent, the
A
sides opposite these angles must be
congruent.
3x - 7
x+15
3x – 7 = x + 15
2x = 22
50  C
B 50 
X = 11
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