Download 4-5

NAME ______________________________________________ DATE
4-5
____________ PERIOD _____
Study Guide and Intervention
Proving Congruence—ASA, AAS
ASA Postulate
The Angle-Side-Angle (ASA) Postulate lets you show that two triangles
are congruent.
If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle, then the triangles are congruent.
ASA Postulate
Example
Find the missing congruent parts so that the triangles can be
proved congruent by the ASA Postulate. Then write the triangle congruence.
a.
B
E
A
C D
F
Two pairs of corresponding angles are congruent, A D and C F. If the
included sides A
C
and D
F
are congruent, then ABC DEF by the ASA Postulate.
b. S
X
R
T
W
Y
R Y and S
R
X
Y
. If S X, then RST YXW by the ASA Postulate.
Exercises
What corresponding parts must be congruent in order to prove that the triangles
are congruent by the ASA Postulate? Write the triangle congruence statement.
1.
2.
C
D
B
E
W
A
Z
5.
B
6.
V
R
B
Y
A
4. A
3.
X
E
D
B
C
D
T
U
C
A
C
E
Lesson 4-5
D
S
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Glencoe Geometry
NAME ______________________________________________ DATE
4-5
____________ PERIOD _____
Study Guide and Intervention
(continued)
Proving Congruence—ASA, AAS
AAS Theorem Another way to show that two triangles are congruent is the AngleAngle-Side (AAS) Theorem.
AAS Theorem
If two angles and a nonincluded side of one triangle are congruent to the corresponding two
angles and side of a second triangle, then the two triangles are congruent.
You now have five ways to show that two triangles are congruent.
• definition of triangle congruence
• ASA Postulate
• SSS Postulate
• AAS Theorem
• SAS Postulate
Example
In the diagram, BCA DCA. Which sides
are congruent? Which additional pair of corresponding parts
needs to be congruent for the triangles to be congruent by
the AAS Postulate?
C
A
A
C
by the Reflexive Property of congruence. The congruent
angles cannot be 1 and 2, because A
C
would be the included side.
If B D, then ABC ADC by the AAS Theorem.
B
A
1
2
C
D
Exercises
In Exercises 1 and 2, draw and label ABC and DEF. Indicate which additional
pair of corresponding parts needs to be congruent for the triangles to be
congruent by the AAS Theorem.
1. A D; B E
2. BC EF; A D
B
C
F
E
A
C
D
B
F
A
3. Write a flow proof.
Given: S U; T
R
bisects STU.
Prove: SRT URT
TR bisects STU.
Given
E
D
S
R
T
U
STR UTR
Def.of bisector
S U
SRT URT
SRT URT
Given
AAS
CPCTC
RT RT
Refl. Prop. of ©
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208
Glencoe Geometry
NAME ______________________________________________ DATE
4-5
____________ PERIOD _____
Skills Practice
Proving Congruence—ASA, AAS
Write a flow proof.
1. Given: N L
J
K
M
K
Prove: JKN MKL
J
L
K
N
M
N L
Given
JK MK
JKN MKL
Given
AAS
JKN MKL
Vertical are .
2. Given: AB
C
B
A C
D
B
bisects ABC.
Prove: A
D
C
D
B
D
A
C
AB CB
Given
A C
ABD CBD
AD CD
Given
ASA
CPCTC
DB bisects ABC.
ABD CBD
Def. of bisector
Given
3. Write a paragraph proof.
F
G
Lesson 4-5
Given: DE
|| F
G
E G
Prove: DFG FDE
E
D
©
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209
Glencoe Geometry
NAME ______________________________________________ DATE
4-5
____________ PERIOD _____
Practice
Proving Congruence—ASA, AAS
1. Write a flow proof.
Given: S is the midpoint of Q
T
.
R
Q
|| T
U
Prove: QSR TSU
S is the
midpoint of QT.
Given
QR || TU
Given
R
T
S
Q
U
QS TS
Def.of midpoint
Q T
Alt. Int. are .
QSR TSU
ASA
QSR TSU
Vertical are .
2. Write a paragraph proof.
Given: D F
G
E
bisects DEF.
Prove: D
G
F
G
D
G
E
F
ARCHITECTURE For Exercises 3 and 4, use the following
information.
An architect used the window design in the diagram when remodeling
an art studio. A
B
and C
B
each measure 3 feet.
B
A
D
C
3. Suppose D is the midpoint of A
C
. Determine whether ABD CBD.
Justify your answer.
4. Suppose A C. Determine whether ABD CBD. Justify your answer.
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Glencoe Geometry
NAME ______________________________________________ DATE
4-6
____________ PERIOD _____
Study Guide and Intervention
Properties of Isosceles Triangles An isosceles triangle has two congruent sides.
The angle formed by these sides is called the vertex angle. The other two angles are called
base angles. You can prove a theorem and its converse about isosceles triangles.
A
• If two sides of a triangle are congruent, then the angles opposite
those sides are congruent. (Isosceles Triangle Theorem)
• If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
B
C
If A
B
C
B
, then A C.
If A C, then A
B
C
B
.
Example 1
Example 2
Find x.
Find x.
S
C
(4x 5)
A
(5x 10)
B
3x 13
R
BC BA, so
mA mC.
5x 10 4x 5
x 10 5
x 15
T
2x
mS mT, so
SR TR.
3x 13 2x
3x 2x 13
x 13
Isos. Triangle Theorem
Substitution
Subtract 4x from each side.
Add 10 to each side.
Converse of Isos. Thm.
Substitution
Add 13 to each side.
Subtract 2x from each side.
Exercises
Find x.
1.
R
P
40
2x 2. S
2x 6
T
3x 6
3.
W
V
Q
4. D
P
K
T (6x 6)
2x Q
5. G
Y
3x 6.
B
Z
T
30
3x 3x D
7. Write a two-column proof.
Given: 1 2
Prove: A
B
C
B
L
R
x
S
B
A
1
3
C
D
2
E
Statements
©
Glencoe/McGraw-Hill
Reasons
213
Glencoe Geometry
Lesson 4-6
Isosceles Triangles
NAME ______________________________________________ DATE
4-6
____________ PERIOD _____
Study Guide and Intervention
(continued)
Isosceles Triangles
Properties of Equilateral Triangles
An equilateral triangle has three congruent
sides. The Isosceles Triangle Theorem can be used to prove two properties of equilateral
triangles.
1. A triangle is equilateral if and only if it is equiangular.
2. Each angle of an equilateral triangle measures 60°.
Example
Prove that if a line is parallel to one side
of an equilateral triangle, then it forms another equilateral
triangle.
A
P 1
Proof:
2 Q
B
C
Statements
Reasons
Q
|| B
C
.
1. ABC is equilateral; P
2. mA mB mC 60
3. 1 B, 2 C
4. m1 60, m2 60
5. APQ is equilateral.
1. Given
2. Each of an equilateral measures 60°.
3. If || lines, then corres. s are .
4. Substitution
5. If a is equiangular, then it is equilateral.
Exercises
Find x.
1.
2.
D
6x 5
F
6x 4.
4x
V
5.
Q
40
60
L
Y
4x 4
Glencoe/McGraw-Hill
4x 60 H
O
A
D
1
2
B
C
Proof:
©
R
M
7. Write a two-column proof.
Given: ABC is equilateral; 1 2.
Prove: ADB CDB
Statements
KLM is equilateral.
6.
X
Z
K
M
H
3x 8 60
R
3x N
5x
J
E
P
3. L
G
Reasons
214
Glencoe Geometry
NAME ______________________________________________ DATE
4-6
____________ PERIOD _____
Skills Practice
Isosceles Triangles
Refer to the figure.
Lesson 4-6
C
1. If A
C
A
D
, name two congruent angles.
B
D
2. If B
E
B
C
, name two congruent angles.
E
A
3. If EBA EAB, name two congruent segments.
4. If CED CDE, name two congruent segments.
ABF is isosceles, CDF is equilateral, and mAFD 150.
Find each measure.
5. mCFD
6. mAFB
7. mABF
8. mA
A
E
F
B
L
9. If mRLP 100, find mBRL.
10. If mLPR 34, find mB.
R
11. Write a two-column proof.
P
D
E
Given: CD
C
G
E
D
G
F
Prove: C
E
C
F
Glencoe/McGraw-Hill
D
35
In the figure, P
L
R
L
and L
R
B
R
.
©
C
B
C
F
G
215
Glencoe Geometry
NAME ______________________________________________ DATE
4-6
____________ PERIOD _____
Practice
Isosceles Triangles
Refer to the figure.
R
1. If R
V
R
T
, name two congruent angles.
S
V
2. If R
S
S
V
, name two congruent angles.
T
U
3. If SRT STR, name two congruent segments.
4. If STV SVT, name two congruent segments.
Triangles GHM and HJM are isosceles, with G
H
M
H
and H
J
M
J
. Triangle KLM is equilateral, and mHMK 50.
Find each measure.
J
K
L
M
H
5. mKML
6. mHMG
7. mGHM
G
8. If mHJM 145, find mMHJ.
9. If mG 67, find mGHM.
10. Write a two-column proof.
Given: DE
|| B
C
1 2
Prove: A
B
A
C
E
2
3
C
A
1
D
4
B
11. SPORTS A pennant for the sports teams at Lincoln High
School is in the shape of an isosceles triangle. If the measure
of the vertex angle is 18, find the measure of each base angle.
©
Glencoe/McGraw-Hill
216
n
col
Lin
ks
Haw
Glencoe Geometry