Name:
Ms. D’Amato
Date:
Block:
Proving Triangles are Congruent
18.
Be able to identify:
1.
an angle included between 2 sides (called the
)
2.
a side included between 2 angles (called the
)
A
Q
C
B
P
R
̅̅̅̅ .
B is said to be “included between” ̅̅̅̅
𝐴𝐵 and 𝐵𝐶
̅̅̅̅
𝐴𝐶 is said to be “included between” A and C.
Examples:
19.
1.
̅̅̅̅ and ̅̅̅̅
What angle is included between 𝑄𝑅
𝑃𝑅 ?
2.
̅̅̅̅ ?
What is the “included angle” between ̅̅̅̅
𝐴𝐶 & 𝐵𝐶
3.
What side is included between P and Q?
The definition of congruent s requires that all 6 corresponding angles and sides be congruent.
There are 5 other ways to prove 2 s congruent that require only half as much information. 
A.
- If three sides of one triangle are
congruent to three sides of a second triangle, then the two triangles are congruent.
B.
- If two sides and the included angle of
one triangle are congruent to two sides and the included angle of a second triangle, then
the two triangles are congruent.
Example 1:
Example 2:
B
A
P
D
E
C
F
ABC  
by
X
Q
R
PQR  
Y
by
Z
In the following examples you must:
a. mark vertical s  and, if lines //, other  s
b. mark reflexive s or reflexive sides .
Examples:
Are the 2 s ? Write yes or no. If yes, which method proves the congruence, SSS or
SAS?
1.
2.
_______ _______
4.
3.
_______ _______
5.
_______ _______
7.
6.
_______ _______
8.
_______ _______
C.
_______ _______
_______ _______
9.
_______ _______
_______ _______
- If two angles and the included side of
one triangle are congruent to two angles and the included side of a second triangle, then
the two triangles are congruent.
Notice that the side must be “included between” the 2 angles.
Examples:
Are the s  by the ASA Postulate?
1.
2.
_______
D.
Examples:
3.
_______
_______
_______
- If two angles and a non-included side
of one triangle are congruent to two angles and the corresponding non-included side of
a second triangle, then the two triangles are congruent.
Are the s below  by the AAS theorem?
1.
2.
_______
E.
4.
3.
_______
4.
_______
_______
- If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse and a leg of a second right triangle, then the
two triangles are congruent.
WARNINGS:
(1)
AAA does not prove 2 s . Why are angles alone not sufficient?
(2)
SSA (two sides and a not-included angle) does not prove 2 s .
Two different sized s might be formed. (See the diagrams below.)
5”
40
4"
5”
40
4”
20.
We often prove 2 triangles congruent in order to show that one of the six parts of the
definition is true.
Review: By the definition of congruent s, if ABC  SNO, then
__________
__________
__________
__________
__________
__________
*We need 3 of these 6 statements in order to prove the triangles congruent. The other 3 statements
can be stated true after proving the s .*
T
Example:
Given:
E is the midpoint of ̅̅̅̅
𝑀𝐽.
̅̅̅̅̅  𝐽𝑇
̅̅̅
𝑀𝑇
Prove:
MET  JET
M
Statements
E
Reasons
1.
1. Given
2.
2. Definition of a midpoint
3.
3. Given
4.
4. Reflexive property
5. TME  
5.
6. MET  
6.
CPCTC means
Note: Def of  s means the same as CPCTC
J