EXAMPLE 1
Identify congruent triangles
Can the triangles be proven congruent with the
information given in the diagram? If so, state the
postulate or theorem you would use.
SOLUTION
a.
The vertical angles are congruent, so two pairs of
angles and a pair of non-included sides are
congruent. The triangles are congruent by the AAS
Congruence Theorem.
EXAMPLE 1
Identify congruent triangles
b.
There is not enough information to prove the
triangles are congruent, because no sides are
known to be congruent.
c.
Two pairs of angles and their included sides are
congruent. The triangles are congruent by the ASA
Congruence Postulate.
EXAMPLE 2
Prove the AAS Congruence Theorem
Prove the Angle-Angle-Side Congruence Theorem.
Write a proof.
GIVEN
PROVE
A
D,
ABC
C
DEF
F, BC
EF
GUIDED PRACTICE
1.
for Examples 1 and 2
In the diagram at the right, what
postulate or theorem can you use to
RST
VUT ? Explain.
prove that
SOLUTION
STATEMENTS
REASONS
S
U
Given
RS
UV
Given
RTS
UTV
The vertical angles
are congruent
GUIDED PRACTICE
for Examples 1 and 2
ANSWER
UTV are congruent because
Therefore RTS
vertical angles are congruent so two pairs of angles
and a pair of non included side are congruent. The
triangle are congruent by AAS Congruence Theorem.
for Examples 1 and 2
GUIDED PRACTICE
Rewrite the proof of the Triangle Sum Theorem
on page 219 as a flow proof.
2.
ABC
GIVEN
PROVE m
1+m
2+m
3 = 180°
STATEMENTS
1. Draw BD parallel to AC .
2. m 4 + m 2 + m 5 = 180°
REASONS
1. Parallel Postulate
2. Angle Addition Postulate and
definition of straight angle
3.
1
4,
3
4. m
1= m
4,m
5. m
1+m
2+m
3. Alternate Interior Angles
5
3= m
5
3 = 180°
Theorem
4. Definition of congruent
angles
5. Substitution Property
of Equality
EXAMPLE 3
Write a flow proof
In the diagram, CE
BD and  CAB
Write a flow proof to show
GIVEN
PROVE
CE
BD,  CAB
ABE
ADE
ABE
CAD
CAD.
ADE
EXAMPLE 4
Standardized Test Practice
EXAMPLE 4
Standardized Test Practice
The locations of tower A,
tower B, and the fire form a
triangle. The dispatcher
knows the distance from
tower A to tower B and the
measures of
A and B. So,
the measures of two angles
and an included side of the
triangle are known.
By the ASA Congruence Postulate, all triangles with
these measures are congruent. So, the triangle formed
is unique and the fire location is given by the third
vertex. Two lookouts are needed to locate the fire.
EXAMPLE 4
Standardized Test Practice
ANSWER
The correct answer is B.
for Examples 3 and 4
GUIDED PRACTICE
3.
In Example 3, suppose ABE
ADE is also
given. What theorem or postulate besides ASA
ABE
ADE?
can you use to prove that
SOLUTION
STATEMENTS
REASONS
ABE
ADE
AEB
AED
Both are right
angle triangle.
BD
ABE
DB
ADE
Given
Definition of right
triangle
Reflexive Property of
Congruence
AAS Congruence Theorem
GUIDED PRACTICE
4.
for Examples 3 and 4
What If? In Example 4, suppose a fire occurs
directly between tower B and tower C. Could
towers B and C be used to locate the fire? Explain
SOLUTION
Proved by ASA congruence
The locations of tower B, tower C, and the fire form a
triangle. The dispatcher knows the distance from tower B
to tower C and the measures of B and
C. So, he
knows the measures of two angles and an included side of
the triangle.
GUIDED PRACTICE
for Examples 3 and 4
By the ASA Congruence Postulate, all triangles with
these measures are congruent. No triangle is formed by
the location of the fire and tower, so the fire could be
anywhere between tower B and C.