Download 4-5 Objective: Prove Triangles Congruent by ASA and AAS

4-5
Objective: Prove Triangles
Congruent by ASA and AAS
Quick Review over SAS and HL
ASA- Angle Side Angle
• Angle Side Angle Congruence Postulate. If
two angles and the included side of two
angles are congruent to two angles and
the included side of a second triangle,
then the two triangles are congruent.
– Notice the side is in between the angles
– See Page 249
AAS- Angle Angle Side
• 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.
– Has to go in order around the triangle: Angle
Angle Side
– See Page 249
Important to Note
• AAA is NOT a characteristic to show that
two triangles are congruent.
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.
Extra Example 1
Flow Proofs
• Making a proof like a flow chart
• Use arrows to show the flow of a logical
argument.
– Still start with given
• Use arrows to show what you can conclude from
the given
– Still need reasons
• Write below the flow chart boxes
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
AAS;
RTS
UTV because they are vertical angles.
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
GUIDED PRACTICE
3.
for Examples 3 and 4
In Example 3, suppose ABE
ADE is also
given. What theorem or postulate besides ASA
ABE
ADE?
can you use to prove that
ANSWER
AAS Congruence Theorem.
SSS, SAS, HL, ASA, and AAS
• Top of page 252
Daily Homework Quiz
For use after Lesson 4.5
Tell whether each pair of triangle are congruent by
SAS, ASA, SSS, AAS or HL. If it is not possible to prove
the triangle congruent, write not necessarily
congruent.
1.
Daily Homework Quiz
For use after Lesson 4.5
Tell whether each pair of triangle are congruent by
SAS, ASA, SSS, AAS or HL. If it is not possible to prove
the triangle congruent, write not necessarily
congruent.
2.
Daily Homework Quiz
Write flow proof.
Given : BD bisects ABC,
Prove :
ABD
CBD
3.
For use after Lesson 4.5
A
C
Daily Homework Quiz
For use after Lesson 4.5
Tell whether each pair of triangle are congruent by
SAS, ASA, SSS, AAS or HL. If it is not possible to prove
the triangle congruent, write not necessarily
congruent.
1.
ANSWER
ASA .
Daily Homework Quiz
For use after Lesson 4.5
Tell whether each pair of triangle are congruent by
SAS, ASA, SSS, AAS or HL. If it is not possible to prove
the triangle congruent, write not necessarily
congruent.
2.
ANSWER
not necessarily congruent .
Daily Homework Quiz
Write flow proof.
Given : BD bisects ABC,
Prove :
ABD
CBD
3.
For use after Lesson 4.5
A
C
Daily Homework Quiz
ANSWER
For use after Lesson 4.5
Homework
• 3-10, 14 –21, 23 – 25, 31 – 34, 41 - 43