Download 4.5 Prove Triangles Congruent by ASA and AAS

4.5
Prove Triangles Congruent
by ASA and AAS
Goal
Your Notes
p Use two more methods to prove congruences.
VOCABULARY
Flow proof
POSTULATE 21: ANGLE-SIDE-ANGLE (ASA)
CONGRUENCE POSTULATE
If two angles and the included side of one triangle are
congruent to two angles and the included side of a
second triangle, then the two triangles are congruent.
If Angle ∠A >
}
Side AC >
Angle ∠C >
then
,
B
E
, and
,
nABC >
A
C
D
F
.
THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS)
CONGRUENCE THEOREM
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.
If Angle ∠A >
Angle ∠C >
}
Side BC >
then
104
Lesson 4.5 • Geometry Notetaking Guide
nABC >
,
B
E
, and
,
A
C
D
F
.
Copyright © Holt McDougal. All rights reserved.
4.5
Prove Triangles Congruent
by ASA and AAS
Goal
Your Notes
p Use two more methods to prove congruences.
VOCABULARY
Flow proof A flow proof uses arrows to show the
flow of a logical argument.
POSTULATE 21: ANGLE-SIDE-ANGLE (ASA)
CONGRUENCE POSTULATE
If two angles and the included side of one triangle are
congruent to two angles and the included side of a
second triangle, then the two triangles are congruent.
If Angle ∠A > ∠D ,
}
}
Side AC > DF , and
Angle ∠C > ∠F ,
then
B
A
E
C
D
F
nABC > nDEF .
THEOREM 4.6: ANGLE-ANGLE-SIDE (AAS)
CONGRUENCE THEOREM
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.
If Angle ∠A > ∠D ,
Angle ∠C > ∠F , and
}
}
Side BC > EF ,
then
104
Lesson 4.5 • Geometry Notetaking Guide
B
A
E
C
D
F
nABC > nDEF .
Copyright © Holt McDougal. All rights reserved.
Your Notes
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.
b.
a.
c.
Solution
a. There is not enough information to prove the triangles
are known to be
are congruent, because no
congruent.
pair of
b. Two pairs of angles and a
sides are congruent. The triangles are congruent by
.
the
c. The vertical angles are congruent, so two pairs of
angles and their
are congruent. The
triangles are congruent by the
.
Checkpoint Can nSTW and nVWT be proven congruent
with the information given in the diagram? If so, state
the postulate or theorem you would use.
1.
S
V
60°
60°
U
T
2.
S
W
T
W
Copyright © Holt McDougal. All rights reserved.
V
Lesson 4.5 • Geometry Notetaking Guide
105
Your Notes
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.
b.
a.
c.
Solution
a. There is not enough information to prove the triangles
are congruent, because no sides are known to be
congruent.
b. Two pairs of angles and a non-included pair of
sides are congruent. The triangles are congruent by
the AAS Congruence Theorem .
c. The vertical angles are congruent, so two pairs of
angles and their included sides are congruent. The
triangles are congruent by the ASA Congruence
Postulate .
Checkpoint Can nSTW and nVWT be proven congruent
with the information given in the diagram? If so, state
the postulate or theorem you would use.
1.
S
V
60°
60°
U
T
W
Yes; AAS Congruence Theorem
2.
S
T
W
V
Yes; ASA Congruence Postulate
Copyright © Holt McDougal. All rights reserved.
Lesson 4.5 • Geometry Notetaking Guide
105
Your Notes
Write a flow proof
Example 2
In the diagram, ∠1 > ∠4 and
}
CF bisects ∠ACE. Write a flow
proof to show nCBF > nCDF.
Solution
}
Given ∠1 > ∠4, CF bisects ∠ACE.
C
B 2
1
A
3 D
4
F
E
Prove nCBF > nCDF
∠1 > ∠4
. }
CF bisects.
. ∠ACE.
∠1 and ∠2 are
∠3 and ∠4 are
Linear Pair Postulate
∠2 >
}
}
CF > CF
∠ACF >
Def. of ∠ bisector
Congruent
Supps. Thm.
∠CBF > ∠CDF
Checkpoint Complete the following exercise.
}
3. In Example 2, suppose it is given that CF bisects
∠ACE and ∠BFD. Write a flow proof to show
nCBF > nCDF.
106 Lesson 4.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Write a flow proof
Example 2
In the diagram, ∠1 > ∠4 and
}
CF bisects ∠ACE. Write a flow
proof to show nCBF > nCDF.
Solution
}
Given ∠1 > ∠4, CF bisects ∠ACE.
C
B 2
1
A
3 D
4
F
E
Prove nCBF > nCDF
∠1 > ∠4
∠1 and ∠2 are supplements . }
CF bisects.
∠3 and ∠4 are supplements . ∠ACE.
Given
Given
Linear Pair Postulate
}
}
CF > CF
Reflexive Prop.
∠2 > ∠3
Congruent
Supps. Thm.
∠ACF > ∠ECF
Def. of ∠ bisector
∠CBF > ∠CDF
AAS Congruence Theorem
Checkpoint Complete the following exercise.
}
3. In Example 2, suppose it is given that CF bisects
∠ACE and ∠BFD. Write a flow proof to show
nCBF > nCDF.
}
}
CF bisects ∠ACE.
CF bisects ∠BFD.
Given
∠ACF > ∠ECF
Def. of ∠
bisector
Given
}
}
CF > CF
Reflexive Prop.
∠BFC > ∠DFC
Def. of ∠
bisector
∠CBF > ∠CDF
ASA Congruence Postulate
106 Lesson 4.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Choose a postulate or theorem
Games You and a friend are trying to find a flag hidden in
the woods. Your friend is standing 75 feet away from you.
When facing each other, the angle from you to the flag is
728 and the angle from your friend to the flag is 538. Is
there enough information to locate the flag?
Solution
The locations of you, your friend, and
the flag form a triangle. The measures
of
and an
of the triangle are known.
72°
75 ft
53°
By the
, all triangles with
these measures are congruent. So, the triangle formed
is unique and the flag location is given by the
.
Checkpoint Complete the following exercise.
4. Theater Two actors are standing apart from each
other on the edge of a stage. Spotlights are located
and pointed as shown in the diagram. Can one of
the actors move to another location on the stage
without changing any of the angles of the triangle,
without changing the distance to the other actor, and
without requiring a spotlight to move?
20 ft
40°
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 4.5 • Geometry Notetaking Guide
107
Your Notes
Example 3
Choose a postulate or theorem
Games You and a friend are trying to find a flag hidden in
the woods. Your friend is standing 75 feet away from you.
When facing each other, the angle from you to the flag is
728 and the angle from your friend to the flag is 538. Is
there enough information to locate the flag?
Solution
The locations of you, your friend, and
the flag form a triangle. The measures
of two angles and an included side
of the triangle are known.
72°
75 ft
53°
By the ASA Congruence Postulate , all triangles with
these measures are congruent. So, the triangle formed
is unique and the flag location is given by the third
vertex .
Checkpoint Complete the following exercise.
4. Theater Two actors are standing apart from each
other on the edge of a stage. Spotlights are located
and pointed as shown in the diagram. Can one of
the actors move to another location on the stage
without changing any of the angles of the triangle,
without changing the distance to the other actor, and
without requiring a spotlight to move?
20 ft
40°
Homework
No. The measures of two angles and a
nonincluded side of the triangle are known.
By the AAS Congruence Theorem, all triangles
with these measures are congruent. To create
a different congruent triangle, one of the actors
would have to move to a location that is off the
stage, and the spotlight following that actor
would have to move.
Copyright © Holt McDougal. All rights reserved.
Lesson 4.5 • Geometry Notetaking Guide
107