Download Identifying Congruent Figures

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
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 ABC  PQR , which reads “triangle
ABC is congruent to triangle PQR.” The notation shows the congruence and the
correspondence.
Corresponding Angles
Corresponding Sides
AP
BQ
CR
AB  PQ
BC  QR
CA  RP
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
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
DEF 
RST .
The congruent angles and sides are as follows.
Angles:  D   R,  E   S,  F   T
Sides: DE  RS , EF  ST , FD  TR
Example
Using Properties of Congruent Figures
In the diagram, NPLM  EFGH.
Find the value of x.
SOLUTION
You know that LM  GH .
So, LM = GH.
8 = 2x – 3
11 = 2 x
5.5 = x
Example
Identifying Congruent Figures
The Third Angles Theorem below follows from the Triangle Sum Theorem.
THEOREM
Theorem 4.3 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
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
60˚ = (2 x + 30)˚
Third Angles Theorem
Substitute.
30 = 2 x
Subtract 30 from each side.
15 = x
Divide each side by 2.
Example
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.
RP  MN, PQ  NQ , and QR  QM
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
Proving Two Triangles are Congruent
Prove that
AEB 
DEC .
A
B
E
C
GIVEN
PROVE
D
AB || DC , AB  DC, E is the midpoint of BC and AD.
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
Proving Two Triangles are Congruent
AEB 
Prove that
DEC .
A
B
E
SOLUTION
D
C
Statements
Reasons
AB || DC , AB  DC
Given
 EAB   EDC,
 ABE   DCE
Alternate Interior Angles Theorem
 AEB   DEC
Vertical Angles Theorem
E is the midpoint of AD,
E is the midpoint of BC
Given
AE  DE , BE  CE
Definition of midpoint
AEB 
DEC
Definition of congruent triangles
Example
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
B
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles
If ABC  DEF , then DEF  ABC .
Transitive Property of Congruent Triangles
If ABC  DEF and DEF  JKL , then ABC 
A
C
E
D
F
L
JKL . J
K
Goal 2