1. Warm-Up Exercises Tell whether it is possible to draw each triangle.
Acute scalene triangle
Obtuse equilateral triangle
Right isosceles triangle
Scalene equiangular triangle
Right scalene triangle

2. Identifying Congruent Figures
Two geometric figures are congruent if they have exactly the same size and shape.
Each of the red figures is congruent to the other red figures.
None of the blue figures is congruent to another blue figure.
Goal 1

3. Identifying Congruent Figures
When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.
For the triangles below, you can write  , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence.
ABC
PQR
Corresponding Angles
Corresponding Sides
 A   P
AB  PQ
 B   Q
BC  QR
 C   R RP
CA 
There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write 
BCA
QRP
Goal 1

4. Naming Congruent Parts
The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts.
SOLUTION
The diagram indicates that 
The congruent angles and sides are as follows.
DEF
RST
Angles:
 D   R,  E   S,  F   T
Sides:
 , 
 , RS DE TR FD ST EF
Example

5. Using Properties of Congruent Figures
In the diagram, NPLM  EFGH.
Find the value of x.
SOLUTION
You know that  GH LM
So, LM = GH.
8 = 2 x – 3
11 = 2 x
5.5 = x
Example

6. Using Properties of Congruent Figures
In the diagram, NPLM  EFGH.
Find the value of x.
Find the value of y.
SOLUTION
SOLUTION
You know that  GH LM
You know that  N   E.
So, LM = GH.
So, m N = m E.
8 = 2 x – 3
72˚ = (7y + 9)˚
11 = 2 x
63 = 7y
5.5 = x
9 = y
Example

7. Identifying Congruent Figures
The Third Angles Theorem below follows from the Triangle Sum Theorem.
THEOREM
Theorem Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If  A   D and  B   E, then  C   F.
Goal 1

8. Using the Third Angles Theorem
Find the value of x.
SOLUTION
In the diagram,  N   R and  L   S.
From the Third Angles Theorem, you know that  M   T.
So, m M = m T.
From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚.
m M = m T
Third Angles Theorem
60˚ = (2 x + 30)˚
Substitute.
30 = 2 x
Subtract 30 from each side.
15 = x
Divide each side by 2.
Example

9. Proving Triangles are Congruent
Decide whether the triangles are congruent. Justify your reasoning.
SOLUTION
Paragraph Proof
From the diagram, you are given that all three corresponding sides are congruent.
 , NQ PQ MN RP  QM QR
and
Because  P and  N have the same measures,  P   N.
By the Vertical Angles Theorem, you know that  PQR   NQM.
By the Third Angles Theorem,  R   M.
So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, 
PQR
NQM
Goal 2

10.  . Proving Two Triangles are Congruent Prove that  . AEB DEC GIVEN
|| , DC AB
 ,
E is the midpoint of BC and AD.
PROVE 
AEB
DEC
Plan for Proof Use the fact that  AEB and  DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.
Example

11. Proving Two Triangles are Congruent
Prove that 
AEB
DEC A B C D E
SOLUTION
Statements
Reasons
|| , DC AB 
Given
 EAB   EDC,
 ABE   DCE
Alternate Interior Angles Theorem
 AEB   DEC
All vertical angles are congruent
E is the midpoint of AD,
E is the midpoint of BC
Given
 , DE AE  CE BE
Definition of midpoint 
AEB
DEC
Definition of congruent triangles
Example

12. Proving Triangles are Congruent
In this lesson, you have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. In upcoming lessons, you will learn more efficient ways of proving that triangles are congruent. The properties below will be useful in such proofs.
THEOREM A B C
Theorem Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
Every triangle is congruent to itself. D E F
Symmetric Property of Congruent Triangles
If  , then 
ABC
DEF J K L
Transitive Property of Congruent Triangles
If  and  , then 
JKL
ABC
DEF
Goal 2

13. Classwork: p. 205 #1-9
Assignment: pp #11-29 odd, 30-33, 35, 38, 39, odd