Transcript Prove Triangles Congruent by ASA & AAS

Prove Triangles Congruent
by ASA & AAS
Lesson 4.10
Use two more methods to prove
congruences
Vocabulary



A flow proof uses arrows to show the flow of a
logical argument.
ASA Congruence Postulate: If two angles and
the included side of one triangle are congruent to
two angles & included side of a second triangle,
then the two triangles are congruent
AAS Congruence Theorem: If two angles and a
non-included side of one triangle are congruent
to two angles and the corresponding nonincluded side of a second triangle, then the two
triangles are congruent.
Triangle Congruence Review
Postulates we can use
CAN’T USE
Warm-Up Exercises
Lesson 4.5, For use with pages 249-255
Tell whether the pair of triangles is congruent or not
and why.
ANSWER
Yes; HL
Thm.
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.
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
ASA Congruence Postulate
AAS Congruence Theorem
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
RST
UTV
VUT
The vertical angles
are congruent
AAS
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